{"text":"\\section{Introduction}\nBoolean network (BN), proposed by Kauffman in 1969 \\cite{Kauffman1969}, is an ideal mathematical model of simulating the gene regulation networks.\nIt quantitates the interactions among genes within cells (or within a particular genome).\nThe expression, replication, transcription and other activities of genes can be directly reflected by system states and functions \\cite{Kauffmanbook1993}.\nBN has prompted many researchers to find and ask for similar models.\nAs a result, a large number of models were born.\nFor example, some genes continuously adjust the glucose consumption of cells and so provide the fuel by which they grow and multiply.\nFor analyzing such a biological system, Boolean control network (BCN) becomes a proper model \\cite{Datta2004,Huang2000}.\nOne of the main tools for studying BNs and BCNs is called the semi-tensor product (STP) of matrices, which was proposed by Prof. Cheng \\cite{Cheng2010BNlinear,Chengbook2011,Chengbook2012}.\nIts basic idea is to describe the system behavior as a discrete time algebra form,\nby which, some classical control ideas are incorporated into the analysis of BNs \\cite{LiangJinBN2020,LiF-IBNS2012,LiRcompleteBN2012,Learning2018,ZhuJBN2015} and the control design of BCNs \\cite{LiHLyapunov2019,LuJstab2018,WuYcontrol2018,Yuregular2019,ZhongJieTrank2019,QZhucontrol2019}, as people have seen in recent years.\n\nMany wild animals carry multiple viruses that have no effect on the animals themselves, but may be both high contagious and deadly to human beings.\nAntiviral immunity plays a key role against virus diseases.\nIts research involves the pathologic manifestations, symptoms and detection technologies of viral disease, which is the major cause of network identification being currently an important topic.\nNetwork identification aims to find the methods or algorithms for constructing the dynamics of systems.\nFor an unknown biological system or an environment where some viruses survive, only input-output data can be obtained, however, their changes can reflect some particular functions and features of a system.\nHence these data are directly used to build the model describing the original complicated network.\nSome early results considered identification of the network transition mappings \\cite{identification2000Akutsu,identification1998Liang,identification2006Nam,identification2005Pal}.\nUnder the framework of STP, the identification of BNs can be equivalently transformed into the identification of related structure matrices, which was proposed in \\cite{ModelConstruction2011} and was extended to BCNs in \\cite{IdentificationofBCN2011}.\n\\cite{IdentificationofBCN2011} noticed that, a BCN is identifiable if and only if it is controllable and O3-observable.\nThis O3-observability originates from one of five branching paths to the development of observability.\nWe list these five definitions of observability below.\n\\begin{definition}\\label{Def}\nA BCN is Oi-observable, {\\rm($i=1,2,3,4,5$)}, if\n\\begin{itemize}\n\\item[{\\rm(O1)}] {\\rm \\cite{OBCN1}} for any two distinct states $x(0)\\neq \\bar{x}(0)$, there exists an input sequence $(u(0), u(1), \\ldots)$, such that the corresponding output sequences are distinct: $(y(0), y(1), $ $\\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O2)}] {\\rm \\cite{OBCN2}} for any a state $x(0)$ there exists an input sequence $(u(0)$, $u(1)$, $\\ldots)$, such that for any $\\bar{x}(0)\\neq x(0)$, the corresponding output sequences are distinct: $(y(0), y(1), \\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O3)}] {\\rm \\cite{OBCN3}} there exists an input sequence $(u(0), u(1),\\ldots)$, such that for any two distinct states $x(0)\\neq \\bar{x}(0)$, the corresponding output sequences are distinct: $(y(0), y(1), \\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O4)}] {\\rm \\cite{OBCN4}} for any two distinct states $x(0)\\neq \\bar{x}(0)$ and for any input sequence $(u(0),u(1),\\ldots)$, the corresponding output sequences are distinct:\n    $(y(0),y(1),$ $\\ldots)$$\\neq$ $(\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O5)}] {\\rm \\cite{OBCN5}} there exists an output-feedback loop $u(t)=f_t(y(t))$ $(u(t)=f(y(t))$ for static control), such that for any two distinct states $x(0)\\neq \\bar{x}(0)$, the corresponding output sequences are distinct: $(y(0), y(1), \\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$.\n\\end{itemize}\n\\end{definition}\n\nMost of the criteria and methods for judging the first four kinds of observability (O1-O4) are not ideal, some are sufficient conditions, some are too complex to apply.\n\\cite{OBCNZhang20161} proposed a unified approach based on finite automata to determine these four observabilities, and presented corresponding four necessary and sufficient conditions.\nThis automata approach is more suitable for BCNs with fewer state nodes and input nodes due to the high complexity of constructing deterministic finite automata.\n\\cite{OBCNCheng20162} concentrated on the most general observability (O1), and presented a matrix-based approach with lower complexity by STP.\nMathematically speaking, the requirement of input sequences used to recognize the initial state $x(0)$ gradually increases from O1-observability (the most general form) to O4-observability (the sharpest form). Hence, O4 $\\Longrightarrow$ O3 $\\Longrightarrow$ O2 $\\Longrightarrow$ O1, which is shown as a relation diagram in \\cite{OBCNZhang20161}.\nIn particular, determining O3-observability is NP-hard \\cite{OBCN3}.\nO5-observability that recognizes the initial state via output feedback, called output-feedback observability, was first proposed in \\cite{OBCNGuoCCC2017}. This one is much sharper than the first two observabilities.\n\\cite{OBCN5} used paralled interconnected two identical BCNs to determine O5-observability, by converting the observability problem of the original BCN to the set reachability problem of the interconnected BCN.\n\nAs mentioned above, the identification of BCNs requires O3-observability.\nA natural question arises: what about the most general form, O1-observability?\nMotivated by that, we further develop the identification problem for BCNs in this paper.\nMain contributions are summarized as follows:\n\n(1) Three important theoretical results are obtained: (3a) A BN is uniquely identifiable if it is observable; (3b) A BCN is uniquely identifiable if it is O1-observable. It is worth pointing out that O1-observability is the most general one of the existing observability terms. (3c) A BN or BCN may be identifiable, but not observable.\n\n(2) In combination with the phenomena in medical detection, we propose two new concepts: single sample and multiple samples to deal with the identification problem of BCNs.\nBased on them, the identification problem is divided into four situations.\nWe point out that the existing works on identification are actually special cases of these four situations.\n\n(3) By virtue of the observability property, we form a one-to-one correspondence between the state and the output sequence.\nThen four simple criteria to determine the identifiability and four effective algorithms to construct the structure matrices are proposed.\n\n\n\nThe rest of the paper is organized as follows. Section II contains preliminary notations, fundamental definitions and problem formulation.\nSection III presents main results on identification of BNs and BCNs, including several discriminant methods for the identification property, several identification algorithms to construct the structure matrices and illustrative examples. Remarks are given to show some challenging and interesting future research.\nFinally, a table describes the relationships and comparisons of the results obtained in this paper, and Section IV concludes the paper.\n\n\n\\section{Preliminaries}\n\n\\subsection{Semi-tensor product}\nThis section gives some necessary preliminaries. More details can be referred to \\cite{Chengbook2012}.\nFirst, some notations are listed below:\n\\begin{itemize}\n\\item[$\\bullet$] $\\mathbb{N}=\\{0,1,2,\\ldots\\}$: the natural number set.\n\\item[$\\bullet$] $[a,b]_{\\mathbb{N}}$: all the natural numbers from $a$ to $b$.\n\\item[$\\bullet$] $\\delta_n^i$: the $i$th column of the identity matrix $I_n$.\n\\item[$\\bullet$] $\\Delta_n:=\\{\\delta_n^1,\\delta_n^2,\\ldots,\\delta_n^n\\}$.\n\\item[$\\bullet$] $\\mathbf{1}_n:=\\sum_{i=1}^n\\delta_n^i$.\n\\item[$\\bullet$] $[\\delta_n^{i_1}~\\delta_n^{i_2}~\\cdots ~\\delta_n^{i_m}]:=$ $\\delta_n[i_1~i_2 ~\\cdots~i_m]$.\n\\item[$\\bullet$] $(\\delta_n^{i_1},\\delta_n^{i_2},\\ldots,\\delta_n^{i_m}):=$ $\\delta_n(i_1,i_2,\\ldots,i_m)$.\n\\item[$\\bullet$] $Col_i(M)$: the $i$th column of matrix $M$.\n\\item[$\\bullet$] $Col(M)$: the set of all columns of $M$.\n\\item[$\\bullet$] $\\mathcal{L}_{m\\times n}$: $=\\{M| M\\in \\mathbb{R}^{m\\times n},Col(M)\\subseteq \\Delta_{m}\\}$.\n\\item[$\\bullet$] $[M]_{i,j}$: the $(i,j)$th entry of matrix $M$.\n\\item[$\\bullet$] $M^{\\mathrm{T}}$: the transpose of matrix $M$.\n\\item[$\\bullet$] Kronecker product: $A\\otimes B=([A]_{i,j}\\times B)$.\n\\item[$\\bullet$] K-R product: $A\\ast B=C$, $Col_l(C)=Col_l(A)\\otimes Col_l(B)$.\n\n\\end{itemize}\n\n\n\\begin{definition}{\\rm\\cite{Chengbook2012}}\nThe semi-tensor product (STP) of two matrices $A\\in\\mathbb{R}^{m\\times n}$ and $B\\in\\mathbb{R}^{p\\times q}$ is\n\\begin{equation*}\nA\\ltimes B=(A\\otimes I_{\\frac{s}{n}})(B\\otimes I_{\\frac{s}{p}}),\n\\end{equation*}\nwhere $s$ is the least common multiple of $n$ and $p$.\n\\end{definition}\n\nObviously, the STP becomes the conventional matrix product if $n=p$. Hence the symbol $\\ltimes$ is omitted in the sequel.\n\n\\begin{lemma}\\label{Lem1}{\\rm\\cite{Chengbook2012}}\nLet $f(x_1,\\ldots,x_n)$ be a Boolean function, where $x_1,\\ldots,x_n$ are Boolean variables.\nWithin the framework of vector form, $f$ can be converted into $f:\\Delta_{2^n} \\rightarrow \\Delta_2$, and there exists a unique matrix $M_f\\in \\mathcal{L}_{2\\times 2^n}$, called the structure matrix of $f$, such that\n\\begin{equation*}\nf(x_1,\\ldots,x_n)=M_f\\ltimes x_1\\ltimes \\cdots\\ltimes x_n.\n\\end{equation*}\n\\end{lemma}\n\nConsider a BCN with $n$ state nodes, $m$ input nodes and $l$ output nodes as follows:\n\\begin{equation}\\label{BCN1}\n\\left\\{\n\\begin{array}{ll}\nx_i(t+1)=f_i(u_1(t),\\ldots,u_m(t),x_1(t),\\ldots,x_n(t)),\\\\\ny_j(t)=h_j(x_1(t),\\ldots,x_n(t)),\\\\\ni\\in [1,n]_{\\mathbb{N}},~j\\in [1,l]_{\\mathbb{N}},~t\\in \\mathbb{N},\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f_i:\\mathcal{D}^{m+n}\\rightarrow\\mathcal{D}$ and $h_j:\\mathcal{D}^{n}\\rightarrow\\mathcal{D}$ are logical functions, $x_i\\in \\mathcal{D}$, $u_j\\in\\mathcal{D}$ and $y_k\\in\\mathcal{D}$ are the state, input and output of the system, respectively.\n\nFrom Lemma \\ref{Lem1}, each logical function $f_i$ ($h_j$) has unique structure matrix $M_{f_i}$ ($M_{h_j}$), then BCN \\eqref{BCN1} can be equivalently transformed into an algebraic form as follows \\cite{Chengbook2012}:\n\\begin{equation}\\label{BCN2}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=Fu(t)x(t),\\\\\ny(t)=Hx(t),\n\\end{array}\n\\right.\n\\end{equation}\nwhere $F=M_{f_1}\\ast M_{f_2}\\ast\\cdots\\ast M_{f_n}$, $H=M_{h_1}\\ast M_{h_2}\\ast\\cdots\\ast M_{h_l}$, $x(t)=\\ltimes_{i=1}^nx_i(t)$, $u(t)=\\ltimes_{j=1}^mu_j(t)$, $y(t)=\\ltimes_{k=1}^ly_k(t)$.\nThis form is called the algebraic state space representation of \\eqref{BCN1}.\n\n\n\\subsection{Problem statement}\n\nThe identification problem of BCN \\eqref{BCN2} is to construct two structure matrices $F$ and $H$ via available data. Denote\n\\begin{align*}\nU_i(p_i):=&\\{u_i(t)\\}_{t=0}^{p_i}=(u_i(0),u_i(1),u_i(2),\\ldots,u_i(p_i)), \\\\\nX_i(p_i):=&\\{x_i(t)\\}_{t=0}^{p_i}=(x_i(0),x_i(1),x_i(2),\\ldots,x_i(p_i)), \\\\\nY_i(p_i):=&\\{y_i(t)\\}_{t=0}^{p_i}=(y_i(0),y_i(1),y_i(2),\\ldots,y_i(p_i)),\n\\end{align*}\nand\n\\begin{align*}\n\\{U_i(p_i)\\}:=&\\{u_i(0),u_i(1),u_i(2),\\ldots,u_i(p_i)\\}, \\\\\n\\{X_i(p_i)\\}:=&\\{x_i(0),x_i(1),x_i(2),\\ldots,x_i(p_i)\\}, \\\\\n\\{Y_i(p_i)\\}:=&\\{y_i(0),y_i(1),y_i(2),\\ldots,y_i(p_i)\\}.\n\\end{align*}\n\n\n\\begin{definition}\nA BCN \\eqref{BCN2} is said to be identifiable, if its two structure matrices $F$ and $H$ can be determined via available data: input data $U_1(p_1),U_2(p_2),\\ldots,$ $U_k(p_k)$ and observed data $Y_1(p_1),Y_2(p_2),\\ldots,Y_k(p_k)$.\n\\end{definition}\n\n\nA coordinate transformation $\\omega=Gx$ could convert \\eqref{BCN2} into the following algebraic form:\n\\begin{align}\\label{BCN3}\n\\left\\{\n\\begin{array}{ll}\n\\omega(t+1)=GF(I_{2^m}\\otimes G^{\\mathrm{T}})u(t)\\omega(t)=:\\widehat{F}u(t)\\omega(t), \\\\\ny(t)=HG^{\\mathrm{T}}\\omega(t)=:\\widehat{H}\\omega(t).\n\\end{array}\n\\right.\n\\end{align}\nDue to the arbitrariness of state recognition, \\eqref{BCN2} and \\eqref{BCN3} are considered to be identical in the same input-output data frame, so the set of all possible $(\\widehat{F},\\widehat{H})$ becomes the equivalence class of $(F,H)$.\nA identifiable BCN is also said to be $uniquely$ $identifiable$ in the sense of equivalence.\n\n\n\\begin{assumption}\\label{Assum1}\nThis paper assumes the available data is sufficient.\nIn other words, the input data and the observed data contain all possible situations which the system could generate.\n\\end{assumption}\n\nGenerally speaking, densely populated cities are good places for virus or infectious diseases, which could spread easily from person to person.\nThe Centers for Disease Control and Prevention can collect a large number of samples from different patients infected by the same pathogen.\nHence, Assumption \\ref{Assum1} is reasonable and its implementation requires $multiple$ $samples$ from large numbers of patients (urine sample or blood sample or cheek swab), not a $single$ $sample$ from one patient, since a single sample may exhibit only part of characteristics of the virus.\nMultiple samples mean that the observed data may be generated from different initial states, while, single sample means that the observed data is generated from some initial state.\n\nOn the basis of the statement above, the identification of BNs and BCNs can be divided into four cases:\n\\begin{itemize}\n\\item[Case 1]: the identification process of single sample in the BN records one group of output data $Y(p)$.\n\n\\item[Case 2]: the identification process of multiple samples in the BN records $k$ groups of output data $Y_1(p_1), Y_2(p_2), \\ldots, Y_k(p_k)$.\n\n\\item[Case 3]: the identification process of single sample in the BCN records $r$ groups of input-output data $U_1(p_1),Y_1(p_1),U_2(p_2),Y_2(p_2),\\ldots, U_r(p_r),Y_r(p_r)$.\n\n\\item[Case 4]: the identification process of multiple samples in the BCN records $rk$ groups of input-output data $U_1^i(p_1),Y_1^i(p_1),U_2^i(p_2),Y_2^i(p_2),\\ldots, U_r^i(p_r),Y_r^i(p_r)$, $i\\in [1,k]_{\\mathbb{N}}$.\n\n\\end{itemize}\nBoth Cases 1 and 3 collect the blood sample from only one patient, while Cases 2 and 4 collect from $k$ patients.\nCase 3 divides the blood sample into multiple portions ($r$ portions) for testing with a variety of reagents. That is to say, $r$ groups of input-output data are generated from the same initial state, i.e., $x_1(0)=x_2(0)=\\cdots=x_r(0)$ (in Case 3). Similarly, $x_1^{i}(0)=x_2^i(0)=\\cdots=x_r^i(0), i\\in[1,k]_{\\mathbb{N}}$ in Case 4.\n\n\n\n\n\n\n\\section{Identification of BNs and BCNs}\n\n\n\\subsection{Identification of BNs}\n\\cite{ModelConstruction2011} investigated the identification of the following BN:\n\\begin{align}\\label{BN1}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=Fx(t),\\\\\ny(t)=x(t),\n\\end{array}\n\\right.\n\\end{align}\nin which the observed data is presented directly by the system state.\nWith a group of observed data $(x_1(0)=\\delta_{2^n}^{i_0},x_1(1)=\\delta_{2^n}^{i_1},\\ldots)$, the $i_0$th column of $F$ can be identified as $Col_{i_0}(F)=\\delta_{2^n}^{i_1}$ and hence the next result is obtained.\n\n\\begin{lemma}\\label{Lem2}{\\rm \\cite{ModelConstruction2011}}\n{\\rm (}Multiple samples{\\rm)} BN \\eqref{BN1} is uniquely identifiable, if and only if the observed data contains all possible states:\n\\begin{align}\n\\{Y_1(p_1)\\}\\cup\\{Y_2(p_2)\\}\\cup\\cdots\\cup\\{Y_k(p_k)\\}=\\Delta_{2^n}.\n\\end{align}\n\\end{lemma}\n\nIt is noted that the observed data considered in \\cite{ModelConstruction2011} may consist of several output sequences (i.e., multiple samples), which is reasonable because a system may contains multiple attractors and multiple attractors mean multiple state trajectories.\nWhen the system state cannot be directly observed, BN \\eqref{BN1} becomes\n\\begin{align}\\label{BN2}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=Fx(t),\\\\\ny(t)=Hx(t).\n\\end{array}\n\\right.\n\\end{align}\nIn the process of identifying this system, it is the most important to distinguish the states.\n\n\n\\begin{definition}\nIn BN \\eqref{BN2}, a state pair $(x(0),\\bar{x}(0))$, $x(0)\\neq \\bar{x}(0)$ is said to be distinguishable if the corresponding output sequences generated by them are distinct: $(y(0),y(1),\\ldots)\\neq (\\bar{y}(0),\\bar{y}(1),\\ldots)$. \\eqref{BN2} is said to be observable if any state pair is distinguishable.\n\\end{definition}\n\nObservability means that $2^n$ distinct initial states generate $2^n$ distinct groups of observed data, i.e.,\n\\begin{align}\n(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,HF^{2}\\delta_{2^n}^i,\\ldots)\\neq (H\\delta_{2^n}^{i'},HF\\delta_{2^n}^{i'},HF^{2}\\delta_{2}^{i'},\\ldots),~i\\neq i'.\n\\end{align}\nSince each state trajectory will fall into an attractor in $2^n$ steps, the subsequent state trajectory and output trajectory will repeat the previous data. Lemma 1 and Proposition 1 in \\cite{OBCN4} show the following result.\n\\begin{proposition}\\label{Pro3}{\\rm\\cite{OBCN4}}\nBN \\eqref{BN2} is observable if and only if for any $i\\neq i'$,\n\\begin{align}\\label{Pro3-1}\n(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,\\ldots, HF^{2^n-1}\\delta_{2^n}^i)\\neq (H\\delta_{2^n}^{i'},HF\\delta_{2^n}^{i'},\\ldots,HF^{2^n-1}\\delta_{2}^{i'}).\n\\end{align}\n\\end{proposition}\nWe call $(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,\\ldots,HF^{2^n-1}\\delta_{2^n}^i)$ the $effective$ $output$ $sequence$ of state $\\delta_{2^n}^i$.\nAn effective output sequence corresponds to a state and its length is $2^n$ steps.\nUnder the case of Assumption \\ref{Assum1}, if the system is observable, $2^n$ distinct effective output sequences can be found by searching and comparing all $2^n$-step output sequences from sufficient observed data.\n\nAssume that the following $k$ groups of observed data are sufficient,\n\\begin{align}\\label{Th1-data}\nY_j(T_j)=(y_j(0),y_j(1),\\ldots,y_j(T_j)),~j\\in [1,k]_{\\mathbb{N}}.\n\\end{align}\nLet $Y_s^j$ represent the $s$th $2^n$-step output sequence to show up in $Y_j(T_j)$:\n\\begin{align}\\label{Th1-data1}\nY_s^j=&(y_j(s-1),y_j(s),\\ldots,y_j(s+2^n-2)),~s\\in [1,T_j']_{\\mathbb{N}},\n\\end{align}\nwhere $T_j'=T_j-2^n+2$.\nThen by retrieval from \\eqref{Th1-data}, an algorithm (Algorithm \\ref{Alg:A}) to find $2^n$ distinct effective output sequences is established, and this algorithm names the $i$th effective output sequence that occurs in Algorithm \\ref{Alg:A} as $Y_i$, $(i\\in[1,2^n]_{\\mathbb{N}})$.\n\n\\begin{algorithm}[H]\n\\caption{Retrieve all distinct effective output sequences}\n\\label{Alg:A}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n     \\REQUIRE data \\eqref{Th1-data}.\n     \\ENSURE $Y_1,Y_2,\\ldots,Y_{2^n}$.\n    \\STATE{set $Y=\\emptyset$ and $i=1$}\n    \\FOR{$j=1; j<k; j++$}\n    \\FOR{$s=1; s<T_j'; s++$}\n        \\IF {$Y_s^j\\in Y$}\n            \\STATE break;\n        \\ELSE\n            \\STATE {$Y_i=Y_s^j$, $Y=Y\\cup Y_i^j$, $i=i+1$;}\n        \\ENDIF\n    \\ENDFOR\n    \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{theorem}\\label{Th1}\n{\\rm(}Multiple samples{\\rm)} BN \\eqref{BN2} is uniquely identifiable if it is observable.\n\\end{theorem}\n\\begin{proof} From the analysis above, if \\eqref{BN2} is observable, all distinct effective output sequences $Y_i,i\\in[1,2^n]_{\\mathbb{N}}$ can be obtained by Algorithm \\ref{Alg:A} and enough observed data \\eqref{Th1-data}.\n\n\nIdentify $Y_i$ as the effective output sequence of state $\\delta_{2^n}^i$, $i\\in[1,2^n]_{\\mathbb{N}}$, then the following $k$ state sequences:\n\\begin{align}\\label{Th1-data2}\nX_j(T_j')=(x_j(0),x_j(1),\\ldots,x_j(T_j')),~j\\in[1,k]_{\\mathbb{N}}\n\\end{align}\ncan be derived by\n\\begin{align}\\label{Th1-data3}\nx_j(t)=\\delta_{2^n}^{i_t},~t\\in[0,T_j'],~{\\rm if}~(y_j(t),y_j(t+1),\\ldots,y_j(t+2^n-1))=Y_{i_t}.\n\\end{align}\nCombining the observed data \\eqref{Th1-data} and the state data \\eqref{Th1-data2}, $F$ and $H$ can be constructed by\n\\begin{align}\\label{Th1-data4}\n\\left\\{\n\\begin{array}{lll}\nx_j(t+1)=Fx_j(t),~t\\in[0,T_j'-1]_{\\mathbb{N}},~j\\in[1,k]_{\\mathbb{N}},\\\\\nH=[Y_1(0)~Y_2(0)~\\cdots~Y_{2^n}(0)],\n\\end{array}\n\\right.\n\\end{align}\nwhere $Y_i(0)$ represents the first element in $Y_i$, $i\\in[1,2^n]_{\\mathbb{N}}$.\n\nNote that the appointment $\\delta_{2^n}^i$ of the effective output sequence $Y_i$ can be changed in any order, due to the arbitrariness of state recognition.\nThen another $(\\widehat{F},\\widehat{H})$ is derived, which is analogous to the case of \\eqref{BCN3}.\nOne kind of order corresponds to one coordinate transformation $w=Gx$, and $(\\widehat{F},\\widehat{H})$ can be written as:\n\\begin{align}\\label{BN3}\n\\left\\{\n\\begin{array}{ll}\nw(t+1)=GFG^{\\mathrm{T}}w(t)=:\\widehat{F}w(t),\\\\\ny(t)=Hx(t)=HG^{\\mathrm{T}}w(t)=:\\widehat{H}w(t),\n\\end{array}\n\\right.\n\\end{align}\nwhich implies both $(F,H)$ and $(\\widehat{F},\\widehat{H})$ belong to an equivalence class.\nHence, BN \\eqref{BN2} is uniquely identifiable.\n\\end{proof}\n\nOn the basis of Theorem \\ref{Th1}, an algorithm (Algorithm \\ref{Alg:1}) to identify $(F,H)$ is established.\n\n\\begin{algorithm}\n\\caption{Identify $(F,H)$ for BN \\eqref{BN2} (Deal with Case 2).}\n\\label{Alg:1}\n{\\textbf{Input:}} data \\eqref{Th1-data}.\n\n{\\textbf{Output:}} $F, H$.\n\n\\begin{description}\n\n\\item[Step $1$]: Find all distinct effective output sequences by Algorithm \\ref{Alg:A}.\n\n\\item[Step $2$]: For $j\\in[1,k]_{\\mathbb{N}}$ and $t\\in[0,T_j']$, identify the state sequence \\eqref{Th1-data2} by \\eqref{Th1-data3}.\n\n\\item[Step $3$]: Construct $F$ and $H$ based on \\eqref{Th1-data4}.\n\\end{description}\n\\end{algorithm}\n\n\n\n\\begin{example}\\label{Eg1}\nConsider a BN with $3$ state codes and $1$ output code, and assume there are two groups of observed data:\n\\begin{align}\\label{Eg1-0}\n\\left\\{\n\\begin{array}{ll}\nY_1(14)=\\delta_2(2,1,1,2,2,2,1,2,2,2,1,2,2,2,1),\\\\\n~~~~~~~~=(y_1(0),y_1(1),y_1(2),y_1(3),\\ldots,y_1(14)),\\\\\nY_2(12)=\\delta_2(1,2,1,2,1,2,1,2,1,2,1,2,1),\\\\\n~~~~~~~~=(y_2(0),y_2(1),y_2(2),y_2(3),\\ldots,y_2(12)).\n\\end{array}\n\\right.\n\\end{align}\n\\end{example}\n\nStep~$1$: All distinct effective output sequences are\n\\begin{align}\\label{Eg1-1}\n\\left\\{\n\\begin{array}{llll}\nY_1=(y_1(0),y_1(1),\\ldots,y_1(7))&=\\delta_2(2,1,1,2,2,2,1,2), \\\\\nY_2=(y_1(1),y_1(2),\\ldots,y_1(8))&=\\delta_2(1,1,2,2,2,1,2,2), \\\\\nY_3=(y_1(2),y_1(3),\\ldots,y_1(9))&=\\delta_2(1,2,2,2,1,2,2,2), \\\\\nY_4=(y_1(3),y_1(4),\\ldots,y_1(10))&=\\delta_2(2,2,2,1,2,2,2,1), \\\\\nY_5=(y_1(4),y_1(5),\\ldots,y_1(11))&=\\delta_2(2,2,1,2,2,2,1,2), \\\\\nY_6=(y_1(5),y_1(6),\\ldots,y_1(12))&=\\delta_2(2,1,2,2,2,1,2,2), \\\\\nY_7=(y_2(0),y_2(1),\\ldots,y_2(7))&=\\delta_2(1,2,1,2,1,2,1,2), \\\\\nY_8=(y_2(1),y_2(2),\\ldots,y_2(8))&=\\delta_2(2,1,2,1,2,1,2,1).\n\\end{array}\n\\right.\n\\end{align}\n\nStep~$2$: Two state sequences are identified as\n\\begin{align}\\label{Eg1-2}\n\\left\\{\n\\begin{array}{lll}\nX_1(8)=(x_1(0),x_1(1),x_1(2),\\ldots,x_1(8))\\\\\n~~~~~~~=\\delta_8(1,2,3,4,5,6,3,4,5),\\\\\nX_2(5)=(x_2(0),x_2(1),x_2(2),x_2(3),x_2(4),x_2(5))\\\\\n~~~~~~~=\\delta_8(7,8,7,8,7,8).\n\\end{array}\n\\right.\n\\end{align}\n\nStep~$3$: Based on~$x_1(t+1)=Fx_1(t)$, $t\\in [0,8]_{\\mathbb{N}}$, we get\n\\begin{align*}\nx_1(1)=Fx_1(0)&\\Rightarrow\\delta_8^2=F\\delta_8^1,x_1(4)=Fx_1(3)\\Rightarrow\\delta_8^5=F\\delta_8^4,\\\\\nx_1(2)=Fx_1(1)&\\Rightarrow\\delta_8^3=F\\delta_8^2,x_1(5)=Fx_1(4)\\Rightarrow\\delta_8^6=F\\delta_8^5,\\\\\nx_1(3)=Fx_1(2)&\\Rightarrow\\delta_8^4=F\\delta_8^3,x_1(6)=Fx_1(5)\\Rightarrow\\delta_8^3=F\\delta_8^6.\n\\end{align*}\nSimilarly, from~$x_2(t+1)=Fx_2(t)$, $t\\in [0,5]_{\\mathbb{N}}$, we get\n\\begin{align*}\nx_2(1)=Fx_2(0)&\\Rightarrow\\delta_8^8=F\\delta_8^7,x_2(2)=Fx_2(1)\\Rightarrow\\delta_8^7=F\\delta_8^8.\n\\end{align*}\nIt is clear that $F=\\delta_8[2~3~4~5~6~3~8~7]$.\nOn the other hand,\n\\begin{align*}\nH=&[Y_1(0)~Y_2(0)~\\cdots~Y_{8}(0)] \\\\\n =&[y_1(0)~y_1(1)~y_1(2)~y_1(3)~y_1(4)~y_1(5)~y_2(0)~y_2(1)] \\\\\n =&\\delta_2[2~1~1~2~2~2~1~2].\n\\end{align*}\nTo sum up, the system is identified as:\n\\begin{align}\\label{Eg1-4}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=\\delta_8[2~3~4~5~6~3~8~7]x(t),\\\\\ny(t)=\\delta_2[2~1~1~2~2~2~1~2]x(t).\n\\end{array}\n\\right.\n\\end{align}\n\nIf we identify $Y_7\\sim\\delta_8^1$, $Y_8\\sim\\delta_8^2$, $Y_1\\sim\\delta_8^3$, $Y_2\\sim\\delta_8^4$, $Y_3\\sim\\delta_8^5$, $Y_4\\sim\\delta_8^6$, $Y_5\\sim\\delta_8^7$ and $Y_6\\sim\\delta_8^8$, then the corresponding BN becomes:\n\\begin{align}\\label{Eg1-5}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=\\delta_8[2~1~4~5~6~7~8~5]x(t),\\\\\ny(t)=\\delta_2[1~2~2~1~1~2~2~2]x(t).\n\\end{array}\n\\right.\n\\end{align}\nAlthough two systems \\eqref{Eg1-4} and \\eqref{Eg1-5} are derived from different appointments of effective output sequences, both of them belong to an equivalence class.\n\nFrom this example, we can discern a special case: the observed multiple data are exactly $2^n$ distinct sequences $(y_i(0),y_i(1),\\cdots,y_i(2^N-1))$, $i=1,2,\\cdots,2^n$. The BN can also be identified by appropriately extending the length of the observed data (further collection adds one more element to each sequence).\nIn addition, Algorithms \\ref{Alg:A} and \\ref{Alg:1} also work with insufficient data, in which case, there exist some identical groups of observed data induced by some states that are indistinguishable, such that two structure matrices constructed are of low dimensions ($F\\in \\mathcal{L}_{2^n\\times s}, H\\in \\mathcal{L}_{2^l\\times s}, s < 2^n$).\n\n\nIn previous studies \\cite{OBCN4,WBNAHS2020}, the effective output sequences are used to judge the observability, which is usually implemented by a tool called observability matrix:\n\\begin{align}\\label{O0}\n\\mathcal{O}=[(PH)^{\\mathrm{T}}~(PHF)^{\\mathrm{T}}~\\cdots ~(PHF^{2^n-1})^{\\mathrm{T}}]^{\\mathrm{T}},\n\\end{align}\nwhere $P=[1~2~\\cdots~2^l]$ is a row vector consisting of $[1,2^l]_{\\mathbb{N}}$ and aims to extract the superscripts of all column vectors in $H$. For example, if $H=\\delta_2[1~2~2~1]$, $PH=[1~2~2~1]$.\nThe effective output sequence of state $\\delta_{2^n}^i$ is recorded in the $i$th column of $\\mathcal{O}$:\n$Col_i(\\mathcal{O})=[PH\\delta_{2^n}^i~PHF\\delta_{2^n}^i~\\cdots~$ $PHF^{2^n-1}\\delta_{2^n}^i]^{\\mathrm{T}}$.\nTherefore, finding $2^n$ effective output sequences is equivalent to constructing the observability matrix.\n\n\\begin{example}\nRecall Example \\ref{Eg1}.\n\\end{example}\n\n$Y_i$ is identified as the of state $\\delta_{2^n}^i$. Hence, we have\n\\begin{small}\n\\begin{align*}\n\\mathcal{O}=&\n\\left[\n\\begin{array}{cccccccccc}\nPy_1(0)&Py_1(1)&\\cdots&Py_1(5) &Py_2(0)&Py_2(1)     \\\\\nPy_1(1)&Py_1(2)&\\cdots&Py_1(6) &Py_2(1)&Py_2(2)     \\\\\n\\vdots&\\vdots&\\ddots&\\vdots &\\vdots&\\vdots  \\\\\nPy_1(7)&Py_1(8)&\\cdots&Py_1(12) &Py_2(7)&Py_2(8)\n\\end{array}\n\\right] \\\\\n=&\\left[\n\\begin{matrix\n2&1&1&2&2&2 &1&2     \\\\\n1&1&2&2&2&1 &2&1     \\\\\n1&2&2&2&1&2 &1&2  \\\\\n2&2&2&1&2&2 &2&1  \\\\\n2&2&1&2&2&2 &1&2  \\\\\n2&1&2&2&2&1 &2&1  \\\\\n1&2&2&2&1&2 &1&2  \\\\\n2&2&2&1&2&2 &2&1\n\\end{matrix}\n\\right],\n\\end{align*}\n\\end{small}\nwhich is the observability matrix of system \\eqref{Eg1-1}.\n\n\n\n\\begin{proposition}\\label{Pro1}\nIf $\\mathcal{O}$ and $\\widehat{\\mathcal{O}}$ are the observability matrix of \\eqref{BN2} and \\eqref{BN3}, respectively, then\n\\begin{align*}\n\\mathcal{O}=\\widehat{\\mathcal{O}}G~{or}~\\mathcal{O}G^{\\mathrm{T}}=\\widehat{\\mathcal{O}}.\n\\end{align*}\n\\end{proposition}\n\\begin{proof}\nFrom \\eqref{O0}, it is clear that:\n\\begin{align}\\label{O01}\n\\widehat{\\mathcal{O}}=[(P\\widehat{H})^{\\mathrm{T}}~(P\\widehat{H}\\widehat{F})^{\\mathrm{T}}~\n\\cdots~(P\\widehat{H}\\widehat{F}^{2^n-1})^{\\mathrm{T}}]^{\\mathrm{T}}.\n\\end{align}\nCombining $\\widehat{H}=HG^{\\mathrm{T}}$ and $\\widehat{F}=GFG^{\\mathrm{T}}$, we get\n\\begin{align}\n\\widehat{\\mathcal{O}}=&[(PHG^{\\mathrm{T}})^{\\mathrm{T}}~(PHG^{\\mathrm{T}}GFG^{\\mathrm{T}})^{\\mathrm{T}}~\n\\cdots~(PHG^{\\mathrm{T}}(GFG^{\\mathrm{T}})^{2^n-1})^{\\mathrm{T}}]^{\\mathrm{T}}\\notag \\\\\n=&[(PHG^{\\mathrm{T}})^{\\mathrm{T}}~(PHFG^{\\mathrm{T}})^{\\mathrm{T}}~\\cdots\n~(PHF^{2^n-1}G^{\\mathrm{T}})^{\\mathrm{T}}]^{\\mathrm{T}},    \\notag \\\\\n\\widehat{\\mathcal{O}}G=&[(PHG^{\\mathrm{T}})^{\\mathrm{T}}~(PHFG^{\\mathrm{T}})^{\\mathrm{T}}~\\cdots\n~(PHF^{2^n-1}G^{\\mathrm{T}})^{\\mathrm{T}}]^{\\mathrm{T}}G   \\notag \\\\\n=&[(PHG^{\\mathrm{T}}G)^{\\mathrm{T}}~(PHFG^{\\mathrm{T}}G)^{\\mathrm{T}}~\\cdots~(PHF^{2^n-1}G^{\\mathrm{T}}G)^{\\mathrm{T}}]^{\\mathrm{T}}  \\notag \\\\\n=&[(PH)^{\\mathrm{T}}~(PHF)^{\\mathrm{T}}~\\cdots ~(PHF^{2^n-1})^{\\mathrm{T}}]^{\\mathrm{T}}=\\mathcal{O}. \\notag\n\\end{align}\n\\end{proof}\n\nIntroducing the observability matrix $\\mathcal{O}$ facilitates MATLAB programming.\nAfter constructing such an observability matrix from observed data, $x(t)$ can be determined as $\\delta_{2^n}^i$ if $[Py(t)~Py(t+1)~\\cdots~Py(t+2^n-1)]^{\\mathrm{T}}=Col_{i}(\\mathcal{O})$.\nProposition \\ref{Pro1} shows that the observability matrix of some system becomes that of another system by elementary column transformations, which further explains that different orders \\eqref{Th1-data2} (or observability matrices) give rise to different systems $(F,H)$.\nHowever, these systems belong to an equivalence class.\n\n\\begin{remark}\\label{Rem1}\nIf BN \\eqref{BN2} is identifiable, then is it observable?\nConsider a BN with $F=\\delta_4[3~3~4~4]$ and $H=\\delta_2[1~1~2~1]$. Its state transition diagram is shown in Fig. \\ref{fig:1}.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=2.5in]{unobservableBN.eps}\n\\end{center}\n\\caption{The state transition diagram of an unobservable BN.}\n\\label{fig:1}\n\\end{figure}\nThis BN is unobservable since $\\delta_4^1$ and $\\delta_4^2$ have same effective output sequence.\n\nWhen the system is unknown, with sufficient data $Y(8)=\\delta_2(1,2,1,1,1,1,1,1,1)$ we can only find three distinct output sequences\n\\begin{align*}\nY_1=\\delta_2(1,2,1,1),~Y_2=\\delta_2(2,1,1,1),~Y_3=\\delta_2(1,1,1,1).\n\\end{align*}\nThen the corresponding state sequence can be determined as $X(5)=\\delta_4(1,2,3,3,3,3)$.\nIf we know $Y_4=Y_1$, $x(1)=Fx(0)$ produces $\\delta_4^2=F\\delta_4^1$ and $\\delta_4^2=F\\delta_4^4$.\nIt follows that $F=\\delta_4[2~3~3~2]$ and $H=\\delta_2[1~2~1~1]$. This system and the original system coincide in the sense of equivalence.\nThis example shows that an unobservable system may be identifiable.\n\\end{remark}\n\n\\begin{remark}\nTo determine the observability, the dimensions of observability matrix $\\mathcal{O}$ could be $(2^n-1)\\times 2^n$ (see \\cite{OBCN4,WBNAHS2020}).\nThat is to say, the condition \\eqref{Pro3-1} in Proposition \\ref{Pro3} can be improved as\n\\begin{align}\n(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,\\ldots,HF^{2^n-2}\\delta_{2^n}^i)\\neq (H\\delta_{2^n}^{i'},HF\\delta_{2^n}^{i'},\\ldots,HF^{2^n-2}\\delta_{2^n}^{i'}),\n\\end{align}\nwhich implies that the length of the effective output sequence can be further reduced to $(2^n-1)$ steps.\nIn a $2^n$-step output sequence $Y$, its first $(2^n-1)$ steps and last $(2^n-1)$ steps, respectively denoted by $f_e(Y)$ and $l_e(Y)$, reflect two effective output sequences, and hence they reflect two consecutive states.\nAssume $Y_i$ and $Y_{i'}$ are two output sequences stemming from $\\delta_{2^n}^i$ and $\\delta_{2^n}^{i'}$, respectively.\nIf $l_e(Y_i)=f_e(Y_{i'})$, then state $\\delta_{2^n}^{i'}$ follows state $\\delta_{2^n}^i$, i.e., $\\delta_{2^n}^{i'}=F\\delta_{2^n}^i$.\nSo, comparing $Y_1,Y_2,\\ldots,Y_{2^n}$ generated by Algorithm \\ref{Alg:A}, we can directly construct $F$ and $H$.\nFor example, \\eqref{Eg1-1} satisfies\n\\begin{align*}\n&l_e(Y_1)=f_e(Y_2),~l_e(Y_2)=f_e(Y_3),~l_e(Y_3)=f_e(Y_4),~l_e(Y_4)=f_e(Y_5),\\\\\n&l_e(Y_5)=f_e(Y_6),~l_e(Y_6)=f_e(Y_3),~l_e(Y_7)=f_e(Y_8),~l_e(Y_8)=f_e(Y_7).\n\\end{align*}\nThen, $F\\delta_8^1=\\delta_8^2,F\\delta_8^2=\\delta_8^3,F\\delta_8^3=\\delta_8^4,F\\delta_8^4=\\delta_8^5, F\\delta_8^5=\\delta_8^6,F\\delta_8^6=\\delta_8^3,F\\delta_8^7=\\delta_8^8,F\\delta_8^8=\\delta_8^7$.\nObviously, this process is simpler than Algorithm \\ref{Alg:1}.\nHowever, extending this method to BCNs is not easy and requires consideration of the input sequence.\n\\end{remark}\n\n\n\n\n\\subsection{Identification of BCNs}\n\nThe identification problem of BCNs is complicated because more attractors will emerge as the control is embedded.\nConsidering BCN \\eqref{BCN2}, the identification of $F\\in \\mathcal{L}_{2^{n}\\times 2^{m+n}}$ needs the input-state data $(U_1(p_1), X_1(p_1), \\ldots,U_k(p_k),X_k(p_k))$ that could cover all the possibilities of $\\Delta_{2^{m+n}}$.\nThen the identification problem is equivalent to checking\n\\begin{align}\n\\bigcup_{i=1}^k\\{U_i(p_i)X_i(p_i)\\}=\\Delta_{2^{m+n}},\n\\end{align}\nwhere $U_i(p_i)X_i(p_i)$ represents $(u_i(0)x_i(0),u_i(1)x_i(1),\\ldots,u_i(p_i)x_i(p_i))$.\n\n\nAn ideal situation that only one group $(U_1(p_1), X_1(p_1))$ (not $k$ groups) is used to solve the identification problem of $F$, which was divided into two cases $H=I_{2^n}$ and $H\\neq I_{2^n}$, and was analyzed in \\cite{IdentificationofBCN2011}.\nThe observed data considered in \\cite{IdentificationofBCN2011} can be seen as a special case of Case 3.\nNext we slightly explain the method of \\cite{IdentificationofBCN2011} and use $(U(p), X(p), Y(p))$ instead of $(U_1(p_1),X_1(p_1),Y_1(p_1))$ for simplicity.\n\nUnder a designable input sequence $U(p)$, the input-state pair $(u(t),x(t))$ to traverse all the possibilities of $\\Delta_{2^{m+n}}$ can be guaranteed by controllability.\nBCN \\eqref{BCN2} is said to be $controllable$, if for any initial state $x(0)=x_0\\in \\Delta_{2^n}$ and any destination state $x_d\\in \\Delta_{2^n}$, there exists an input sequence $U(T)$, such that $x(T+1)=x_d$ \\cite{OBCN2}.\nWhen the system state is directly observable $(H=I_{2^n})$, the following lemma is immediate.\n\\begin{lemma}\\label{Lem3}{\\rm \\cite{IdentificationofBCN2011}}\nLet $H$ be the identity matrix $I_{2^n}$, then BCN \\eqref{BCN2} is uniquely identifiable if and only if \\eqref{BCN2} is controllable.\n\\end{lemma}\n\nIf a system is controllable, a single test sample could traverse all the possibilities.\nHence Lemma \\ref{Lem3} is based on only one group of input-output data.\nFor the general case of $H\\neq I_{2^n}$, the observability is involved again.\n\nThe O3-observability property of BCN \\eqref{BCN2} can be viewed as the observability property of BN \\eqref{BN2} to some extent.\nGiven an input sequence, a BCN is subject to a fixed evolutionary mechanism, similar to a BN.\nTherefore, our main purpose is to find all distinct output sequences generated by an input sequence that makes \\eqref{BCN2} O3-observable.\n\\cite{IdentificationofBCN2011} designed an enough input sequence $U(T')$ which could determine the state sequence $X(T)$ $(T<T')$ by combining the corresponding output sequence $Y(T'+1)$, where $X(T)$ is the previous part of $X(T'+1)$ generated by $U(T')$.\n\n\\begin{lemma}\\label{Lem4}{\\rm \\cite{IdentificationofBCN2011}}\nBCN \\eqref{BCN2} is uniquely identifiable from input-output data if and only if it is controllable and O3-observable.\n\\end{lemma}\n\nLemma \\ref{Lem4} is also based on only one group of input-output data.\nBy controllability, there exists an input sequence $U(T')$, such that the system runs along the following state trajectory $X(T'+1)$ \\cite{IdentificationofBCN2011}:\n\\begin{align}\n&~~X(T'+1)=~(\\underbrace{x(0),x(1),x(2),\\ldots,x(T)}_{controllability}, \\notag\\\\\n&\\underbrace{x(T+1),x(T+2),\\ldots,\\overset{x(0)}{\\overset{\\shortparallel}{x(t_{i_0}}}}_{controllability}\\hspace{-1mm}\\underbrace{), x(t_{i_0}+1),x(t_{i_0}+2),\\ldots,x(t_{i_0}+p+1)}_{O3-observability},\\notag\\\\\n&\\underbrace{x(t_{i_0}+p+2),x(t_{i_0}+p+3),\\ldots,\\overset{x(1)}{\\overset{\\shortparallel}{x(t_{i_1}}}}_{controllability}\\hspace{-1mm}\\underbrace{),x(t_{i_1}+1),x(t_{i_1}+2),\\ldots,x(t_{i_1}+p+1)}_{O3-observability},\n\\notag \\\\\n\\label{XT-0}&\\cdots, x(T'+1)),\n\\end{align}\nwhere $T'+1=t_{i_T}+p$, and $X(T)$ satisfies\n\\begin{align}\\label{XT-01}\n\\hspace{-0.2cm}\n\\{U(T)X(T)\\}=\\Delta_{2^{m+n}}.\n\\end{align}\n\nTo explain the purpose of driving the system along this state trajectory, we take the second and third rows as an example.\nThe state transition from $x(T+1)$ to $x(t_{i_0})=x(0)$ aims to find $x(0)$ by controllability, and that from $x(t_{i_0})$ to $x(t_{i_0}+p+1)$ aims to determine the value of $x(0)$ by O3-observability.\nContinuing this process, BCN \\eqref{BCN2} can be identified if these moments $t_{i_0}, t_{i_1},\\ldots, t_{i_{T}}$ are in place.\nObviously, it is difficult to design such an input sequence $U(T')$ before the system is known, especially for the input sequence $(u(T+1),u(T+2),\\ldots,u(T'))$ after time $T$, which involves timing accuracy.\n\nHere we provide a method to identify $(F,H)$ with the help of multiple samples (Case 3), which is different from \\cite{IdentificationofBCN2011} that used a single one.\nAn input sequence is called an O3-$test$ of BCN \\eqref{BCN2} if it makes \\eqref{BCN2} O3-observable.\nFrom \\eqref{XT-0}, we know that the input sequence $(u(T+1),u(T+2),\\ldots,u(T'))$ aims to find and identify each state in $(x(0),x(1),\\ldots,x(T))$.\nNext, we simplify this process with the help of Case 3.\n\nAssume that $(u_0',u_1',$ $\\ldots,u_p')$ is an O3-test, and $(u_0,u_1,\\ldots,u_T)$ satisfies\n\\begin{align}\\label{Alg:2-1}\n\\{u(t)x(t)|u(t)=u_t,t\\in[0,T]_{\\mathbb{N}}\\}=\\Delta_{2^{m+n}}.\n\\end{align}\nThen an O3-test could infiltrate the state at each time step by the following input sequences:\n   \\begin{align}\\label{Alg:2-3}\n   \\left\\{\n   \\begin{array}{lll}\n   U_0(p)=(u_0',u_1',\\ldots,u_p'),~~~~~~~~~~~~~~\\\\\n   U_1(p+1)=(u_0,u_0',u_1',\\ldots,u_p'), \\\\\n   U_2(p+2)=(u_0,u_1,u_0',u_1',\\ldots,u_p'),\\\\\n       ~~~~~~\\cdots\\\\\n   U_{T+1}(p+T+1)=(u_0,u_1,\\ldots,u_T,u_0',u_1',\\ldots,u_p').\n   \\end{array}\n   \\right.\n   \\end{align}\nDenote the corresponding observed data stemming from $x_0(0), x_1(0),\\ldots, x_{T+1}(0)$ ($x_0(0)=x_1(0)=\\cdots=x_{T+1}(0)$) as\n    \\begin{align}\\label{Alg:2-4}\n    \\left\\{\n    \\begin{array}{ll}\n    Y_0(p+1)=(y_0(0),y_0(1),\\ldots,y_0(p+1)), ~~~~~ \\\\\n    Y_1(p+2)=(y_1(0),y_1(1),\\ldots,y_1(p+2)), \\\\\n    ~~~~~~\\cdots \\\\\n    Y_{T+1}(p+T+2)=(y_{T+1}(0),y_{T+1}(1),\\ldots,y_{T+1}(p+T+2)).\n    \\end{array}\n    \\right.\n    \\end{align}\nSince the system is O3-observable with respect to $(u_0',u_1',\\ldots,u_p')$, we can find all distinct output sequences by retrieval from\n    \\begin{align}\\label{Alg:2-5}\n    Y^j=(y_j(j),y_j(j+1),\\ldots,y_j(j+p+1)),~j\\in [0,T+1]_{\\mathbb{N}}.\n    \\end{align}\nThe corresponding retrieval algorithm (Algorithm \\ref{Alg:B}) is given.\n\\begin{algorithm}[H]\n\\caption{Retrieve all distinct output sequences}\n\\label{Alg:B}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n     \\REQUIRE data \\eqref{Alg:2-4}.\n     \\ENSURE $Y_1,Y_2,\\ldots,Y_{2^n}$.\n    \\STATE{set $Y=\\emptyset$ and $i=1$}\n    \\FOR{$j=0; j<T+1; j++$}\n        \\IF {$Y^j\\in Y$}\n            \\STATE continue;\n        \\ELSE\n            \\STATE {$Y_i=Y^j$, $Y=Y\\cup \\{Y_i\\}$, $i=i+1$;}\n        \\ENDIF\n    \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nIdentify $Y_i$ as the output sequence of state $\\delta_{2^n}^i$ with respect to $(u_0',u_1',\\ldots,u_p')$, then BCN \\eqref{BCN2} can be identified by input-output data \\eqref{Alg:2-3} and \\eqref{Alg:2-4}.\nOn the basis of the discussion above, we provide an algorithm (Algorithm \\ref{Alg:2}) to identify $(F,H)$.\n\n\\begin{algorithm}\n\\caption{Identify $(F,H)$ for a controllable and O3-observable BCN \\eqref{BCN2} (Deal with Case 3).}\n\\label{Alg:2}\n{\\textbf{Input:}} an O3-test $(u_0',u_1',\\ldots,u_p')$, and $(u_0,u_1,\\ldots,u_T)$ satisfying \\eqref{Alg:2-1}.\n\n{\\textbf{Output:}} $F, H$.\n\t\n\\begin{description}\n\\item[Step $1$]: Construct input sequences \\eqref{Alg:2-3} and record the observed data \\eqref{Alg:2-4}.\n\n\\item[Step $2$]: Find all distinct output sequences $Y_1,Y_2,\\ldots,Y_{2^n}$ by Algorithm \\ref{Alg:B}.\n\n\\item[Step $3$]: Identify the state sequence $X_{T+1}(T+1)=(x_{T+1}(0),x_{T+1}(1),\\ldots,x_{T+1}(T+1))$\n    by\n    \\begin{align*}\n    x_{T+1}(t)=\\delta_{2^n}^{i_t},~t\\in[0,T+1]_{\\mathbb{N}},~{\\rm if}~(y_{t}(t),y_{t}(t+1),\\ldots,y_{t}(t+p+1))=Y_{i_t}.\n    \\end{align*}\n\n\\item[Step $4$]: Construct $F$ and $H$ based on\n    \\begin{align}\\label{Alg:2-6}\n    \\left\\{\n    \\begin{array}{lll}\n    x_{T+1}(t+1)=Fu_tx_{T+1}(t),~t\\in [0,T], \\\\\n    H=[Y_1(0)~Y_2(0)~\\cdots~Y_{2^n}(0)],\n    \\end{array}\n    \\right.\n    \\end{align}\n    where $Y_i(0)$ represents the first element in $Y_i$, $i\\in[1,2^n]_{\\mathbb{N}}$.\n\\end{description}\n\\end{algorithm}\n\n\nThe following example originates from literature \\cite{IdentificationofBCN2011}, in which an identification process with single input sequence is shown.\nWe use it to show how to use Algorithm \\ref{Alg:2}.\n\\begin{example}\\label{Eg2}\n{\\rm\\cite{IdentificationofBCN2011}}\nConsider an O3-observable and controllable BCN \\eqref{BCN2} with\n\\begin{align}\\label{Eg2-1}\n\\left\\{\n\\begin{array}{ll}\nF=&\\delta_4[2~4~1~1~2~3~2~2], \\\\\nH=&\\delta_2[2~1~1~2].\n\\end{array}\n\\right.\n\\end{align}\n\\end{example}\n\nFix the initial state $x(0)=\\delta_4^1$ and set\n\\begin{align*}\n\\left\\{\n\\begin{array}{ll}\n(u_0,u_1,\\ldots,u_{10})=\\delta_2(1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2), \\\\\n(u_0',u_1')=\\delta_2(1,1).\n\\end{array}\n\\right.\n\\end{align*}\nCase 3 divides the test sample into enough portions to be tested $(x_0(0)=x_1(0)=\\cdots=x_r(0)=\\delta_4^1)$.\n\nStep 1: 12 groups of input sequence are constructed and are infiltrated to 12 test samples: $x_k(0)=\\delta_4^1, k\\in[0,11]$. Then the corresponding output sequences are\n\\begin{align*}\nY_0(2)     =&\\delta_2(\\underline{2,1,2}),\\\\\nY_1(3)     =&\\delta_2(2,\\underline{1,2,2}),\\\\\nY_2(4)     =&\\delta_2(2,1,\\underline{2,2,1}),    \\\\\nY_3(5)     =&\\delta_2(2,1,2,2,1,2),\\\\\nY_4(6)     =&\\delta_2(2,1,2,2,1,2,2),\\\\\nY_5(7)     =&\\delta_2(2,1,2,2,1,\\underline{1,2,1})\\\\\nY_6(8)     =&\\delta_2(2,1,2,2,1,1,2,1,2), \\\\\nY_7(9)     =&\\delta_2(2,1,2,2,1,1,2,1,2,2),\\\\\nY_8(10)    =&\\delta_2(2,1,2,2,1,1,2,1,2,2,1),  \\\\\nY_9(11)    =&\\delta_2(2,1,2,2,1,1,2,1,2,1,2,2), \\\\\nY_{10}(12) =&\\delta_2(2,1,2,2,1,1,2,1,2,1,1,2,1), \\\\\nY_{11}(13) =&\\delta_2(2,1,2,2,1,1,2,1,2,1,1,1,2,2).\n\\end{align*}\n\nStep 2: All distinct output sequences generated by $(u_0',u_1')$ are\n\\begin{align*}\nY_1=&(y_0(0),y_0(1),y_0(2))=\\delta_2(2,1,2),\\\\\nY_2=&(y_1(1),y_1(2),y_1(3))=\\delta_2(1,2,2),\\\\\nY_3=&(y_2(2),y_2(3),y_2(4))=\\delta_2(2,2,1),\\\\\nY_4=&(y_5(5),y_5(6),y_5(7))=\\delta_2(1,2,1),\n\\end{align*}\nwhich are the last three elements of $Y_0(2)$, $Y_1(3)$, $Y_2(4)$ and $Y_5(7)$, respectively.\n\nStep 3: Since\n\\begin{align*}\n&(y_0(0),y_0(1),y_0(2))=Y_1,~(y_1(1),y_1(2),y_1(3))=Y_2,\\\\\n&~~~~~~~~~~~~~~~~~~~\\cdots~~~~~~~~~~~~~~~~~~~~~~~~\\cdots\\\\\n&(y_{10}(10),y_{10}(11),y_{10}(12))=Y_4,~(y_{11}(11),y_{11}(12),y_{11}(13))=Y_2,\n\\end{align*}\none gets the state sequence\n\\begin{align*}\nX_{11}(11)=&(x_{11}(0),x_{11}(1),\\ldots,x_{11}(11)) \\\\\n          =&\\delta_4(1, 2, 3, 1, 2, 4, 1, 2, 3, 2, 4, 2).\n\\end{align*}\n\nStep 4: Based on \\eqref{Alg:2-6}, this BCN is identified as\n\\begin{align}\\label{Eg2-2}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=\\delta_4[2~3~1~1~2~4~2~2]u(t)x(t), \\\\\ny(t)=\\delta_2[2~1~2~1]x(t).\n\\end{array}\n\\right.\n\\end{align}\n\nIn the sense of O3-observability, the observability matrix of BCN \\eqref{BCN2} is analogous to the observability matrix of BN \\eqref{BN2}, and can be written as:\n\\begin{align}\n\\mathcal{O}=\n\\left[\n\\begin{array}{llll}\nPH \\\\\nPHFu_0'\\\\\nPHFu_1'Fu_0'\\\\\n~~~\\vdots\\\\\nPHFu_p'\\cdots Fu_1'Fu_0'\n\\end{array}\n\\right]_{(p+2)\\times 2^n}\n\\end{align}\nwhich consists of $2^n$ distinct output sequences stemming from $2^n$ initial states under the input sequence $(u_0',u_1',\\ldots,u_p')$. For example, the observability matrix $\\mathcal{O}$ of \\eqref{Eg2-2} is\n\\begin{align}\n\\mathcal{O}=\n\\left[\n\\begin{array}{ccccc}\n2&1&2&1\\\\\n1&2&2&2\\\\\n2&2&1&1\n\\end{array}\n\\right],\n\\end{align}\nwhere the $i$th column of $\\mathcal{O}$ corresponds with $Y_i$, $i\\in[1,4]_{\\mathbb{N}}$.\n\n\nFrom the discussion above, one sees that, the identification problem is transformed into three conditions:\n\\begin{description}\n\\item[1)]The state trajectory of the system covers the entire state space;\n\n\\item[2)]There exists a method to distinguish all states;\n\n\\item[3)]There exist methods to implement 1) and 2).\n\\end{description}\nThese three conditions can be guaranteed by controllability, observability, and Assumption \\ref{Assum1}, respectively.\nUnder the case of Assumption \\ref{Assum1}, we can find $2^n$ distinct effective output sequences, or $2^n$ distinct output sequences generated by an O3-test, (or say the observability matrix $\\mathcal{O}$), based on available data.\nIf such output sequences are found or the observability matrix is constructed, the state sequence will be determined, and then the identification problem will be solved.\n\nCase 3 relaxes the limitation on the number of input sequences and output sequences.\nAlthough $(T+2)$ input sequences are used in Algorithm \\ref{Alg:2}, it can be implemented in biological experiments by cloning or segmentation, like diagnosis of disease in blood samples.\nInspired by the idea of multiple input sequences, next Lemma \\ref{Lem4} will be extended to the most general form O1-observability.\n\nConsidering an O1-observable BCN \\eqref{BCN2}, there exist $N:=2^{2n-1}-2^{n-1}$ input sequences $U_{(i,i')}(p)$, $1\\leq i\\neq i' \\leq 2^n$, such that $\\delta_{2^n}^i$ and $\\delta_{2^n}^{i'}$ can be distinguishable by $U_{(i,i')}(p)$. Arrange these input sequences as\n\\begin{align}\\label{O1-test}\nU_{s(i,i')}(p)=(u_{s(i,i')}(0),\\ldots,u_{s(i,i')}(p))=U_{(i,i')}(p),\n\\end{align}\nwhere the function $s(i,i')$ is defined by the following rule:\n\\begin{align}\ns(i,i')=\\left\\{\n\\begin{array}{ll}\ni'-i,~i=1, \\\\\n\\sum_{j=1}^{i-1}(n-j)+i'-i,~i\\geq 2.\n\\end{array}\n\\right.\n\\end{align}\nWe call \\eqref{O1-test} an O1-$test$ of BCN \\eqref{BCN2}.\nLet $Y_{i,s}=(y_i^s(0),y_i^s(1),\\ldots,y_i^s(p+1))$ represent the output sequence stemming from $\\delta_{2^n}^i$ with respect to $U_s(p)$, where $y_i^s(t)=HFu_s(t-1)\\cdots Fu_s(1)Fu_s(0)\\delta_{2^n}^i$.\nConstruct data arrays $D_i$, $i\\in [1,2^n]_{\\mathbb{N}}$ as\n\\begin{align}\\label{Di}\nD_i=(Y_{i,1}, Y_{i,2}, \\ldots, Y_{i,N}).\n\\end{align}\nThen for any distinct state $\\delta_{2^n}^i$ and $\\delta_{2^n}^{i'}$, we have\n$Y_{i,s(i,i')}\\neq Y_{i',s(i,i')}$ and $D_i\\neq D_{i'}$.\n\n\\begin{proposition}\\label{Pro2}\nBCN \\eqref{BCN2} is O1-observable if and only if there exist $2^n$ data arrays constructed by \\eqref{Di} satisfying\n$D_{i}\\neq D_{i'},~1\\leq i\\neq i'\\leq 2^{n}$.\n\\end{proposition}\n\nProposition \\ref{Pro2} shows that $2^n$ distinct states correspond to $2^n$ distinct data arrays in the sense of O1-observability.\nHence, analogous to the method used in O3-observability, by infiltrating O1-test \\eqref{O1-test} into $x(0),x(1),\\ldots, x(T+1)$ at each time step, we construct input sequences $U_0^s,U_1^s,\\ldots,U_{T+1}^s$, $s\\in [1,N]_{\\mathbb{N}}$ as\n   \\begin{align}\\label{Alg:3-2}\n   \\left\\{\n   \\begin{array}{lll}\n   U_0^s=(u_s(0),u_s(1),\\ldots,u_s(p)),~~~~~\\\\\n   U_1^s=(u_0,u_s(0),u_s(1),\\ldots,u_s(p)), \\\\\n   U_2^s=(u_0,u_1,u_s(0),u_s(1),\\ldots,u_s(p)),\\\\\n    ~~~~\\cdots\\\\\n   U_{T+1}^s=(u_0,u_1,\\ldots,u_T,u_s(0),u_s(1),\\ldots,u_s(p)),\n   \\end{array}\n   \\right.\n   \\end{align}\nand denote the corresponding observed data as,\n    \\begin{align}\\label{Alg:3-3}\n    \\left\\{\n    \\begin{array}{lll}\n    Y_0^s=(y_0^s(0),y_0^s(1),\\ldots,y_0^s(p+1)),~~~~~\\\\\n    Y_1^s=(y_1^s(0),y_1^s(1),\\ldots,y_1^s(p+2)),\\\\\n    Y_2^s=(y_2^s(0),y_2^s(1),\\ldots,y_2^s(p+3)),\\\\\n        ~~~~\\cdots \\\\\n    Y_{T+1}^s=(y_{T+1}^s(0),y_{T+1}^s(1),\\ldots,y_{T+1}^s(T+p+2)),\n    \\end{array}\n    \\right.\n    \\end{align}\nwhere $(u_0,u_1,\\ldots,u_T)$ satisfies \\eqref{Alg:2-1}.\n\nAll data arrays generated by O1-test are recorded in\n    \\begin{align}\\label{Alg:3-1}\n    D_s^j=(y_j^s(j),y_j^s(j+1),\\ldots,y_j^s(j+p+1)),~j\\in [0,T+1]_{\\mathbb{N}}.\n    \\end{align}\nAccording the following retrieval algorithm (Algorithm \\ref{Alg:C}), we can find all distinct data arrays $D_i$, $i\\in[1,2^n]_{\\mathbb{N}}$.\n\\begin{algorithm}[H]\n\\caption{Retrieve all distinct data arrays}\n\\label{Alg:C}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n     \\REQUIRE data \\eqref{Alg:3-1}.\n     \\ENSURE $D_1,D_2,\\ldots,D_{2^n}$.\n    \\STATE{set $D=\\emptyset$ and $i=1$}\n    \\FOR{$j=0; j<T+1; j++$}\n        \\STATE{$D^j=(D_1^j,D_2^j,\\ldots,D^j_{N})$}\n        \\IF {$D^j\\in D$}\n            \\STATE continue;\n        \\ELSE\n            \\STATE {$D_i=D^j$, $D=D\\cup \\{D_i\\}$, $i=i+1$;}\n        \\ENDIF\n    \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nIdentify $O_i$ as the data array stemming from $\\delta_{2^n}^i$, then the state sequence $(x(0),$ $x(1),\\ldots,x(T+1))$ can be identified by\n\\begin{align}\\label{Alg:3-4}\nx(t)=\\delta_{2^n}^{i_t},~{\\rm if}~D^t=D_{i_t}.\n\\end{align}\nCombining $(u_0,u_1,\\ldots,u_T)$, $(x(0),x(1),\\ldots, x(T+1))$ and $(y_{T+1}^1(0),\\ldots, y_{T+1}^1(T+1))$, $F$ and $H$ can be easily obtained.\n\n\\begin{theorem}\\label{Th2} {\\rm(}Single sample{\\rm)}\nBCN \\eqref{BCN2} is uniquely identifiable if it is controllable and O1-observable.\n\\end{theorem}\n\nOn the basis of the discussion above, we give an algorithm (Algorithm \\ref{Alg:3}) to identify BCN \\eqref{BCN2} which is O1-observable.\n\\begin{algorithm}\n\\caption{Identify $(F,H)$ for a controllable and O1-observable BCN \\eqref{BCN2} (Deal with Case 3).}\n\\label{Alg:3}\n{\\textbf{Input:}} O1-test: $U_1(p), U_2(p),\\ldots,U_{N}(p)$, and $(u_0,u_1,\\ldots,u_T)$ satisfying \\eqref{O1-test}.\n\n{\\textbf{Output:}} $F, H$.\n\n\n\\begin{description}\n\n\\item[Step $1$]: Construct input sequences \\eqref{Alg:3-2} and record the observed data \\eqref{Alg:3-3}.\n\n\\item[Step $2$]: Find all distinct data arrays by Algorithm \\ref{Alg:C}.\n\n\\item[Step $3$]: Identify the state sequence $X(T+1)=(x(0),x(1),\\ldots,x(T+1))$\n    by \\eqref{Alg:3-4}.\n\\item[Step $4$]: Construct $F$ and $H$ based on $X(T+1)$, $(u_0,u_1,\\ldots,u_T)$, $(y_{T+1}^1(0),\\ldots,$ $y_{T+1}^1(T+1))$ and\n    \\begin{align}\\label{Alg:3-5}\n    \\left\\{\n    \\begin{array}{lll}\n    x(t+1)=Fu_tx(t),  \\\\\n    y_{T+1}^1(t)=Hx(t), \\\\\n    t\\in [0,T].\n    \\end{array}\n    \\right.\n    \\end{align}\n\n\\end{description}\n\\end{algorithm}\n\nAlgorithm \\ref{Alg:3} is not a complex process, for the method used in \\cite{IdentificationofBCN2011}, should be relatively straightforward.\nThe core of the identification problem is the complete determination of state sequence $X(T+1)$.\nCompared with \\cite{IdentificationofBCN2011} (see \\eqref{XT-0}), the method proposed here considers only the first $T+1$ states and inputs, and avoids the design of subsequent inputs $u(T+1),u(T+2),\\ldots$.\nNote that the determination of the state sequence depends on the appointment of $D_i$. Hence $(F,H)$ generated by Algorithm \\ref{Alg:3} belongs to the equivalence class of the original system.\n\n\\begin{remark}\nOne idea for proving the necessity is to employ Lemma \\ref{Lem4} and a fact that O3-observability implies O1-observability.\nLogically, this is misleading, because, Theorem \\ref{Th2} is based on multiple input sequences, and Lemma \\ref{Lem4} is based on single input sequence.\nHence, we have\nidentifiability $\\Longleftarrow$ controllability $+$ O1-observability $\\Longleftarrow$ controllability $+$ O3-observability.\n\\end{remark}\n\n\nAs predicted, the implement of multiple input sequences (Case 3) allows more numerous systems to be identified.\nNext we consider whether the conditions of Theorem \\ref{Th2} be further relaxed. The answer is yes.\nIn Case 4, a large number of samples from different patients are collected and each sample is divided into multiple portions.\nIf the state sequences $X_j^i(p)=(x_j^i(0),x_j^i(1),\\ldots,x_j^i(p))$ and the input sequences $U_j^i(p)=(u_j^i(0),u_j^i(1),\\ldots,u_j^i(p))$ cover all the possibilities of $\\Delta_{2^{m+n}}$, i.e.,\n\\begin{align}\n\\bigcup_{i=1,j=1}^{k,r}\\{U_j^i(p)X_j^i(p)\\}=\\Delta_{2^{m+n}},\n\\end{align}\nthen by infiltrating the O1-test to all states $x_j^i(t), i\\in[1,k]_{\\mathbb{N}}, j\\in[1,r]_{\\mathbb{N}}, t\\in[0,p]_{\\mathbb{N}}$, the identification problem can be solved.\n\n\\begin{definition}\nConsider BCN \\eqref{BCN2} with a set of initial states $P_0$ and a set of destination states $P_d$.\n\n{\\rm i)} $P_d$ is set-$P_0$ controllable, if for any $x_d\\in P_d$, there exist a state $x(0)=x_0\\in P_0$ and an input sequence $U(T)$, such that $x(T+1)=x_d$;\n\n{\\rm ii)} \\eqref{BCN2} is set-$P_0$ controllable, if $\\Delta_{2^n}$ is set-$P_0$ controllable.\n\n{\\rm iii)} A state $x_0$ is controllable, if \\eqref{BCN2} is set-$P_0$ controllable and $P_0$ is a singleton set $P_0=\\{x_0\\}$.\n\n\\end{definition}\n\n\n\\begin{theorem}\\label{Th3} {\\rm(}Multiple samples{\\rm)}\nSuppose the set of initial states is $P_0$, then BCN \\eqref{BCN2} is uniquely identifiable if it is set-$P_0$ controllable and O1-observable.\n\\end{theorem}\n\n\\begin{proof}\nWithout loss of generality, assume $P_0=\\{\\delta_{2^n}^1,\\delta_{2^n}^2,\\ldots,\\delta_{2^n}^k\\}$, and the state space is split into $\\Delta_{2^n}=P_d^1\\cup P_d^2\\cup \\cdots \\cup P_d^k$, such that $P_d^a$ is set-$\\{\\delta_{2^n}^a\\}$ controllable, $a\\in[1,k]_{\\mathbb{N}}$. ($P_0$ may contain more (than $k$) states and we only use those which make the above condition true.)\n\nConsider state $\\delta_{2^n}^a$ and set $P_d^a$, if $P_d^a=\\delta_{2^n}\\{d_1^a,d_2^a,\\ldots,d_{\\theta_a}^a\\}$, then for any state $\\delta_{2^n}^i\\in P_d^a$, there exists an input sequence\n\\begin{align}\nU_{a\\rightarrow i}(p_i^a)=(u_{a\\rightarrow i}(0),u_{a\\rightarrow i}(1),\\ldots,u_{a\\rightarrow i}(p_i^a)),\n\\end{align}\nsuch that $Fu_{a\\rightarrow i}(p_i^a)\\cdots Fu_{a\\rightarrow i}(1)Fu_{a\\rightarrow i}(0)\\delta_{2^n}^a=\\delta_{2^n}^i$.\n\nConstruct an input sequence as\n\\begin{align}\n\\label{Th3-2} U^{j}_{a\\rightarrow i}(p_i^a+1)=&(u_{a\\rightarrow i}^j(0),u_{a\\rightarrow i}^j(1),\\ldots,u_{a\\rightarrow i}^j(p_i^a),u_{a\\rightarrow i}^j(p_i^a+1)) \\notag \\\\\n=&:(u_{a\\rightarrow i}(0),u_{a\\rightarrow i}(1),\\ldots,u_{a\\rightarrow i}(p_i^a),\\delta_{2^m}^j),\n\\end{align}\nunder which, the state sequence stemming from $\\delta_{2^n}^a$ becomes\n\\begin{align}\nX^{j}_{a\\rightarrow i}(p_i^a+2)=&(x_{a\\rightarrow i}^j(0),x_{a\\rightarrow i}^j(1),\\ldots,x_{a\\rightarrow i}^j(p_i^a+1),x_{a\\rightarrow i}^j(p_i^a+2)) \\notag \\\\\n  =&:(\\delta_{2^n}^a,x_{a\\rightarrow i}(1),\\ldots,\\delta_{2^n}^i,x_{a\\rightarrow i}^j(p_i^a+2)),\n\\end{align}\nthen we have\n\\begin{align}\n\\delta_{2^m}^j\\delta_{2^n}^{i}=u_{a\\rightarrow i}^j(p_i^a+1)x_{a\\rightarrow i}^j(p_i^a+1)\\in \\{U^{j}_{a\\rightarrow i}(p_i+1)X^j_{a\\rightarrow i}(p_i+1)\\}.\n\\end{align}\n\nIt follows that\n\\begin{align}\n\\Delta_{2^{m+n}}=&\\bigcup_{j=1}^{2^m}\\big\\{\\delta_{2^m}^j\\delta_{2^n}^i| \\delta_{2^n}^i\\in \\Delta_{2^n}\\big\\} \\notag \\\\\n=&\\bigcup_{j=1}^{2^m}\\bigcup_{a=1}^{k}\\big\\{\\delta_{2^m}^j\\delta_{2^n}^i| \\delta_{2^n}^i\\in P_d^a\\}\\big\\} \\notag \\\\\n\\subseteq& \\bigcup_{j=1}^{2^m}\\bigcup_{a=1}^{k}\\big\\{U^{j}_{a\\rightarrow i}(p_i+1)X^j_{a\\rightarrow i}(p_i+1)|i\\in\\{d_1^a,d_2^a,\\ldots,d_{\\theta_a}^a\\}\\big\\}\\notag \\\\\n=&\\bigcup_{j=1}^{2^m}\\bigcup_{a=1}^{k}\\bigcup_{i=1}^{2^n}\\big\\{U^{j}_{a\\rightarrow i}(p_i+1)X^j_{a\\rightarrow i}(p_i+1)\\big\\},\n\\end{align}\nwhich implies that enough input-state data could cover $\\Delta_{2^{m+n}}$.\n\nBy infiltrating the O1-test \\eqref{O1-test} to all states in $X^{j}_{a\\rightarrow i}(p_i^a+2)$, BCN \\eqref{BCN2} can be identified.\n\n\\end{proof}\n\n\n\\begin{corollary}\\label{Cor3-1}{\\rm(}Multiple samples{\\rm)}\nSuppose BCN \\eqref{BCN2} is O1-observable and the initial state set is $P_0$. If $P_d$ is the maximum set which is set-$P_0$ controllable, then $|P_d|$ states can be identified.\n\\end{corollary}\n\nTheorem \\ref{Th3} and Corollary \\ref{Cor3-1} show that, the maximum level of identifying BCN \\eqref{BCN2} depends on the relationship of all possible states stemming from the initial state set.\nIf the controllability is not available, the identification problem can still be well done with the same method.\nIn densely populated cities, the Centers for Disease Control and Prevention can collect a large number of samples, which most likely contain all possible initial states.\nThat is to say, all possible test samples endowed with $2^n$ distinct initial states are collected.\nIn Case 4, assume that $U_s$ \\eqref{O1-test} is an O1-test of the system, and $x_1^i(0)=x_2^i(0)=\\cdots=x_r^i(0)=\\delta_{2^n}^i$, $i\\in[1,2^n]_{\\mathbb{N}}$.\nConstruct input sequences\n\\begin{align}\\label{Alg:4-1}\n\\left\\{\n\\begin{array}{lll}\nU_s^i(p)       =(u_s^i(0),u_s^i(1),\\ldots,u_s^i(p))=U_s,~s\\in[1,N]_{\\mathbb{N}}, \\\\\nU_{jN+s}^i(p+1)  =(u_{jN+s}^i(0),u_{jN+s}^i(1),\\ldots,u_{jN+s}^i(p+1))\\\\\n~~~~~~~~~~~~~~~~~~=(\\delta_{2^m}^j,U_s),~j\\in[1,2^m]_{\\mathbb{N}},~s\\in[1,N]_{\\mathbb{N}},\n\\end{array}\n\\right.\n\\end{align}\nthen we have\n   \\begin{align}\n   \\Delta_{2^{m+n}}=&\\bigcup_{j=1}^{2^m}\\bigcup_{i=1}^{2^n}\\{\\delta_{2^m}^j\\delta_{2^n}^i\\} \\notag \\\\\n                   =&\\bigcup_{j=1}^{2^m}\\bigcup_{i=1}^{2^n}\\big\\{u_{jN+s}^i(0)x_{jN+s}^i(0)\\big\\} \\notag \\\\\n                   \\subseteq &\\bigcup_{j=1}^{2^m}\\bigcup_{i=1}^{2^n}\\big\\{U_{jN+s}^i(p) X_{jN+s}^i(p)\\big\\}.\n   \\end{align}\nThen the system can be easily identified analogous to Theorem \\ref{Th3}.\nHence we have the following result under the case of Assumption 1.\n\\begin{theorem}\\label{Th4} {\\rm(}Multiple samples{\\rm)}\nBCN \\eqref{BCN2} is uniquely identifiable if it is O1-observable.\n\\end{theorem}\n\nOn the basis of the analysis above, the corresponding identification algorithm (Algorithm \\ref{Alg:4}) can be established.\n\\begin{algorithm}\n\\caption{Identify an O1-observable BCN \\eqref{BCN2} (Deal with Case 4).}\n\\label{Alg:4}\n{\\textbf{Input:}} O1-test \\eqref{O1-test}: $U_1(p), U_2(p),\\ldots,U_{N}(p)$.\n\n{\\textbf{Output:}} $F,H$.\n\n\\begin{description}\n\n\\item[Step $1$]: Construct input sequences \\eqref{Alg:4-1} and record the observed data\n   \\begin{align}\\label{Alg:4-2}\n   \\left\\{\n   \\begin{array}{lll}\n   Y_s^i(p+1)=(y_s^i(0),y_s^i(1),\\ldots,y_s^i(p+1)),~s\\in[1,N]_{\\mathbb{N}}, \\\\\nY_{jN+s}^i(p+2)  =(Y_{jN+s}^i(0),\\ldots,Y_{jN+s}^i(p+2)),~j\\in[1,2^m]_{\\mathbb{N}},~s\\in[1,N]_{\\mathbb{N}}.\n\\end{array}\n\\right.\n\\end{align}\n\n\\item[Step $2$]: Find all distinct data arrays by Algorithm \\ref{Alg:D}.\n\n\\item[Step $3$]: Identify the states\n    $x_1^i(0)$, $x_{jN+1}^i(1)$, $j\\in [1,2^m]_{\\mathbb{N}}, i\\in[1,2^n]_{\\mathbb{N}}$, by\n    \\begin{align}\n    \\left\\{\n    \\begin{array}{lll}\n    x_1^i(0)=\\delta_{2^n}^{a_i},~{\\rm if}~(Y_1^i(p+1),\\ldots,Y_N^i(p+1))=D_{a_i}, \\\\\n    x_{Nj+1}^i(1)=\\delta_{2^n}^{a_i},~{\\rm if}~(\\overline{Y}_{jN+1}^i,\\ldots,\\overline{Y}_{jN+s}^i)=D_{a_i},\n    \\end{array}\n    \\right.\n    \\end{align}\n    where $\\overline{Y}_{jN+s}^i=(y_{jN+s}^i(1),\\ldots,y_{jN+s}^i(p+2))$.\n\\item[Step $4$]: Construct $F$ and $H$ based on\n    \\begin{align}\\label{Alg:4-4}\n    \\left\\{\n    \\begin{array}{lll}\n    x_{Nj+1}^i(1)=F\\delta_{2^m}^jx_{Nj+1}^i(0)=F\\delta_{2^m}^jx_{1}^i(0),  \\\\\n    y_{1}^i(0)=Hx_{1}^i(0).\n    \\end{array}\n    \\right.\n    \\end{align}\n\n\\end{description}\n\n\\end{algorithm}\n\n\n\\begin{algorithm}[H]\n\\caption{Retrieve all distinct data arrays}\n\\label{Alg:D}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n     \\REQUIRE data \\eqref{Alg:4-2}.\n     \\ENSURE $D_1,D_2,\\ldots,D_{2^n}$.\n    \\STATE{set $D=\\emptyset$ and $a=1$}\n    \\FOR{$i=0; i<2^n; i++$}\n    \\FOR{$j=0; j<2^n; j++$}\n        \\IF {$j=0$}\n        \\STATE{$D_s^j=Y_s^i(p+1)$}\n        \\ELSE\n           \\STATE{$D_s^j=(y_{jN+s}^i(1),\\ldots,y_{jN+s}^i(p+2))$}\n        \\ENDIF\n        \\STATE{$D^j=(D_1^j,D_2^j,\\ldots,D^j_{N})$}\n        \\IF {$D^j\\in D$}\n            \\STATE continue;\n        \\ELSE\n            \\STATE {$D_a=D^j$, $D=D\\cup D_a$, $a=a+1$;}\n        \\ENDIF\n    \\ENDFOR\n    \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{remark}\nThere are two ways to construct the structure matrix $H$. One is based on the state sequence and the output sequence, like \\eqref{Alg:3-5} and \\eqref{Alg:4-4} (in Algorithms \\ref{Alg:3} and \\ref{Alg:4}), the other is based on the first elements of the effective output sequences or the output sequences generated by O3-test or the data arrays generated by O1-test, like \\eqref{Th1-data4} and \\eqref{Alg:2-6} (in Algorithms \\ref{Alg:1} and \\ref{Alg:2}).\n\\end{remark}\n\n\\begin{remark}\nIt is important to point out that, the input sequence required for identification is assumed to be known, whether in {\\rm\\cite{IdentificationofBCN2011}} {\\rm(}used O3-observability{\\rm)} or our paper {\\rm(}O1-observability{\\rm)}.\nFor unknown systems, different input sequences are used to implement Algorithm \\ref{Alg:4} (Algorithm \\ref{Alg:2}, Algorithm \\ref{Alg:3}) until the system is fully identified.\nThe former {\\rm(}O3-observability{\\rm)} covers an NP-hard problem{\\rm \\cite{OBCN3}}, which has been mentioned in the introduction part. So does our method relax, in a sense, the difficulty of dealing with the identification problem? In other words, is the difficulty of finding an O1-test also NP-hard?\n\\end{remark}\n\n\\begin{example}\\label{Eg4}\nConsider the reduced model for the lac operon in the bacterium Escherichia coli.\nThis BCN has three state nodes $\\{x_1,x_2,x_3\\}$ and three input nodes $\\{u_1,u_2,u_3\\}$.\n$x_1$: lac mRNA, $x_2$: lactose in high concentration, $x_3$: lactose in medium concentration, $u_1$: extracellular glucose, $u_2$: high extracellular lactose, and $u_3$: medium extracellular lactose.\nThe dynamics of this system can be written as \\cite{lacoperon2011}\n\\begin{align}\\label{Eg4-1}\n\\left\\{\n\\begin{array}{ll}\nx_1(t+1)=\\neg u_1(t)\\wedge (x_2(t)\\vee x_3(t)), \\\\\nx_2(t+1)=\\neg u_1(t)\\wedge u_2(t) \\wedge x_1(t), \\\\\nx_3(t+1)=\\neg u_1(t)\\wedge (u_2(t)\\vee (u_3(t)\\wedge x_1(t))).\n\\end{array}\n\\right.\n\\end{align}\nThis BCN is O1-observable when the outputs are considered as \\cite{SetcontrolCheng2018}\n\\begin{align}\\label{Eg4-2}\n\\left\\{\n\\begin{array}{ll}\ny_1(t)=x_1(t)\\vee \\neg x_2(t)\\vee x_3(t),\\\\\ny_2(t)=\\neg x_1(t)\\vee x_2(t) \\wedge \\neg x_3(t),\\\\\ny_3(t)=\\neg x_1(t)\\wedge \\neg x_2(t)\\vee x_3(t).\n\\end{array}\n\\right.\n\\end{align}\n\\end{example}\nIts algebraic form is\n\\begin{align}\\label{Eg4-3}\n\\left\\{\n\\begin{array}{ll}\nF=\\delta_8[8~8~8~8~8~8~8~8 ~8~8~8~8~8~8~8~8 \\\\\n~~~~~~~~~8~8~8~8~8~8~8~8 ~8~8~8~8~8~8~8~8 \\\\\n~~~~~~~~~1~1~1~5~3~3~3~7 ~1~1~1~5~3~3~3~7 \\\\\n~~~~~~~~~3~3~3~7~4~4~4~8 ~4~4~4~8~4~4~4~8],\\\\\nH=\\delta_8[8~6~3~6~5~6~7~6].\n\\end{array}\n\\right.\n\\end{align}\n\nHere we analyze the identification problem of this system. Fix the initial state $x_1^i(0)=\\cdots=x_r^i(0)=\\delta_8^i$, $i\\in[1,8]_{\\mathbb{N}}$ and choose the following O1-test\n\\begin{align}\\label{Eg4-O1-test}\nU_s=\\left\\{\n\\begin{array}{lll}\n(\\delta_8^5),~s\\in S,\\\\\n(\\delta_8^1),~s\\in S^c,\n\\end{array}\n\\right.\n\\end{align}\nwhere $S=\\{9,10,11,12,13,20,21,22,27\\}$ and $S^c=[1,28]_{\\mathbb{N}}\\setminus \\{9,10,11,12,13,20,21,$ $22,27\\}$.\n\nStep 1: Construct input sequences\n\\begin{align*}\n\\left\\{\n\\begin{array}{lll}\nU_s^i(0)       =(u_s^i(0))=U_s,~s\\in[1,28]_{\\mathbb{N}}, \\\\\nU_{28j+s}^i(1)  =(u_{28j+s}^i(0),u_{28j+s}^i(1))=(\\delta_8^j,U_s),~j\\in[1,8]_{\\mathbb{N}}~s\\in[1,28]_{\\mathbb{N}},\n\\end{array}\n\\right.\n\\end{align*}\nthen the output sequences stemming from states $x_1^1(0),x_2^1(0),\\ldots,x_r^1(0)$ are\n\\begin{align*}\n\\left\\{\n\\begin{array}{lll}\nY_s^1(1)   =\\delta_8(\\underline{8,6}),~s\\in S^c,\\\\\nY_s^1(1)   =\\delta_8(\\underline{8,8}),~s\\in S,\\\\\nY_{28+s}^1(2) =Y_{56+s}^1(2)=Y_{84+s}^1(2) =Y_{112+s}^1(2) =\\delta_8(8,\\underline{6,6}),~s\\in S^c,   \\\\\nY_{28+s}^1(2) =Y_{56+s}^1(2)=Y_{84+s}^1(2) =Y_{112+s}^1(2) =\\delta_8(8,\\underline{6,7}),~s\\in S,   \\\\\nY_{140+s}^1(2) =Y_{168+s}^1(2) =\\delta_8(8,{8,6}),~s\\in S^c,   \\\\\nY_{140+s}^1(2) =Y_{168+s}^1(2) =\\delta_8(8,{8,8}),~s\\in S,   \\\\\nY_{196+s}^1(2) =\\delta_8(8,\\underline{3,6}),~s\\in S^c,   \\\\\nY_{196+s}^1(2) =\\delta_8(8,\\underline{3,8}),~s\\in S,   \\\\\nY_{224+s}^1(2) =\\delta_8(8,\\underline{6,6}),~s\\in S^c,   \\\\\nY_{224+s}^1(2) =\\delta_8(8,\\underline{6,5}),~s\\in S,\n\\end{array}\n\\right.\n\\end{align*}\nand others are shown in Appendix \\ref{appA}.\n\n\n\n\nStep 2: All distinct data arrays generated by the O1-test \\eqref{Eg4-O1-test} are\n\\begin{align*}\nD_1=& (D_1^1,D_2^1,\\ldots,D_{28}^1),~\\left\\{\n\\begin{array}{lll}\nD_s^1=\\delta_8(8,6),~s\\in S^c,\\\\\nD_s^1=\\delta_8(8,8),~s\\in S,\n\\end{array}\n\\right. \\\\\nD_2=& (D_1^2,D_2^2,\\ldots,D_{28}^2),~\\left\\{\n\\begin{array}{lll}\nD_s^2=\\delta_8(6,6),~s\\in S^c,\\\\\nD_s^2=\\delta_8(6,7),~s\\in S,\n\\end{array}\n\\right. \\\\\n&\\cdots \\\\\nD_8=& (D_1^8,D_2^8,\\ldots,D_{28}^8),~\\left\\{\n\\begin{array}{lll}\nD_s^8=\\delta_8(6,6),~s\\in S^c,\\\\\nD_s^8=\\delta_8(6,3),~s\\in S,\n\\end{array}\n\\right.\n\\end{align*}\nwhich are the last two elements of $Y_s^1(1)$, $Y_{28+s}^1(2)$, $Y_{196+s}^1(2)$, $Y_{224+s}^1(2)$, $Y_{s}^2(2)$, $Y_{196+s}^4(2)$, $Y_{s}^5(1)$ and $Y_{s}^6(1)$, respectively.\n\nStep 3: Identify $D_i$ as the data array stemming from $\\delta_8^i$, $i\\in[1,8]_{\\mathbb{N}}$, then from Step 2, we have\n\\begin{align*}\nx_{28+1}^1(1)=Fu_{28+1}^1(0)x_{28+1}^1(0)\\Rightarrow &~ \\delta_8^2=F\\delta_8^1\\delta_8^1, \\\\\nx_{56+1}^1(1)=Fu_{56+1}^1(0)x_{56+1}^1(0)\\Rightarrow &~ \\delta_8^2=F\\delta_8^2\\delta_8^1, \\\\\nx_{84+1}^1(1)=Fu_{84+1}^1(0)x_{84+1}^1(0)\\Rightarrow &~ \\delta_8^2=F\\delta_8^3\\delta_8^1, \\\\\nx_{112+1}^1(1)=Fu_{112+1}^1(0)x_{112+1}^1(0)\\Rightarrow &~ \\delta_8^2=F\\delta_8^4\\delta_8^1, \\\\\nx_{140+1}^1(1)=Fu_{140+1}^1(0)x_{140+1}^1(0)\\Rightarrow &~ \\delta_8^1=F\\delta_8^5\\delta_8^1, \\\\\nx_{168+1}^1(1)=Fu_{168+1}^1(0)x_{168+1}^1(0)\\Rightarrow &~ \\delta_8^1=F\\delta_8^6\\delta_8^1, \\\\\nx_{196+1}^1(1)=Fu_{196+1}^1(0)x_{196+1}^1(0)\\Rightarrow &~ \\delta_8^3=F\\delta_8^7\\delta_8^1, \\\\\nx_{224+1}^1(1)=Fu_{224+1}^1(0)x_{224+1}^1(0)\\Rightarrow &~ \\delta_8^4=F\\delta_8^8\\delta_8^1, \\\\\n\\cdots &\n\\end{align*}\nand\n\\begin{align*}\n&y_{1}^1(0)=Hx_{1}^1(0)\\Rightarrow~ \\delta_8^8=H\\delta_8^1,~y_{1}^2(0)=Hx_{1}^2(0)\\Rightarrow ~ \\delta_8^6=H\\delta_8^5, \\\\\n&y_{1}^3(0)=Hx_{1}^3(0)\\Rightarrow~ \\delta_8^3=H\\delta_8^3,~y_{1}^4(0)=Hx_{1}^4(0)\\Rightarrow ~ \\delta_8^6=H\\delta_8^4, \\\\\n&y_{1}^5(0)=Hx_{1}^5(0)\\Rightarrow~ \\delta_8^5=H\\delta_8^7,~y_{1}^6(0)=Hx_{1}^6(0)\\Rightarrow ~ \\delta_8^6=H\\delta_8^8, \\\\\n&y_{1}^7(0)=Hx_{1}^7(0)\\Rightarrow~ \\delta_8^7=H\\delta_8^6,~y_{1}^8(0)=Hx_{1}^8(0)\\Rightarrow~ \\delta_8^6=H\\delta_8^2.\n\\end{align*}\nThis BCN therefore is identified as\n\\begin{align}\\label{Eg4-4}\n\\left\\{\n\\begin{array}{ll}\nF=\\delta_8[2~2~2~2~2~2~2~2 ~2~2~2~2~2~2~2~2 \\\\\n ~~~~~~~~~2~2~2~2~2~2~2~2 ~2~2~2~2~2~2~2~2 \\\\\n ~~~~~~~~~1~6~1~7~1~3~3~3 ~1~6~1~7~1~3~3~3 \\\\\n ~~~~~~~~~3~2~3~6~3~4~4~4 ~4~2~4~2~4~4~4~4],\\\\\nH=\\delta_8[8~6~3~6~6~7~5~6].\n\\end{array}\n\\right.\n\\end{align}\n\n\\begin{remark}\nA BN may be identifiable but not observable, as mentioned in Remark \\ref{Rem1}.\nSimilar to the method in BNs (see Remark \\ref{Rem1}), it is still possible to identify an unobservable BCN.\nTherefore, observability is a stronger property than identifiability in BNs and BCNs.\n\\end{remark}\n\n\\begin{remark}\nAbout the identification algorithm in the Matlab programming, the storage method and storage space of the identification standard affect the execution time of the algorithm, because the programming involves the storage and retrieval of data.\nHence, it is vital to plan for storage space needs at the beginning of the design phase.\nCompared with $2^n$ distinct effective output sequences or $2^n$ distinct output sequences generated by an O3-test or the corresponding observability matrix, $2^n$ distinct data arrays generated by an O1-test require more storage space. How to adjust the storage method and how to reduce the storage space are two challenging and interesting topics.\n\\end{remark}\n\nUp to now, we have provided several ways to deal with different situations.\nThe comparison of them is needed and is shown in Table I, where Case 3' considers one group of input-output data, a special case of Case 3.\n\n\n\\begin{table*}[!htbp]\n\\centering\n\\begin{tabular}{c | c | c | c |c }\n\\hline\nApproach & System & Case & Condition 1 & Condition 2 \\\\\n\\hline\nLemma \\ref{Lem2} \\cite{ModelConstruction2011} & BN & Case 2&  & $H=I_{2^l}$ \\\\\n\\cdashline{1-5}[1pt\/2pt]\nTheorem \\ref{Th1} & \\multirow{2}{*}{BN} & \\multirow{2}{*}{Case 2}  &  & \\multirow{2}{*}{Observability} \\\\\nAlgorithm \\ref{Alg:1}  &     &  &    &   \\\\\n\\cdashline{1-5}[1pt\/2pt]\nLemma \\ref{Lem3}\\cite{IdentificationofBCN2011} & BCN & Case 3 & controllability & $H=I_{2^l}$ \\\\\n\\cdashline{1-5}[1pt\/2pt]\nLemma \\ref{Lem4}\\cite{IdentificationofBCN2011} & \\multirow{2}{*}{BCN} & Case $3'$ & \\multirow{2}{*}{controllability} & \\multirow{2}{*}{O3-observability}  \\\\\nAlgorithm \\ref{Alg:2} &     & Case 3   &  &   \\\\\n\\cdashline{1-5}[1pt\/2pt]\nTheorem \\ref{Th2} &   \\multirow{2}{*}{BCN}   & \\multirow{2}{*}{Case 3}   &\\multirow{2}{*}{controllability} & \\multirow{2}{*}{O1-observability}  \\\\\nAlgorithm \\ref{Alg:3}   &   &  &   & \\\\\n\\cdashline{1-5}[1pt\/2pt]\nTheorem \\ref{Th3} &  BCN  & Case 4   & Set controllability & O1-observability   \\\\\n\\cdashline{1-5}[1pt\/2pt]\nTheorem \\ref{Th4} &  \\multirow{2}{*}{BCN}  & \\multirow{2}{*}{Case4}   &     & \\multirow{2}{*}{O1-observability} \\\\\nAlgorithm \\ref{Alg:4} &   &  &   & \\\\\n\\hline\n\\end{tabular}\n\\caption{A comparison table of various methods}\n\\end{table*}\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this paper, we systematically explored the identification problem of BNs and BCNs, gained new cognition.\nBased on the practical application, we built a new analytical framework that consists of single sample and multiple samples, and then divided the identification problem into four situations.\nFour simple criteria were proposed for determining the identifiability of BNs and BCNs, and the corresponding identification algorithms were provided to identify related structure matrices.\nIt is worth noting that these algorithms are easy to implement by MATLAB.\nUnder this analytical framework, we found three novel and important results:\n(1) A BN is uniquely identifiable if it is observable; (2) A BCN is uniquely identifiable if it is O1-observable;\n(3) The necessity of (1) or (2) does not hold.\nAt last, we presented a table to reveal the relationships of the proposed results, which could be used to further analyze the relationships of identifiability, controllability, O1-observability and O3-observability, as a foreshadowing.\n\nThe authors believe that this new analytical framework is a very powerful explanation tool.\nOn the basis of this paper, there are several natural and interesting problems remaining for further study. For example: (1) If a BN or BCN is unobservable, some states produce same output sequence, which implies that some columns in the structure matrices $F$ and $H$ are indeterminate. A very natural question is whether an unobservable system can be identified, or, how to identify an unobservable system?\n(2) The assumption (Assumption \\ref{Assum1}) that the input-output data is sufficient is a highlight of this paper.\nHow to generalize this idea to other systems, like singular BNs or switched BCNs?\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{INTRODUCTION }\n\\label{sec:intro}\n\nThe 2D power spectrum (PS) has proven to be a valuable diagnostic quantity in the hunt for the signature of the Epoch of Reionisation (EoR) in the early H\\textsc{i}\\ density field \\citep{Liu2014,Trott2014,Barry2016}.\nWhile the 21cm EoR signal is expected to be well-described by the 1D PS, whose interpretation assumes both homogeneity and isotropy in 3 dimensions,\nits processing through low-frequency radio telescopes, as well as the various foreground contaminants, do not preserve 3D isotropy.\nIndeed, radio interferometers are remarkably complex and exhibit different systematic effects in frequency and spatial modes, corresponding to line-of-sight ($k_{||}$) and perpendicular ($k_\\perp$) modes respectively.\nThe 2D PS maintains the separation of these modes, which is highly useful in identifying and removing\/mitigating systematics.\n\nIt is well known that the dominant component of the raw data arises not from the desired 21 cm signal, but the extremely bright foregrounds \\citep[eg.][]{Shaver1999}. \nMitigation of these foregrounds has become one of the key areas of research in the field \\citep[eg.][]{Bernardi2009,Liu2011,Chapman2012,Chapman2013,DiMatteo2004,Dillon2015,Chapman2015,Jensen2016,Sims2016,Mertens2017,Chapman2016,Ghosh2017,Murray2017}, and a standard picture has emerged which centres on the expected spectral smoothness of the foregrounds in contrast to the EoR signal. \nIn this picture, the foregrounds are constrained to low-$k_{||}$ modes in the 2D PS, leaving high-$k_{||}$ modes relatively free of their contamination (for a clear schematic, see Fig. 1 of \\citet{Liu2014a}).\nVarious methods have been developed to exploit this behaviour, either by entirely \\textit{avoiding} low-$k_{||}$ modes, \\textit{subtracting} spectrally-smooth foreground models (either parametrically or non-parametrically), or a combination of the two \\citep{Chapman2016}.\n\nThese methods have a well-known complication, however: any radio interferometer is by nature \\textit{chromatic}, i.e. its view of the sky differs at different frequencies (via a number of factors including the frequency-dependent antenna response and the geometry of the baselines). \nThis results in different \\textit{perpendicular} scales having different dependency on frequency, i.e. \\textit{parallel} scales -- a ``mode-mixing'' that can result in low-$k_{||}$ power being ``leaked'' into higher $k_{||}$ modes for higher $k_\\perp$. \nThis gives rise to the infamous foreground ``wedge'', and its complementary EoR ``window''.\n\nThe wedge causes issues for foreground mitigation techniques. \nFor example, consider the ``avoidance'' technique: firstly, the existence of the wedge increases the number of modes over which the foregrounds dominate, and therefore further obscures the faint signal. \nSecondly, if one incorrectly specifies the boundary of the wedge, the extra leaked power inside the supposed ``window'' will at best bias results, if not overwhelm them.\n\nThe foreground wedge has been the subject of a great deal of research over the past decade.\nIt was originally reported by \\citet{Datta2010}, though it was alluded to in different form in previous work \\citep[eg.][]{Bowman2009}.\nFollowing this initial report, a handful of papers appeared that included intuitive and pedagogical explanations for why the wedge arises, along with some discussion as to the primary physical factors which determine its shape and amplitude \\citep{Morales2012,Parsons2012,Trott2012,Vedantham2012}.\nThese various explanations are complementary in the sense that they appraise the problem from different perspectives, arriving at similar conclusions.\nFor instance, \\citet[hereafter M12]{Morales2012} explain the foreground wedge in the context of \\textit{residual foreground emission} from point-sources (eg. arising from imperfect source peeling), considering as an example a single point-source at some location on the sky. \nThey argue that the wedge arises due to the combination of the migration of baselines in the $uv$-plane with frequency, and the missing information inherent to a discrete sampling of the $uv$-plane. \nFigure 2 in M12 presents their intuitive picture: the inherent voltage correlations for a given baseline length are sampled by a single baseline which moves across the corrugated correlations at an angle in frequency. \nAfter accounting for smearing from the instrumental beam, the gridded baseline, when Fourier-Transformed over frequency, contains oscillations that introduce power at mode $k_{||} > 0$, dependent on how quickly the baseline moves through the corrugations in frequency.\nThe value of $k_{||}$ at which this power is introduced is simply proportional to the ``slope'' of the baseline in the $|u|\\nu$-plane, which is itself proportional to the baseline length or $k_\\perp$. \nFurthermore, as sources closer to the horizon have higher oscillations in their voltage correlations as a function of baseline length, this mode-coupled $k_{||}$ value is also proportional to the distance of the source from phase-centre: $k_{||} \\propto l k_\\perp$. \nSince sources are constrained to be within the horizon, we have $l_{\\rm max} = \\sin \\theta_{k, {\\rm max}} = 1$, and we are able to define the ``horizon limit'' beyond which we do not expect flat-spectrum foreground sources to contribute.\n \nM12 also argues that the wedge is \\textit{fundamental} in low-frequency interferometric observations, and cannot be avoided (at the relevant modes) merely by clever analysis, such as visibility gridding and weighting schemes.\n\n\\citet{Trott2012} (T12) reformulates the description of M12 in the context of a uniform distribution of faint undetected sources, deriving an exact analytical form  for the expected foreground power under a set of simplifications.\n\nAlternatively, \\citet[hereafter P12]{Parsons2012} describes the emergence of the wedge in terms of the so-called ``delay transform''. \nThe delay transform considers a single baseline at a time, and associates `delay'-modes -- the fourier-dual of the frequency-track of the baseline -- with the line-of-sight modes $k_{||}$. \nThe delays themselves are simply time-delays between the reception of plane-wave emission at the two antennas composing the baseline (see Fig. 1 of P12 for a clear diagram).\nIn this scheme, the arguments are entirely geometric;\nfor a given source, not at phase centre, a longer baseline will correspond to a higher delay. \nSimilarly, for a given baseline, a source closer to the horizon will correspond to a higher delay -- with a physical maximum at the horizon. \nThis leads to the now familiar equation relating the line-of-sight mode at which power from a single source manifests: $\\tau \\sim k_{||} \\propto l k_\\perp$.\nP12 explains that the delay transform ideally maps a flat-spectrum point source to a delta-function in delay space, but that in practice the non-flat spectral properties of both source and instrument add a (hopefully narrow) kernel which can throw power into modes beyond the ``horizon limit'' (cf. Fig. 1 of P12).\nThey thus suggest that designing instruments with maximal spectral and spatial smoothness, as well as reduced field-of-view, and then ignoring modes below the horizon limit, is a useful way to avoid the foreground problem. \nThis has motivated the design of the PAPER \\citep{,Ali2015} and HERA \\citep{DeBoer2016} experiments. \n\nDespite the simplicity of these intuitive descriptions of the emergence of the wedge, its precise amplitude and shape are dependent on a combination of various complex effects. \nAmongst these are unavoidable sky-based effects such as the (spatially varying) spectrum of sources and diffuse emission, angular distribution of sources \\citep{Bowman2009,Trott2016a,Murray2017}, effects of co-ordinate transformation from curved to flat sky \\citep{Thyagarajan2013,Thyagarajan2015,Ghosh2017} and polarization leakage \\citep{Gehlot2018}, as well as spectral characteristics of the instrument, such as the beam attenuation pattern, bandpass, chromatic baselines and chromatic calibration errors \\citep{Bowman2009,Thyagarajan2015,Pober2015,Trott2016a}.\nDue to the complexity of these effects, they are often investigated either by using simulations or via analytic simplifications which elucidate the effects of some subset of the components. \nSeveral works have been devoted to developing general frameworks to model the foreground wedge in order to mitigate it effectively \\citep[eg.][]{Liu2014,Liu2014a,Pober2015,Ghosh2017}.\n\nDespite the breadth of this research, one aspect which seems to have gained little attention is the layout of the antennas (or correspondingly the baselines) themselves,\nand how they might be used to mitigate the wedge.\nThis is perhaps surprising as several of the seminal works on the topic suggest that one way to alleviate mode-mixing is to employ dense $uv$-sampling so that $uv$-samples overlap at various frequencies \\citep[eg.][]{Bowman2009,Morales2012,Parsons2012}.\nWhile a perfect $uv$-sampling is clearly unachievable, which has perhaps led to this avenue being largely ignored, it would seem advantageous to determine the extent of wedge-suppression possible under reasonable constraints. \n\nThe purpose of this paper is to explore this question, which we attack in two parts. First we approach the question of how the wedge relates to the baseline layout, seeking intuitive semi-analytical understanding of the factors involved. Secondly, we ask the more pointed question of how far the wedge might be suppressed merely by choice of layout, limiting ourselves to layouts which might be realistically achieved. For this latter question, we necessarily turn to simple numerical simulations.\nWe approach the questions from a pedagogical view, making simplifications where necessary in order to elucidate conceptual understanding.\n\n\n\n\nThe layout of the paper is as follows. \\S\\ref{sec:framework} introduces the general equations (and assumptions) used throughout this paper to define the expected 2D PS, and the model simplifications we adopt. \n\\S\\ref{sec:classic} presents the classical form of the wedge by solving the general equation for a suitably sparse layout, which is shown to be equivalent to the delay spectrum.\nThis lays the groundwork for \\S\\ref{sec:weighted}, which considers the same family of $uv$-sampling functions, but with increased density, and thus must account for correlations between baselines.\nIt provides a semi-analytic framework to describe the density of baselines required to mitigate the wedge, and also the effects of deviations of the baseline layout from the assumed perfect regularity.\n\\S\\ref{sec:mitigation:arrays} turns to discussion of the consequences of the preceding results for realistic arrays, and analyses explicit wedge reduction for a series of archetypal layouts.\nFinally, \\S\\ref{sec:conclusions} wraps up with a summary of the key arguments and conclusions throughout the paper, and a prospectus for future work.\n\n\\section{Framework for Expected Foreground Power}\n\\label{sec:framework}\nIn this section we derive a (simplified) general equation that describes the \\textit{expected} 2D PS for a given sky distribution and instrument model, following similar lines as \\citet{Trott2012}, \\citet{Liu2014} and \\citet{Trott2016}\\footnote{Readers familiar with these derivations should be able to skim this section lightly, referring to Eqs. \\ref{eq:simple_vis}, \\ref{eq:vis_gridded} and \\ref{eq:power_general} and Tables \\ref{tab:assumptions} and \\ref{tab:models} thereafter.}.\nWe differ from \\citet{Liu2014} in that we express the expected power (closely related to the covariance of visibilities) in the basis of the natural coordinates, ($\\vect{u}$, $\\eta$), rather than baseline vectors and delay (they express their covariance in the latter basis, and reconstitute in cosmologically aligned coordinates via another transformation).\nThis makes sense for our analysis, as we are interested in the conceptual understanding of where foreground power arises from, rather than a numerically efficient power spectrum estimator.\nWe also differ from \\citet{Trott2016} in that we consider correlations between baselines in the expectation of the foreground power, which is necessary in order to properly evaluate the effects of $uv$-sampling.  \n\nWe \\textit{a priori} remark that our framework is not fully general -- it does not include all possible factors.\nThis is in the hope of elucidating our primary goal -- the effect of the antenna layout.\nOne simplification we will enforce for this paper is that we only consider the effect of \\textit{point sources}, not Galactic emission, or extended sources (or the negligible EoR signal for that matter). \nThe extension to these other sources of foreground emission is neither conceptually important for this work nor conceptually difficult (though the details of the formulation can be rather involved, eg. \\citet{Trott2017,Murray2017}).\nA second simplification is that we consider the simple case in which the telescope is pointing instantaneously at zenith.\nThis alleviates complications arising from baseline foreshortening (not entirely, though enough for the conceptual understanding aimed for in this work, cf. \\citet{Thyagarajan2015}).\nFurther, for simplicity, we will assume a perfectly co-planar array. \nThe effect of relaxing these assumptions is expected to modulate power within the wedge, and potentially extend its reach to some degree.\nNevertheless, our focus is on examining the fundamental reason for the wedge, and whether it may be suppressed via appropriate $uv$-sampling -- thus focusing on a simple subset is appropriate. \nWe attempt to exhaustively list the various global assumptions and simplifications we have made in Table \\ref{tab:assumptions}.\nWe will outline further model simplifications and choices as we develop the equations within this section.\n\n\\begin{table*}[!ht]\n\t\\begin{center}\n\t\t\\begin{tabular}{ l }\n\t\t\t\\hline\n\t\t\t\\textbf{Assumptions used in framework} \\\\ \n\t\t\t\\hline\n\t\t\tRestriction to extra-galactic point sources \\\\\n\t\t\tZenith-pointing only \\\\\n\t\t\tCo-planar antenna array \\\\\n\t\t\tFlat-sky approximation \\\\\n\t\t\tNaturally-weighted baselines \\\\\n\t\t\tThermal noise values of all baselines drawn from i.i.d Normal distribution, centred on zero \\\\\n\t\t\tFlat-spectrum sources \\\\\n\t\t\t\\hline\n\t\t\t\\hline\t\n\t\t\\end{tabular}\n\t\\end{center}\n\\caption{Summary of universal assumptions and simplifications used in this paper. \\label{tab:assumptions}}\n\\end{table*}\n\n\\subsection{Single-baseline visibility}\nWe begin with the simple visibility equation for a co-planar array, which defines the signal received by any baseline, in the presence of point sources and thermal noise:\n\\begin{equation}\n\\label{eq:simple_vis}\nV_{i}(\\nu, \\vect{u}_i) = \\phi_\\nu \\left[\\mathcal{N}_{i,\\nu} + \\int d\\vect{l} dS  \\   n(\\vect{l}, S) SB(\\nu, \\vect{l}) e^{-2\\pi i \\vect{u}_i\\cdot \\vect{l}}\\right],\n\\end{equation}\ni.e. the Fourier-transform over the sky of the emission brightness attenuated by the beam $B$ and a frequency taper $\\phi_\\nu$.\n\nThe vector $\\vect{u}_i$ is the baseline length in units of the observational wavelength:\n\\begin{equation}\n\\vect{u}_i = \\vect{b}_i\/\\lambda,\n\\end{equation}\nwherein lies the chromaticity of the $uv$-sampling.\nHenceforth,\nwe let $\\nu_0$ be a reference frequency (this will later be tied to the mid-point of the frequency band of observation without loss of generality), and define $f = \\nu\/\\nu_0$.\nWe also explicitly let $\\vect{u}_i$ denote the value of $\\vect{u}_i$ \\textit{at the reference frequency}.\nThis is illustrated in Fig.~\\ref{fig:baseline_schematic}, which also shows why the delay approximation -- identifying the vertical shaded regions with the diagonal baseline tracks -- is reasonable for small $u$. \n\nIn this paper, we will exclusively use a Gaussian-shaped beam, and predominantly it will be achromatic\\footnote{For demonstration purposes, \\S\\ref{sec:classic} will also use a chromatic Gaussian beam, for which we have\n\t\\begin{equation}\n\t\\sigma_\\nu = 0.42c\/\\nu D = \\sigma_0\/f. \n\t\\end{equation}\n}. \nThe Gaussian beam is thus\n\\begin{equation}\n\tB_\\nu(l) = e^{-l^2\/2\\sigma^2},\n\\end{equation}\nwhere $\\sigma = 0.42c\/\\nu_0 D$ is the beam width \\citep{Trott2016}.\nThe motivation for using a Gaussian beam is that it is the most realistic analytically-tractable form possible, and its use for conceptual studies has precedent \\citep{Liu2014}.\nWe do note that the choice of a smooth Gaussian, which does not have sidelobes, has desirable effects on the form of the wedge, in that it perfectly suppresses horizon sources. \nThis effectively combats the complexities of baseline foreshortening at the horizon due to our appropriated flat-sky approximation. \n\n\nThe emission brightness is written in Eq.~\\ref{eq:simple_vis} as the sum of the flux density of all point-sources in the sky, where the number counts of these sources are given by $n(\\vect{l},S)$. These differential number counts are in general a statistical quantity, as we will typically consider sources below the confusion limit of an instrument. \nTo simplify the calculations to follow, we have followed common pedagogical practice and assumed that the the spectral shape of each source is entirely flat. \nThis simplification is rather heavy, but it should not affect the \\textit{conceptual} understanding of the following calculations.\n\nFurthermore, the observed frequency window is both physically attenuated by the instrument, and will also be tapered within the analysis in order to suppress frequency side-lobes\\footnote{Note that each sub-band also has its own structure. This may be assumed to be a part of $\\phi$, or may be introduced as a secondary convolution \\citep[cf. $\\gamma$ in][]{Liu2014}. We shall ignore it in this work so as not to complicate the key ideas.}. \nIn this paper, we assume that the bandpass is relatively broad compared to the taper, and that its effect can be safely ignored\\footnote{We note that this is a particularly strong simplification. \n\tThe bandpass will in general introduce smaller-scale oscillations into $\\phi$, which tend to broaden the footprint of the foreground power in $\\omega$. This is in some way countered by our use of a broad $\\omega$-space Gaussian taper.}.\nWe normalize the frequency-taper to $\\phi(0) = 1$ \n\\footnote{This assumption, which we employ for simplicity throughout this paper, ties the central frequency of the bandpass to the reference frequency, $\\nu_0$. \nThis is without loss of generality, as we may always shift all frequency-dependent parameters to a new ``reference'' before any analysis.}, and exclusively use a Gaussian taper here for tractibility:\n\\begin{equation}\n\t\\phi(f-1) = e^{-\\tau^2 (f-1)^2},\n\\end{equation}\nwith $\\tau$ an inverse-width (or precision).\n\nMore common choices for the taper are the Blackman-Harris \\citep{Trott2016a} or its self-convolution \\citep{Thyagarajan2016}.\nThese serve to reduce leakage of power into higher modes, and are better choices than a Gaussian in practice. \nWe utilise the Gaussian for analytic simplicity and note that most qualitative conclusions of the paper are insensitive to this choice (note also that there is precedent for choosing such a taper for theoretical studies, in \\citet{Liu2014}).\nThe primary point of difference is in the definition of the ``brick'' (cf. Table \\ref{tab:simple_summary}), which extends to a higher value of $\\omega$ when using a Gaussian.\nThis also affects the position of the emergence of the wedge.\n\n\nTaking the Fourier transform (over $\\nu$), we arrive at\n\\begin{align}\n\t\\label{eq:delay_vis}\n\t\\tilde{V}_i(\\eta, \\vect{u}_i) = \\nu_0 & \\int df\\ \\ e^{-2\\pi i \\nu_0f \\eta} V_i(\\nu, \\vect{u}_i).\n\\end{align}\nNote that we will make the change of variables $\\omega = \\nu_0 \\eta$ for the remainder of this work, where $\\omega$ is dimensionless and makes for simpler theoretical equations.\nThe square of this particular quantity is called the \\textit{delay spectrum}, and we explore it briefly in \\S\\ref{sec:classic}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{figures\/baseline_schematic}\n\t\\caption{Schematic showing the migration of baseline length (in wavelength units) as a function of frequency. The vertical shaded regions indicate the axis of the frequency Fourier Transform, showing that multiple baselines contribute at different frequencies for high $u$. Orange dots show where each baseline is equivalent to the scale at which it would be equated in the delay approximation. At low $u$, a FT along a baseline is almost equivalent to the true FT, showing the delay approximation to be accurate.}\n\t\\label{fig:baseline_schematic}\n\\end{figure}\n\n\\subsection{Multi-baseline visibility}\nTo increase signal-to-noise, we typically \\textit{grid} the discrete baselines in some fashion.\nThe delay spectrum approach only grids measurements after squaring the visibilities, whereas image-based approaches grid the complex visibilities before squaring to form the power.\nWe follow the latter approach in this work, as it is required in order to utilise the benefits of the $uv$-sampling.\nHowever, the approaches can be forced to align with each other under special array layout conditions, and we explore this briefly in \\S\\ref{sec:classic}.\n\nIn essence, the gridding assigns a weight to each baseline, specifying its contribution to a given UV point (i.e. closer points receive higher weights) \n\\footnote{We are also free to choose a baseline weighting which is a function only of the magnitude of $u_i$. Nevertheless, previous work has shown that an optimal choice for the baseline weighting is to have each baseline weighted equally \\citep{Bowman2009,Parsons2012}, and we follow suit here.}. \nThe total estimated visibility at point $\\vect{u}$ is thus the sum of all weighted visibilities, normalised by the total weight.\nLetting $w_\\nu$ denote the weighting function, and defining the total weight as\n\\begin{equation}\n\\label{eq:wnuu}\nW_\\nu(\\vect{u}) = \\sum_{i=1}^{N_{\\rm bl}}w_\\nu(\\vect{u} - f \\vect{u}_i),\n\\end{equation}\nwe have\n\\begin{equation}\n\\label{eq:vis_gridded}\nV(\\nu, \\vect{u}) =  \\frac{1}{W_\\nu(\\vect{u})}\\sum_{i=1}^{N_{\\rm bl}} w_\\nu(\\vect{u} - f\\vect{u}_i) V_i(\\nu,\\vect{u}_i).\n\\end{equation}\nThe Fourier-space visibility is then\n\\begin{align}\n\t\\label{eq:vis_full}\n\t\\tilde{V}(\\omega, \\vect{u}) = \\nu_0 & \\int df\\ \\ e^{-2\\pi i f\\omega} V(\\nu, \\vect{u}),\n\\end{align}\nwhich may be squared to form the grid-based PS.\n\nThe combination of Eqs. \\ref{eq:simple_vis}, \\ref{eq:vis_gridded} and \\ref{eq:vis_full}  provide the basis for all following work.\n\nIt has been shown in T16 that an unbiased gridding of visibilities is determined by inverting a matrix involving the fourier-transformed primary beam\\footnote{In their work (cf. their Eq. 17) it also involves a matrix $\\vect{G}$ which accounts for sky-curvature and other effects which we ignore here.}.\nIndeed, they find that a good approximation to this matrix inversion, which enhances computability considerably, is to use the diagonalized inversion, which corresponds precisely to a weighted average of baselines, with a weighting function given by the fourier-transform of the beam, $\\tilde{B}(\\vect{u})$. \n\nExplicitly, for the Gaussian beam employed in this work, we have\n\\begin{equation}\nw_\\nu(u) =  B(u) = e^{-2\\pi^2 \\sigma^2 u^2}.\n\\end{equation}\n\n\n\\subsection{Statistical properties of the visibility}\nThe visibility is in general a statistical variable, both because of the random thermal noise and the (typically) statistical nature of $n(\\vect{l},S)$.\n$\\tilde{V}$ is in general not Gaussian, nevertheless as we will be dealing with the power spectrum -- a quadratic quantity -- we shall only be required to know up to second-order properties of $\\tilde{V}$ for the purposes of this paper.\nAs long as the thermal noise is independent of the foreground signal, these are simply derived.\n\nOur primary sky model consists of a uniform Poisson process of point sources (uniform in $\\vect{l}$, cf. T16)\\footnote{This is a reasonable approximation for a relatively narrow beam (with no side-lobes) at zenith, where $\\theta \\sim \\vect{l}$. Our adoption of the Gaussian beam means this assumption will have little consequence. In reality, the curved nature of the sky introduces excess brightness towards the horizon, which can result in the ``pitchfork'' structure reported in \\citet{Presley2015,Thyagarajan2015,Thyagarajan2016} (see also \\citet{Kohn2016,Kohn2018})}.\nIn this model, expectation of the frequency-space visibility is obtained simply by replacing the sky emission with the mean flux density,\n\\begin{align}\n\t\\label{eq:su_meanvis}\n\t\\langle V_{i}\\rangle(\\nu, \\vect{u}_i) &= \\phi_\\nu \\int d\\vect{l}\\   \\bar{S} B_\\nu(\\vect{l}) e^{-2\\pi i f \\vect{u}_i \\cdot \\vect{l}}. \\nonumber \\\\\n\t&= \\phi_\\nu \\bar{S} B(f\\vect{u}_i),\n\\end{align}\nand the expected Fourier-space visibility is\n\\begin{align}\n\t\\langle V (\\omega, \\vect{u})\\rangle = \\bar{S} \\nu_0 \\int df \\frac{e^{-2\\pi i f \\omega}\\phi_{\\nu}}{W_\\nu(\\vect{u})} \\sum_{i=1}^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)  B(fu_i). \n\\end{align}\n\nThe variance of the visibility is composed of two parts -- a thermal variance and sky variance -- which are assumed to be independent. \nThe sky variance may be worked out in similar fashion to the procedure outlined in \\cite{Murray2017}.\nThe total covariance (i.e. with thermal noise term) is then\n\\begin{align}\n\\label{eq:su_covvis}\n\\vect{C}_{\\rm bl} &=  \\phi_\\nu^2 \\sigma_N^2 \\delta_{ij} \\delta(\\nu-\\nu')  \\nonumber \\\\\n& + \\phi_\\nu^2 \\mu_2  \\int d\\vect{l}_1 B_{\\nu}B_{\\nu'} e^{2\\pi i \\vect{l} (f' \\vect{u}_j  - f \\vect{u}_i)},\n\\end{align}\nbetween baselines $i$ and $j$, and frequencies $\\nu, \\nu'$, where we have used the assumed flat-spectrum of all sources, and $\\mu_2$ is the second moment of the source count distribution:\n\\begin{equation}\n\\mu_n = \\int dS\\ S^n \\frac{dN}{dS}.\n\\end{equation}\nThus the variance of the Fourier-space gridded visibility is\n\\begin{align}\n\t\\label{eq:general_var}\n\t{\\rm Var}(\\tilde{V}) &=& \\nu_0^2 &\\int df df' \\frac{e^{-2\\pi i \\omega (f-f')}}{W(\\vect{u}) W'(\\vect{u})} \\nonumber \\\\\n\t&& & \\times \\sum_{ij} ^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)B(\\vect{u} - f'\\vect{u}_j) \\vect{C}_{\\rm bl} \\nonumber \\\\\n\t&=& \\nu_0^2 \\sigma_N^2 &\\int df\\frac{ \\phi_\\nu^2 }{W^2_\\nu(\\vect{u})}  \\sum_i^{N_{\\rm bl}} B^2(\\vect{u}-f\\vect{u}_i) \\nonumber \\\\\n\t& &\\ +  \\nu_0^2 &\\int d\\vect{l} df df' \\frac{e^{-2\\pi i \\omega (f-f')}}{W_\\nu(\\vect{u}) W'_\\nu(\\vect{u})} \\nonumber \\\\\n\t& & & \\times \\sum_{ij} ^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)B(\\vect{u} - f'\\vect{u}_j) \\vect{C}_{\\rm sky}.\n\\end{align}\n\nThe first term of this equation defines the thermal noise variance of a grid-point $(\\omega, \\vect{u})$. \nWe omit the term for all following calculations\\footnote{This can be thought of as going to the limit of a perfectly calibrated instrument, or infinite integration time, and thus setting $\\sigma_N \\rightarrow 0$.}, but we define\n\\begin{equation}\n\t\\label{eq:w_theta}\n\tW^2(\\vect{u}) = \\int df \\frac{ \\phi_\\nu^2 }{W^2_\\nu(\\vect{u})}  \\sum_i^{N_{\\rm bl}} B^2(\\vect{u}-f\\vect{u}_i)\n\\end{equation}\nwhich will be used to determine a grid-point's relative weight when averaged to form the 2D PS.\n\n\\subsection{Expected power spectrum}\nArmed with the statistical descriptors of the visibility, we can determine the expected 3D PS:\n\\begin{equation}\n\t\\label{eq:expectation_equation}\n\t\\langle P(\\omega, \\vect{u})\\rangle  \\equiv \\langle \\tilde{V}\\tilde{V}^*\\rangle = {\\rm Var}(\\tilde{V}) + |\\langle \\tilde{V} \\rangle|^2.\n\\end{equation}\nNote that for the statistically uniform sky that we primarily adopt (where uniformity is in $\\vect{l}$), the second term is effectively the transfer function of the instrument (i.e. the beam and bandpass), and can typically be neglected on small perpendicular scales when these characteristics are angularly smooth.\n\nThe 2D PS is given by\n\\begin{equation}\n\t\\label{eq:power_general}\n\t\\langle P(\\omega, u) \\rangle = \\frac{\\int_0^{2\\pi} d\\theta \\ \\langle P(\\omega, \\vect{u})\\rangle W^2(\\vect{u})}{\\int_0^{2\\pi} d\\theta \\ W^2(\\vect{u})} , \n\\end{equation}\nwhere $\\theta$ is the polar angle of $\\vect{u}$.\n\nWe provide a synopsis of the meaning of symbols used in this paper in Table \\ref{tab:models}.\n\n\\begin{table*}[!ht]\n\t\n\t\\begin{center}\n\t\t\\begin{tabular}{ l l l }\n\t\t\t\\hline\n\t\t\t\\textbf{Symbol} & \\textbf{Description} & \\textbf{Models\/Values}  \\\\ \n\t\t\t\\hline\n\t\t\t$\\nu$ & Frequency & \\\\\n\t\t\t$f$ & Normalised Frequency, $\\nu\/\\nu_0$ & \\\\\n\t\t\t$\\vect{l}$ & Cosine-angle of sky co-ordinate, $\\cos\\theta$ & \\\\\n\t\t\t$\\vect{b}$ & Baseline length & \\\\\n\t\t\t$\\vect{u}$ & Fourier-dual of $\\vect{l}$, equivalent to $\\vect{b}\/\\lambda$ & \\\\ \n\t\t\t$\\eta$, $\\omega$ & Fourier-dual (and scaled by $\\nu_0$) of $\\nu$ & \\\\\n\t\t\t$k_\\perp$, $k_{||}$ & Cosmologically-scaled $u$ and $\\eta$ respectively & \\\\\n\t\t\t$V$ & Interferometric Visibility as function of frequency& \\\\\n\t\t\t$\\tilde{V}$ & Frequency FT of $V$ & \\\\\n\t\t\t$S$ & Flux density (subscripted for a particular source) & \\\\\n\t\t\t$I(\\nu,\\vect{l})$ & Sky Intensity &  \\\\\n\t\t\t\n\t\t\t\\hline\n\t\t\t$\\mu_1 \\equiv \\bar{S}$ & Mean brightness of sky  & 1 Jy\/sr \\\\\n\t\t\t$\\mu_2$ & Second moment of source-count distribution & 1 Jy$^2$ \/sr \\\\\n\t\t\t$S_0$ & Flux density of single source in sky & 1 Jy \\\\\n\t\t\t$\\vect{l}_0$ & Position of single source in sky & (1,0) \\\\\n\t\t\t$\\nu_0$ & Reference frequency & 150 MHz \\\\\n\t\t\t$\\sigma$ & Beam-width at $\\nu_0$ & 0.2 rad \\\\\n\t\t\t$\\tau$ & Unitless band-pass precision, $1\/2\\sigma_f^2$ & 100 \\\\\n\t\t\t$D$ & Tile Diameter & 4m \\\\\n\t\t\t$\\phi$ & Frequency Taper & Gaussian \\\\\n\t\t\t$\\phi_B$ & Bandpass & Uniform \\\\\n\t\t\t$\\psi$ & Source spectral shape & Flat \\\\\n\t\t\t\\hline\n\t\t\t$B_\\nu$ & Beam Attenuation & Gaussian (Static; Chromatic)  \\\\  \n\t\t\t$w$ & Visibility-gridding weights & Fourier-beam kernel \\\\\n\t\t\t$n(\\vect{l},S)$ & Source count distribution & Single Source; Stochastic Uniform \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\t\\caption{Summary of symbols and models used throughout this paper. Where possible, parameters list their default value used in plots throughout this paper. Any models list all models explored throughout this paper.  Any Latin-subscripted perpendicular scale (eg. $\\vect{u}_i$) refers to a particular physical baseline at reference frequency. \\label{tab:models}}\n\t\\end{center}\n\\end{table*}\n\n\n\n\n\n\n\n\n\n\\section{Uncorrelated Visibilities}\n\\label{sec:classic}\nThe wedge has been shown to naturally arise in the expected PS when correlations between baselines are either ignored or absent \\citep[eg.][]{Parsons2012,Trott2016}.\nFor example the delay spectrum, which by definition cannot correlate visibility pairs, can be used to provide a simple illustration of the emergence of the wedge. \nTo provide a backdrop for discussion of the effects correlating close baselines, we first turn to this uncorrelated case (within the context of our framework) to illustrate the emergence of the wedge.\n\n\n\nIn order to obtain simple single-baseline measurements of the PS within our framework, we use three conditions: i) baselines sparsely arranged on a set of logarithmic spokes, ii) an artificially narrow gridding kernel, and iii) for simplicity, we evaluate the PS only at reference baseline positions (i.e. $\\vect{u} = \\vect{u}_i$).\n\nAs we shall see, condition (iii) is not really required, but does simplify the procedure slightly.\nCondition (ii) can be more precisely stated as setting the gridding kernel width to approach zero.\nThis is artificial, because we will not enforce the beam width to follow the same limit.\nAlternatively, one may imagine condition (ii) as employing a nearest-baseline weighting method, such that the closest baseline to a point $\\vect{u}$ at a given frequency will be the sole contributor.\nWe shall see that even for standard gridding kernels, this condition will be met for large $u$ if condition (i) is met.\n\nCondition (i) is illustrated in Fig.~\\ref{fig:delay_transform_schematic}.\nThe baselines are arranged in a regular (logarithmic) polar grid, or equivalently, a series of ``spokes\" along which baselines are strung in a logarithmically regular fashion. \nImportantly for this section, the base of the logarithm is large enough such that a single baseline remains the sole contributor to a point co-located with its reference co-ordinate for the entire bandwidth (illustrated by the inset orange bell-curve in Fig.~\\ref{fig:delay_transform_schematic}).\nIn addition, every spoke is equivalent, such that the baselines define a set of concentric rings. \nIn summary then, the layout consists of $N_\\theta \\times N_r$ baselines, with regularly-spaced angular coordinates $\\theta_k = 2\\pi k\/N_\\theta$ and log-spaced radial co-ordinates $u_j = u_{j-1} + \\Delta_u (u_{j-1})$, with $\\Delta_u(u) = u\\Delta$ (with constant $\\Delta$) and arbitrary $u_0$.\nWe shall re-use variants of this simple layout throughout this paper.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{figures\/delay_transform_schematic}\n\t\\caption{Schematic of baseline layout with overlaid averaging\/histogram grid. Each cell of the averaging grid contains a single baseline which defines the \"position\", $\\vect{u}_\\mu$ of the cell at $f=1$. Inset is a figure transforming to frequency space, in which the bandpass\/taper is shown, along with the frequencies at which a particular baseline is at the cell edges and centre. The edges correspond to 3 Gaussian widths of the bandpass.}\n\t\\label{fig:delay_transform_schematic}\n\\end{figure}\n\nThese conditions allow for simple analytic solutions of the expected PS for a range of combinations of sky models and beam shapes.\nWhile emergence of the wedge can in principal be illustrated without specialising to any particular model, we find it illustrative to do so.\nFurthermore, to demonstrate that the wedge arises irrespective of sky model or beam shape, we use two models for each and evaluate the expected PS for three combinations: single-source with static beam, single-source with chromatic beam, and uniform sky with static beam.\nThus in addition to our fiducial choice of static Gaussian beam and uniform point-source sky, we also use a chromatic Gaussian beam (cf. \\S\\ref{sec:framework}) and a single-source sky model. \nThe latter model places a single source of flux density $S_0$ at position $\\vect{l}_0$.\n\nThe solutions are derived in App.~\\ref{app:delay}, and shown in tabulated form in Table \\ref{tab:simple_summary}. \nFor completeness we show the assumed values of various parameters in Table \\ref{tab:models}, and these correspond to the plots in Fig. \\ref{fig:ss_ngp_static}.\n\n\n\\subsection{Discussion of Uncorrelated Examples}\nThe basic form of the three derived solutions is very similar, as can be most clearly seen in the final two columns of Table \\ref{tab:simple_summary}, which provide a schematic view of where the power cuts off in $\\omega$. \nIn each case the standard form of the wedge, in which the power cut-off traces $\\omega \\propto u$, is recovered.\nThe constant of proportionality here (for this choice of coordinates) is given either by $l_0$ in the case of a single source (two leftmost plots, first with $l=1$ and second with $l=0.5$), or the beam width, $\\sigma$ (rightmost plot), in the case of a stochastic uniform sky.\nIn either case, the maximum possible value for these is unity (in fact, for the beam width it must be less than this or else sky-curvature terms become important).\nThus we can draw the standard ``horizon line'' at $\\omega = u$. \n\nWe also find that the lower-left portion of the $\\omega u$-plane forms a ``brick'' whose width is determined primarily --- in our case --- by the taper.\nClearly, the ``brick'' is in general determined by the overall frequency envelope of the instrument and analysis -- i.e. the combination of taper, bandpass, chromaticity of the beam, and spectral structure of the sources. \nIndeed, the effects of the chromaticity of the beam are apparent in the case of a single-source sky; the width of the brick is determined by $p^2 = \\tau^2 + l_0^2\/2\\sigma^2$, which has a dependence on the beam-width. \nIn practice, the taper\/bandpass dominate the spectral response, and $p^2 \\approx \\tau^2$, nevertheless this illustrates that even with a theoretically infinite uniform bandpass, other spectral characteristics of the instrument will limit the ability to sequester power into the lowest $\\omega$ modes. \nIn general, the balancing of the various spectral terms provides the motivation for design criteria on the spectral smoothness of the instrument. \n\nAs has been previously noted, the wedge occupies a greater portion of the $\\omega u$-plane for sources close to the horizon (precisely because their delay transform for the same baseline is larger). \nThus, a beam which is tighter (and doesn't have high-amplitude sidelobes) can effectively attenuate these sources and ameliorate the wedge \\citep{Parsons2012}.\nOf course, such a beam will also attenuate the 21 cm signal, and is therefore not an ideal solution for the problem. \n\nAs has been extensively noted in the literature, many factors affect the precise form of the power in the wedge \\citep[eg.][]{Thyagarajan2013,Thyagarajan2015,Thyagarajan2016,Gehlot2018} --- and most of these we have ignored in this analysis.\nWhile the broad structure remains the same -- a brick with width given by the spectral envelope of the instrument, and linear wedge extending to $\\omega \\approx u$ -- the power within this region may be shifted around or amplified by various factors, and even leaked beyond the horizon line when small-scale spectral features are present in the analysis. \nAn example of this changing of form can be witnessed in Fig.~\\ref{fig:ss_ngp_static} between the single-source and stochastic skies. \nThe smooth attenuation of sources at larger angles causes a smoother cut-off in the wedge.\n\nNevertheless, none of these features can lay claim to being the fundamental reason for the wedge. \nAltering them merely alters the shape or amplitude of the wedge, and not its basic form or existence.\nThe fundamental reason for the wedge is rather the combination of the migration of the baselines with frequency, and the sparsity of the $uv$-sampling. \nWe turn to examining this latter condition for the remainder of this paper.\n\n\n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\begin{tabular}{ l l l l l }\n\t\t\t\\hline\n\t\t\t\\textbf{Sky Dist.} & \\textbf{Beam} & \\textbf{Form of $P(\\omega, u_\\mu)$} & \\textbf{Low $u$} &\\textbf{High $u$}  \\\\ \n\t\t\t\\hline\n\t\t\tSingle-Source & Static & \\( \\displaystyle \\frac{1}{N_\\theta}  \\frac{S_0^2 \\nu_0^2 \\pi}{\\tau^2} \\exp\\left(-\\frac{l_0^2}{\\sigma^2}\\right) \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2(\\omega + u_\\mu l_0 \\cos (2\\pi k\/N_\\theta))^2}{\\tau^2} \\right) \\) &$\\tau\/\\pi\\sqrt{2}$ & $ul_0$ \\\\\n\t\t\tSingle-Source & Chromatic & \\(\\displaystyle \\frac{S_0^2 \\nu_0^2 \\pi}{N_\\theta p^2} e^{-\\tau^2l_0^2\/2\\sigma^2p^2} \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2 (\\omega + u l_0 \\cos (2\\pi k\/N_\\theta))^2}{p^2}\\right) \\) &$p\/\\pi\\sqrt{2}$ & $ul_0$ \\\\\n\t\t\tStochastic Uniform & Static & \\(\\displaystyle \\frac{\\nu_0^2 \\pi}{p_u}\\exp\\left(-\\frac{2\\pi^2\\omega^2}{p_u^2}\\right) \\left[ \\frac{\\bar{S}^2}{p_u} \\exp\\left(-\\frac{2\\tau^2\\pi^2\\sigma^2u^2}{p_u^2}\\right) + \\frac{\\mu_2 \\pi^2\\sigma^2}{\\tau}\\right]\\) & $\\tau\/\\pi\\sqrt{2}$ &  $u\\sigma$ \\\\\n\t\t\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{\\label{tab:simple_summary} Summary of analytic solutions for the simple discrete polar grid layout with histogram gridding of \\S\\ref{sec:classic}. Final two columns display a schematic representation of where the foreground power cuts off in $\\omega$. For simplicity we list the cutoff such that the power is reduced by a factor of $e$ from the total. For a $\\chi$-order of magnitude suppression, multiply the result by $\\chi \\ln 10$. In the table, $p_u^2 = \\tau^2 + 2\\pi^2\\sigma^2u^2$ and $p^2 = \\tau^2 + l_0^2\/2\\sigma^2$.}\n\t\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth,trim=2cm 0cm 2cm 0cm]{figures\/averaged_gridding_examples}\n\t\\caption{2D PS examples using averaged gridding, and different combinations of beam and sky models. Each displays very similar behaviour, however the wedge is\n\t\tsharper in the case of a single source. In each case, a horizontal ``brick'' line is drawn at the theoretical $10^{th}$ magnitude of suppression (cf. Table \\ref{tab:simple_summary}), and the diagonal line is the ``horizon line'': $\\omega = u$. For each, $\\tau$ remains the same, while the two single-source models have sources at different zenith angles, the first at the horizon and the second at $l=0.5$. The clear difference in amplitude arises due to the difference in number of position of sources in the beam. Changing the position of a single source changes height and slope of the wedge line. A stochastic uniform sky has a softer edge for the wedge.}\n\t\\label{fig:ss_ngp_static}\n\\end{figure*}\n\n\n\\section{Dense Logarithmic Polar Grid}\n\\label{sec:weighted}\n\nMuch can be learned about the effects of including visibility correlations by re-using the logarithmic polar grid baseline layout of \\S\\ref{sec:classic}, and we address this class of problems in this section.\nHere we will dispense with condition (ii) --- that the gridding kernel is arbitrarily narrow --- and use a self-consistent kernel width.\nMore importantly, we will dispense with the condition that the layout be ``sparse'', \nallowing an arbitrary radial density of baselines (i.e. arbitrarily low values of $\\Delta$)\\footnote{The adjustment of angular density trivially has no impact, as each ring measures the same expected PS everywhere}. \nIt is precisely as we modify this density that we will identify the effects of $uv$-sampling.\nWe give a schematic of this layout in Fig. \\ref{fig:weighted_gridding_schematic}, noting the extent of each baseline via the Fourier beam kernel, and also the interplay of this scale with the bandpass shape.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/weighted_gridding_schematic}\n\t\\caption{Schematic of weighted gridding, with logarithmic polar grid of baselines. Extent of Fourier-beam kernels are indicated by blue shaded regions around each baseline. Also shown are representative bandpasses, indicating the weight of a baseline centred on the bandpass at $f=1$ as it moves with frequency. The four bandpasses indicate four regimes which may be considered. Note that bandpass indications on the left hand side adopt a lower value of $\\tau$. The bandpass (as represented in $u$-space) naturally expands at higher $u$.}\n\t\\label{fig:weighted_gridding_schematic}\n\\end{figure}\n\nWe note that the limiting case of this derivation -- namely an arbitrarily densely packed set of baselines (i.e. a smooth continuum of baselines) in the radial direction -- is addressed in App. \\ref{app:radial}. It is in line with the predictions of the \\textit{discrete} polar grid results of this section.\n\nThroughout this section we will use a static beam, and consider the stochastic uniform sky (we have already established that such choices do not greatly impact the qualitative form of the solution). \nFurthermore, due to the fact that the term $|\\langle V \\rangle|^2$ has negligible power on small scales, we consider only the variance term of Eq.~\\ref{eq:power_general}.\nDue to the isotropy of the sky and the fact that we use an angularly symmetric layout, we immediately have that $P(\\omega, u) = {\\rm Var}(\\tilde{V}(\\omega, u))$ without requiring an integral over $\\theta$.\nThus for this section we require only the second term of Eq.~\\ref{eq:general_var}, which can alternatively be written:\n\\begin{subequations}\n\t\\label{eq:wg_master}\n\t\\begin{align}\n\t\t{\\rm Var}(\\tilde{V}) &= \\mu_2\\nu_0^2 \\int d^2\\vect{l}\\  e^{-l^2\/\\sigma^2} |I|^2, \\\\\n\t\tI &= \\int df\\ \\frac{\\phi_\\nu}{W_\\nu(\\vect{u})}  \\sum_{i=1}^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)e^{-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)} \\nonumber \\\\\n\t\t&= \\int df\\ \\frac{\\phi_\\nu \\sum_{i=1}^{N_{\\rm bl}} e^{-2\\pi^2\\sigma^2(\\vect{u}-f\\vect{u}_i)^2} e^{-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)}}{\\sum_{i=1}^{N_{\\rm bl}} e^{-2\\pi^2\\sigma^2(\\vect{u}-f\\vect{u}_i)^2}} \n\t\\end{align}\n\\end{subequations}\nThe solution of $I$ here is in general intractable, due primarily to the sum over baselines in the denominator (i.e. $W_\\nu(\\vect{u})$). \n\n\nEq. \\ref{eq:wg_master} may be expanded as follows (with $q^2 = 2\\pi^2 \\sigma^2$):\n\\begin{align}\nI =  \\int df\\ \\frac{\\sum_{i=1}^{N_{\\rm bl}} e^{-f^2(\\tau^2 + q^2 u_i^2) -2f(\\tau^2 + q^2 \\vect{u}\\cdot\\vect{u}_i)-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)}}{e^{\\tau^2} \\sum_{j=1}^{N_{\\rm bl}} e^{-q^2(f^2 u_j^2 - 2f\\vect{u}\\cdot\\vect{u}_j)}} \n\\end{align}\nEvaluating this in general is here still intractable.\nNevertheless we can appreciate the general characteristics of the solution by considering the two limits of $u$. \n\nAt small $u$, the exponential cut-off of the beam kernel requires that only baselines with small $u_i$ have non-negligible impact in the sum.\nThus, for $u \\ll \\tau\/q$, for which only terms with $u_i \\ll \\tau\/q$ can contribute, all components with a dependency on $u_i$ in the numerator disappear. \nFurthermore, since the denominator is clearly a much broader function of frequency than the numerator (as $q^2\\vect{u}\\cdot\\vect{u}_i \\approx q^2u^2 \\ll \\tau^2$), it can be removed from the frequency integral. \nThus we arrive at\n\\begin{align}\nI_{u\\ll\\tau\/q} = \\frac{\\sum_{i=1}^{N_{\\rm bl}}e^{- q^2 u_i^2}}{\\sum_{j=1}^{N_{\\rm bl}} e^{-q^2(u_j^2 - 2\\vect{u}\\cdot\\vect{u}_j)}} \\int df\\ \\phi_\\nu e^{-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)}. \n\\end{align}\nThe frequency integral is identical to that for the sparse grid (cf. Eq. \\ref{eq:stoch_uniform}).\nIn fact, the solution is a complex sum over terms, all of which are very close to the exact solution of Eq.~\\ref{eq:stoch_uniform}, and therefore the behaviour will be almost identical -- i.e. to produce a ``brick'' feature at $u \\ll \\tau\/q$, with an $\\omega$ cut-off at $\\omega \\approx \\tau\/\\sqrt{2}\\pi$. \nRemember that this is irrespective of the density of baselines and the width of the gridding kernel, though its regime of applicability is determined by both the gridding kernel width and taper width.\n\nConversely, at sufficiently large $u$, the baselines are far enough apart from each other that the weighted sum of visibilities is dominated by the single closest baseline (except in rare cases where we consider scales with two or more equidistant baselines, but the rarity of these will make them negligible in the final angular average)\nFor a given logarithmic separation $\\Delta$, this criterion can be considered to be\n\\begin{equation}\n\tu^2 \\gg \\frac{1}{q^2\\Delta^2}.\n\\end{equation}\nNote that baselines being separated enough to consider just one in the baseline sum is not equivalent to them being radially separated enough to only consider the same baseline over all frequencies.\nIt is entirely possible that the baselines will move enough with frequency that they entirely replace one another, while only ever considering one at a time in the baseline sum (cf. Fig. \\ref{fig:weighted_gridding_schematic}). \nWe denote the baseline closest to $\\vect{u}$ at $f$ as $\\vect{u}_f$ (this is meant to be an identifier, so that the baseline's value of $\\vect{u}$ at $f$ is $f\\vect{u}_f$).\nIn this case, we can rewrite $I$:\n\\begin{equation}\n\t\\label{eq:Iequation}\n\tI_{u\\gg 1\/q^2\\Delta^2} = \\int df \\phi_\\nu e^{-2\\pi i f (\\omega + \\vect{l}\\cdot \\vect{u}_f)}.\n\\end{equation}\n\nWe have already encountered the case in which the baselines are separated enough such that only a single baseline contributes across \\textit{all frequencies} (cf. \\S\\ref{sec:classic}), and this gives the classical form for the wedge. \nThis merely shows that if $\\Delta$ is large enough, there will \\textit{always} be a regime of $u$ such that this classical solution holds for a logarithmic polar grid.\n\nAlternatively, we may consider the limit as $\\Delta \\rightarrow 0$ (for which the $u$ regime is at extremely large $u$). \nIn this case, the closest baseline to $\\vect{u}$ will always have $f\\vect{u}_f \\approx \\vect{u}$ (i.e. there will always be a baseline sitting on $\\vect{u}$).\nThen we have\n\\begin{equation}\nI_{u\\gg 1\/q^2\\Delta^2} = e^{-2\\pi i \\vect{l}\\cdot\\vect{u}} \\int df \\phi_\\nu e^{-2\\pi i f \\omega},\n\\end{equation}\nso that the final power spectrum is\n\\begin{equation}\n\tP(\\omega, u) = \\mu_2 \\nu_0^2 \\tilde{B}(u)\\tilde{\\phi}(\\omega).\n\\end{equation}\nThis separable equation clearly does not contain a wedge, rather containing only the ``brick'' determined by the taper, with an exponential cut-off in $u$.\nThough this was shown only for the fictional region $u \\rightarrow \\infty$ in this case, it is really an example of a continuous distribution of baselines along a radial trajectory, which is shown in detail to omit a wedge in App.~\\ref{app:radial}.\n\nOf course, in most cases, the (radial) density of baselines will lie between these extremes, and a natural question is how dense the baselines must be in order to yield a given level of wedge reduction.\n\\ifanalytic\n{\\color{red} A detailed solution to this question is presented in App.~\\ref{app:logsolution}}, but \n\\fi\nWe address this question semi-empirically following our conceptual interpretation of the next subsection. \n\n\\subsection{Conceptual Interpretation}\n\\label{sec:weighted:conceptual}\nTo gain an intuition for the results of the previous subsection, imagine a point $\\vect{u}$ for which we are evaluating the power, and consider only baselines that are along a spoke passing through $\\vect{u}$ (thus reducing the problem to one dimension). \nFigure~\\ref{fig:wedge_rising} gives a schematic representation of this, similar in form to Fig.~\\ref{fig:baseline_schematic}.\nHere we have chosen a very sparse baseline sampling, akin to the layout chosen for the averaged gridding in the previous section, and show only two points of evaluation (centre of the grey regions), which are concurrent with the baselines at $f=1$. \n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 1cm 0cm 0cm]{figures\/wedge_rising}\n\t\\caption{Schematic representation of the mechanics of the emergence of the wedge. Panels should be traced in order of their assigned number. (1.) shows the migration of baselines (black) in $u$ with frequency. Coloured overlaid lines show the ``effective baseline'' traced as an estimate along the points of evaluation, $u$ (grey regions), determined by its weighting kernel. (2.) shows the projection of the effective baseline (coloured) onto the visibility amplitude, accounting for the beam (black line). (3.) shows these results projected (coloured) onto the frequency axis, where they are multiplied by the bandpass (black). The FT of this final curve gives the power spectrum as a function of $\\omega$ for the evaluation point $u$. Wider curves transform to narrower curves.}\n\t\\label{fig:wedge_rising}\n\\end{figure*}\n\nPerforming a FT following the trajectory of a baseline is the delay transform, and at low $u$ this is very close to performing the FT at constant $u$. \nDue to the sparsity of baselines, the effective baseline used in a constant-$u$ transform merely follows the closest baseline.\nThis ``effective baseline'' is illustrated as a coloured shading overlaying the baseline's trajectory. \nAn arrow from this coloured line to the constant-$u$ FT trajectory indicates that it is \\textit{this} value of $u$ that is used in the estimation of the Fourier-space visibility.\n\nThe top panel shows the immediate results of this effective baseline migration. \nThe black curve shows the visibility of the true sky as a function of $u$.\nWe note that as this is a simplified schematic, we show an effective amplitude of the visibility (which is inherently complex). \nThis visibility accounts for the beam attenuation of the instrument, resulting in an exponential curve.\nThe estimated visibility amplitude at any frequency is merely its value as traced vertically from the coloured curve in the lower left panel. \nThat is, in this case, the amplitude is merely traced from left to right as frequency increases, and is indicated by a corresponding coloured line. \nThe greater the value of $u$, the larger the arc-length of this line segment, and therefore the greater the ratio between its minimum and maximum. \n\nThe right-hand panel shows these effects on the frequency axis.\nIn black is the bandpass (or taper). \nTo obtain the frequency-space visibility, this is multiplied by the frequency-dependent sky response from the top panel, and shown as corresponding coloured curves. \nWhile the low-$u$ curve (in blue) is almost constant-amplitude, and therefore barely affects the frequency-space visibility, the high-$u$ curve dramatically suppresses the high-frequency amplitude, effectively causing the frequency-response to be tighter than the natural bandpass (we note that for schematic purposes, we have re-normalised and re-centred the coloured curves).\nThe frequency-space FT of these curves gives the power spectrum for a given $u$ as a function of $\\omega$.\nClearly, tighter curves will transform to wider curves, and hence the ``wedge'' will form when the tightening arising from the effective baseline migration dominates the bandpass (and thence will depend linearly on $u$).\nThis is much the same description as contained in works such as \\citet{Morales2012}, \\citet{Parsons2012} and \\citet{Trott2012}.\n\n\nLet us consider now a very dense array with logarithmically-spaced baselines.\nThis we illustrate in Fig. \\ref{fig:schematic_inf_bl}.\nThis figure is the same in form as Fig. \\ref{fig:wedge_rising}, but clearly has a much larger number of baselines which pass through the point of evaluation, $u$.\nDue to the logarithmic spacing of the baselines, they pass through $u$ at equal intervals of $f$. \nIn this case, the ``effective baseline'', shown as the blue curve in the lower-left panel, periodically swaps from one baseline to another. \nWe recall that this effective baseline is the weighted average position of all baselines, where the weighting kernel is the Fourier-space beam (in this case, a Gaussian).\nSince the baselines are so closely packed, the oscillations created are very small, and it is clear that an infinite number of baselines will yield a truly vertical effective baseline -- corresponding to the true constant-$u$ estimate. \nConsequently, the top-left panel shows that a very small range of visibility amplitudes is covered -- effectively constant over all frequencies.\nThis in turn renders its product with the bandpass to be solely determined by the latter, and therefore the wedge to be completely avoided. \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 1cm 0cm 0cm]{figures\/schematic_inf_bl}\n\t\\caption{The same as Fig. \\ref{fig:wedge_rising}, but for a single evaluation point at high-$u$, with a dense packing of logarithmically-spaced baselines. In this case, the ``effective baseline'' is nearly constant with frequency, and the resulting effect is to negligibly impact the frequency response shape.}\n\t\\label{fig:schematic_inf_bl}\n\\end{figure*}\n\nWhat of a  baseline density between the previous two figures?\nThis is shown in Fig. \\ref{fig:schematic_multi_bl}.\nHere the ``effective baseline'' oscillates between true baselines in a much more marked manner, creating a footprint in $u$ which is much wider than in the previous case.\nProjected onto the visibility amplitude, a much wider range is covered, and that range is covered periodically, with period given by the separation of baselines.\nThus it is no surprise to find that the frequency-space product of the response with the bandpass is oscillatory on small scales, with an overall shape given by the bandpass itself. \nThe FT of such a function can be approximated as the combination of a smooth Gaussian with width inverse to the width of the bandpass, and a high-frequency term given by the period of the oscillations. \nIn this case, in place of a pure wedge, one should obtain a ``bar'' in the 2D PS (along with its harmonics), where the position of the bar in $\\omega$-space is inversely proportional to the separation of the baselines, and its amplitude is proportional to this separation.\nThat is, denser baselines will yield a lower-amplitude bar at higher $\\omega$, eventually leading to a negligible bar, and the complete disappearance of the wedge.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 1cm 0cm 0cm]{figures\/schematic_multi_bl}\n\t\\caption{The same as Fig. \\ref{fig:wedge_rising}, but for a single evaluation point at high-$u$, with a semi-dense packing of logarithmically-spaced baselines. In this case, the ``effective baseline'' oscillates widely with frequency. Consequently the $u$-footprint is widened and the range of visibility amplitude is also widened. The effect is to overlay a regular oscillatory component atop the bandpass, which contributes a high-$\\omega$ ``bar'' (and its harmonics) in addition to the basic ``brick'' yielded by the bandpass.}\n\t\\label{fig:schematic_multi_bl}\n\\end{figure*}\n\n\nIn general, baselines will not be regularly logarithmically separated, and various baseline distributions will yield different versions of this oscillatory structure.\nFor instance, linearly-separated baselines should cause the bar's position to be linearly dependent on $u$.\n\nThe exact results of these intuitions are complicated when accounting for baselines in the 2D plane, some of which may significantly contribute to the weight at $\\vect{u}$ without being on its radial trajectory.\nFurthermore, radial distributions which are not regular will also complicate matters.\nWe will continue to explore these issues in the following subsections; nevertheless, the basic intuition will remain the same.\n\n\n\n\n\\subsection{Wedge Mitigation}\n\nWe now turn to considering the properties of wedge mitigation in our simple polar grid baseline layout.\nWe begin by considering the results of our simple schematic representations of the wedge, in Table \\ref{tab:simple_summary}, which suggest that the wedge only emerges from the ``brick'' at $u>\\tau\/q \\equiv \\check{u}$.\nThis defines a region of interest, in which we may hope to mitigate the wedge.\nNote that for $u > 1\/q\\Delta$, the problem can also essentially be considered as 1D, as constructed in \\S\\ref{sec:weighted:conceptual}, regardless of the angular density.\nWe shall assume this condition throughout this section, noting that deviations from the assumption will always be small.\n\nWithin this region, we ask how a wedge may be \\textit{ensured}.\nWe have already seen that if a single, constant baseline contributes to the sum over baselines, for all frequencies in the bandpass, then we arrive at a wedge.\nThus, we may ensure a wedge by considering the integrated contribution of the two baselines closest to $q$.\nIf the one baseline dominates, then we are assured of a wedge. \nIf the second baseline contributes non-negligibly then we cannot rule out a wedge, but open up the possibility of its mitigation.\nWe denote the threshold contribution as $e^{-t}$.\n\nIn effect, our constraints are defined by the inequality\n\\begin{equation}\n\\label{eq:general_wedge_regime}\n\\frac{\\int df \\exp\\left[-\\tau^2 (f-1)^2 - q^2 (u+\\Delta_u - fu)^2\\right]}{\\int df \\exp\\left[-\\tau^2 (f-1)^2 - q^2 (u+\\Delta_u)^2 (1 - f)^2\\right]} < e^{-t}.\n\\end{equation}\nHere we have assumed that the dominant baseline is co-located with $u$ at $f=1$. \nThe qualitative results are insensitive to this assumption.\n\nThe solution to Eq. \\ref{eq:general_wedge_regime} is \n\\begin{equation}\n\\label{eq:solution_wedge}\n\\frac{1}{2}\\ln\\left(\\frac{\\tau^2 + q^2(u+\\Delta_u)^2}{\\tau^2 + q^2 u^2}\\right) - \\frac{\\tau^2 q^2\\Delta_u^2}{\\tau^2+q^2u^2} < -t.\n\\end{equation}\nIf we consider only scales where a wedge is possible (i.e. $u>\\check{u}$), and maintain that at these scales, $\\Delta_u \\ll u$, then we may use the approximation $\\ln(1+\\delta) \\approx \\delta$ to solve for $\\Delta_u$:\n\\begin{equation}\n\t\\label{eq:solution_for_concentric}\n\t\\Delta_u \\gtrsim \\frac{u}{2\\tau^2}\\left(1+\\sqrt{1+4\\tau^2 t}\\right) \\approx \\frac{u\\sqrt{t}}{\\tau},\n\\end{equation} \nwhere the last approximation assumes $t \\gg 1\/\\tau^2$.\nThat is, a wedge is \\textit{ensured} if the baseline separation is larger than $u\\sqrt{t}\/\\tau$.\n\nThe salient features of this equation are \n\\begin{enumerate}\n\t\\item  $\\Delta_u$ rises proportionally to $u$, so a regular logarithmic spacing for $u > \\check{u}$ ensures consistency of wedge\/non-wedge for all $u$. \n\t\\item $\\Delta_u$ is inversely proportional to $\\tau$, so that larger bandwidths support larger separations before a wedge is ensured.\n\t\\item $\\Delta_u$ is proportional to the root of the threshold, $t$. This is difficult to assess conceptually, as we are \\textit{a priori} uncertain as to what level the primary baseline must contribute to ensure a wedge.\n\\end{enumerate}\n\nTo get a better sense of the kinds of separations required, we note that \n\\begin{equation}\n\\Delta_u = \\Delta_x\/\\lambda_0 \\approx \\frac{\\Delta_x}{2{\\rm m}},\n\\end{equation}\nwhere $\\Delta_x$ is the difference in baseline lengths in distance units (note that this is \\textit{not} distances between antennae, but differences between these distances).\nExpressing this physical separation in units of the tile diameter, $\\Delta_x = \\chi D$, we can express our results in terms of the parameter $\\chi$.\nWe note first that\nfor a (static) Gaussian beam at $\\nu_0\\approx150$ MHz, the tile diameter can be approximately related to the beam width by\n\\begin{equation}\n D \\approx 1{\\rm m}\/\\sigma,\n\\end{equation}\nThus, we let $\\Delta_u = \\chi D\/2{\\rm m} = \\chi\/2\\sigma$.\nIn this case, we have that\n\\begin{equation}\n\\chi \\gtrsim \\frac{2\\sigma u\\sqrt{t}}{\\tau}\n\\end{equation}\nensures a wedge.\nAs a minimum, at $u= \\check{u}$, we have $\\chi > \\sqrt{t}\/\\pi$. \n\nUnfortunately, it is difficult to exactly specify the value of $t$, as it merely represents an order-of-magnitude estimate of the contribution of secondary baselines.\nFurthermore, even if we could specify it, we do not have a good analytic handle on what happens for baseline separations smaller than that given by $\\chi$ -- we cannot simply assume that the wedge will disappear, though we do expect it to disappear at some small separation.\nThus we turn to a numerical\/empirical approach.\n\nIn Fig. \\ref{fig:dense_log_concentric} we show the numerically-calculated power spectra (see App.~\\ref{app:numerical} for details on the numerical algorithm) for our fiducial set of physical parameters, and a range of logarithmic separations. \nEach panel is titled by the value of $t$ and $\\chi$ which correspond to the baseline separations \\textit{at} $\\check{u}$ (which is marked by the vertical dashed line).\nThis clearly shows that a baseline separation of about half the tile diameter is required (taking the minimum, which is at $\\check{u}$) for the wedge to begin to disappear.\nThis occurs at a threshold value of $t \\sim 0.4$, corresponding to the second baseline contributing $\\sim 60\\%$ of the primary baseline over the range of frequencies.\nThese values are roughly instrument-independent .\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth, trim=1cm 1cm 1cm 0cm]{figures\/dense_log_concentric.pdf} \n\t\\caption{2D power spectra for fiducial parameters, with static beam and stochastic sky. The baseline layout is concentric, with logarithmically-increasing separations between each circle. The gridding is weighted according to the Fourier beam kernel. Each panel represents a different regular logarithmic spacing, $\\Delta$. Titles of each panel indicate the order-of-magnitude contribution of the next-closest baseline at $\\check{u}$ (labelled $t$), and the physical separation of baselines as a fraction of the tile diameter at $\\check{u}$. The dashed vertical line marks the scale $\\check{u}$. The colour-scale in each panel is identical, as are the schematic representations of the wedge\/brick shown as black lines (these come from the corresponding row of table \\ref{tab:simple_summary}).}\n\t\\label{fig:dense_log_concentric}\n\\end{figure*}\n\nInterestingly, we also see horizontal ``bars'' as we had predicted from our conceptual consideration of the problem (cf. \\S\\ref{sec:weighted:conceptual}).\nAs predicted, the fundamental bar moves up in $\\omega$ as the baselines become closer, and at some separation we expect the harmonics to effectively disappear.\n\nFinally, we ask whether such a layout is physically feasible.\nIn principle, a perfect polar grid of baselines is unachievable by laying out antennas --- there will always be baselines that are off the grid. \nHowever, the logarithmic polar grid can be achieved by using logarithmic spokes of antennas, and ignoring all baselines that are off the grid (with a great deal of inefficiency).\nIn this case, one cannot physically deploy a layout with $\\chi < 1$, as the antennas will necessarily overlap with themselves. \nWe have found that we require $\\chi \\approx 1\/2$ to mitigate the wedge at $\\check{q}$, rendering this completely infeasible. \nWhile in principle it is possible to design layouts which would enable more closely-spaced baselines, they would come at the cost of reduced layout efficiency, and will be practically infeasible.\nWe will soon (\\S\\ref{sec:mitigation:arrays}) explore how leaving the off-grid baselines in the baseline layout affects results.\n\n\n\\subsubsection{Linear Radial Grid}\n\\label{sec:weighted:linear}\nIt is interesting to consider the case in which radial baselines are regular in linear space. \nEq. \\ref{eq:solution_for_concentric} suggests that in this case, at some point $u>u'$ the separation will become small enough to mitigate the wedge.\nIn fact, letting $\\Delta_u \\equiv \\Delta$, we can explicitly solve for $u'$:\n\\begin{equation}\nu' = \\tau \\Delta\/\\sqrt{t} \\approx 1.5 \\tau \\Delta,\n\\end{equation}\nwhere the last approximation assumes $t=0.4$ defines the transition from wedge to no-wedge, as described above. \nIndeed, if $u' < \\check{u}$, we expect the wedge to be completely mitigated. \nThis is given by the same baseline difference as the logarithmic case, i.e. corresponding to $\\chi \\approx 1\/2$. \nFurthermore, we expect that the bars we saw in the logarithmic case will also be present in the linear case, except that they will not be horizontal, but rather diagonal, as they increase in frequency as $u$ increases.\n\nTo illustrate and check these arguments, we show the linear analogue of Fig. \\ref{fig:dense_log_concentric} in Fig. \\ref{fig:dense_linear_concentric}.\nThe diagonal bars are quite clear in this case. \nWe also see that $t\\approx 0.4$ again roughly delineates the disappearance of the wedge at $\\check{u}$. \nWe note that the vertical lines which appear in the upper panels are due to the fact that in this case, the actual nodes of evaluation, $u_i$, lie at various positions between the radial baselines, rather than being forced to match at $f=1$. \nThis creates oscillatory behavior in $u$, but disappears as the distance between baselines increases. \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth, trim=1cm 1cm 1cm 0cm]{figures\/dense_linear_concentric.pdf} \n\t\\caption{Exactly the same as Figure \\ref{fig:dense_log_concentric}, except that the baselines are spaced regularly in \\textit{linear} space.}\n\t\\label{fig:dense_linear_concentric}\n\\end{figure*}\n\nThis kind of layout is \\textit{less} physically feasible than the logarithmic polar grid, as it requires impractical baseline densities for all $u$.\n\n\n\\subsubsection{Effects of Angular Density}\n\nDue to isotropy, the results of this section are not sensitive to the number of ``spokes'' in the layout.\nNevertheless it is clear that a single spoke does not have the same \\textit{covariance} as a layout with a large number, and when performing inference on a given set of observed data, this covariance is crucial.\nAn alternative way to think of this is that while a single spoke will yield the same mean over a large random set of skies, it will not necessarily give a good account of a single sky, whose distribution is randomly deviated from perfect symmetry. \n\n\n\\subsubsection{Radial Irregularities}\n\\label{sec:weighted:irregular}\nThe precise radial alignment (and regularity) of baselines in the simple polar grid lead to it being impractical layout for wedge mitigation.\nIn this section, we consider a relaxation of the ideal assumptions of radial regularity in favour of random radial placement, which will come in two forms: (i) completely random and (ii) a random offset from logarithmic regularity.\n\nThe advantage of a random array is that it may be perfectly efficient in terms of mapping an antenna layout to the baseline layout --- we need not ignore any pairs within a spoke. \nThus we can achieve a much greater overall baseline density for the same cost.\nConversely, however, we shall see that the lack of radial alignment increases the overall required baseline density to achieve wedge mitigation.\n\nIn our ``completely random'' layout, we allow the baselines to be stochastically placed along radial trajectories, with the same overall density as a logarithmic placement. \nWe find that doing so yields a somewhat surprising result, which is illustrated in Fig. \\ref{fig:random_trajectories}.\nIn this plot, we compare the 2D PS of a regular logarithmic layout in which the baseline separation is $\\Delta_u \\approx 0.08\\sqrt{0.4}u\/\\tau$, (i.e. 12.5 times smaller than required to mitigate the wedge), with a layout whose baseline density (and therefore average separation as a function of $u$) is identical, but in which the baselines are stochastically placed. \nWe also show the result of an over-dense random layout.\nFigure \\ref{fig:random_separations} shows the actual separations between baselines as a function of $u$ for each case.\nEven for the random arrangements, \\textit{all} baselines have separations smaller than the wedge-mitigation threshold.\nWhile we might expect all of them to have near-perfect wedge-mitigation, we find that the random layout yields a subdued, but very present, wedge. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=1cm 0cm 2cm 0cm]{figures\/random_trajectories}\n\t\\caption{2D PS comparison between baseline layouts for three cases: (i) random radial placement, with mean separation proportional to $u$ and $N=10,000$ (left panel); (ii) the same random placement, but with $N=50,000$ (centre panel), and (iii) regular logarithmic placement of equivalent density to case (i) (right panel). Clearly the introduction of stochastic baseline separations re-introduces a wedge.}\n\t\\label{fig:random_trajectories}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/random_separations}\n\t\\caption{Baseline separations, $\\Delta_u$, as a function of $u$, for three cases: (i) random radial placement, with mean separation proportional to $u$ and $N=10,000$ in green; (ii) the same random placement, but with $N=50,000$ in red, and (iii) regular logarithmic placement of equivalent density to case (i) in orange. The blue line shows the threshold separation below which a regular grid significantly mitigates the wedge (cf. Eq. \\ref{eq:solution_for_concentric}).}\n\t\\label{fig:random_separations}\n\\end{figure}\n\n\nThe explanation for this behaviour arises from the ``bars'' that occur for the regular logarithmic polar grid (cf. Fig. \\ref{fig:dense_log_concentric}).\nA regular grid creates oscillations which sit atop the bandpass in frequency-space (cf. \\S\\ref{sec:weighted:conceptual}). \nIf the grid is logarithmic in $u$-space, these oscillations are linear in frequency space, causing neatly-spaced peaks in the power spectrum. \nWhen the baselines have stochastic separations, the oscillations are irregular, and cause a cacophony of ``bars'' above the main ``brick''.\nEssentially, this haphazard distribution of peaks restores a somewhat subdued wedge. \n\nThe level to which it is subdued will depend on the baseline density, however it clearly requires a significant increase in density to match the regular logarithmic grid. \nWe note that it is not primarily the fluctuating \\textit{minimum} separation of baselines that causes the re-emergence of the wedge.\nThis can be clearly understood from Fig. \\ref{fig:random_separations}, in which for the over-dense random layout, the separation very rarely ventures above that of the regular grid. \nThe issue is rather that the unevenness of the random distribution causes higher-order structure in the oscillations that lie atop the bandpass, which emanate as the smeared peaks within the wedge.\n\nTo determine the extent of this effect, we use the same regular set of 10,000 baselines, and randomly offset them by some fraction of their amplitude, according to a normal distribution. \nWe show the results in Fig.~\\ref{fig:random_offsets}.\nEven when the fractional offset is $\\sim 10^{-5}$, the wedge returns, albeit quite subdued (2-3 orders of magnitude).\nAs the offsets increase in magnitude, the wedge is restored to a greater degree, as expected.\nIt would thus seem that any hopes of mitigating the wedge via regular radial arrays are impractical both due to their high density requirements and their strong dependence on strict regularity. \nNevertheless, it is possible that irregularity between spokes will alleviate some of this, and we will explore this further in the following section.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=1.7cm 0cm 3.5cm 0cm]{figures\/random_offsets}\n\t\\caption{2D PS comparison between baseline layouts for a regular logarithmic polar grid, and three cases in which the baselines have been randomly offset from regularity. A subdued wedge clearly returns even for fractional offsets of $10^{-5}$. The amplitude of the wedge increases by a couple of orders of magnitude as the offsets increase in their magnitude.}\n\t\\label{fig:random_offsets}\n\\end{figure}\n\n\\section{Wedge Properties of Real Arrays}\n\\label{sec:mitigation:arrays}\n\n\\begin{table*}[!htb]\n\t\\begin{center}\n\t\t\\begin{tabular}{ l l l }\n\t\t\t\\hline\n\t\t\t\\textbf{Label} & \\textbf{Description} & \\textbf{Varieties}  \\\\ \n\t\t\t\\hline\n\t\t\t\\texttt{circle} & Equi-spaced on circumference of circle, diameter $x_{\\rm max}$ &  \\\\\n\t\t\t\\texttt{circle\\_filled\\_} & Randomly filled circle of diameter $x_{\\rm max}$ & Uniform (\\texttt{\\_0}), Logarithmic (\\texttt{\\_1}) \\\\\n\t\t\t\\texttt{spokes\\_} & Regular radial\/angular spacing, max $x_{\\rm max}$ & Logarithmic\/Linear, $N_{\\rm spokes}$  \\\\\n\t\t\t\\texttt{rlx\\_boundary} & Equi-spaced on boundary of Reuleaux triangle \\citep[eg.][]{Keto1997} &  \\\\\n\t\t\t\\texttt{rlx\\_grid\\_} & Regular concentric Reulaeux triangles & Logarithmic  \\\\\n\t\t\t\\texttt{hexagon} & Regular hexagon, width $x_{\\rm max}$  &  \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{\\label{tab:baseline_layouts}Summary of antenna layouts used in Figs. \\ref{fig:big_baseline_diagram} -- \\ref{fig:big_layout_std}.}\n\t\\end{center}\n\\end{table*}\n\n\nWe have found that the wedge may in principle be avoided by employing a sufficiently radially dense and regular baseline layout.\nIn our previous explorations, we have considered explicit baseline layouts, ignoring the fact that no \\textit{antenna} layout may exactly correspond to such a baseline layout (alternatively, choosing an antenna layout which corresponds to a superset of the desired baseline layout and ignoring the extraneous baselines).\nIn this final exploration, we expand our consideration to several physically-feasible antenna layouts, with full correlation of \\textit{all} antennas.\n\nIn this case, simple ``spoke'' antenna layouts will contain both the regular subset which we have previously considered, and a larger set of irregular baselines. \nThus we will find whether the increased baseline density outweighs the increased irregularity in terms of wedge mitigation (cf. \\S\\ref{sec:weighted:irregular}).\n\nThe kinds of antenna layouts  we employ (with their variants) can be found in Table \\ref{tab:baseline_layouts}, and an illustration of each is found in Fig. \\ref{fig:big_baseline_diagram}.\nWe note that for the linear ``spokes'' layouts, to decrease the redundancy, we use regularly-spaced antennae for half of the spoke, and a single antenna at the far end.\nThis does not apply for the logarithmic spoke layouts, for which each spoke necessarily begins at the centre. \n\n\nWe use the same number of antennae, $N_{\\rm ant}$, for each layout (or as close to this number as possible, given the constraints of some), and place all baselines within a set radius $x_{\\rm max} \\approx 2u_{\\rm max}$.\nWe choose $N_{\\rm ant} = 256$ and $u_{\\rm max} = 800$ for the figures in this section.\nEach layout is first checked for overlapping antennae, with antenna diameters of 4m (corresponding roughly to the MWA tiles), so that the final layout is physically possible. \nWith these choices, comparisons of the power spectra from each array are roughly insensitive to the overall density or ``cost'' of the array, and are rather indicative of the form of the layout itself.\nNote that we also use the tile diameter of 4m to calculate the beam width.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 3cm 0cm 3cm]{figures\/big_baseline_diagram}\n\t\\caption{Plots of antenna and baseline layouts for the definitions in Table \\ref{tab:baseline_layouts}. The orange markers indicate antennas, and blue points represent baselines. The axes are in units of $u$ at a frequency of 150 MHz. Antennae are all spaced at least 4m apart, which corresponds to the size of an antenna.}\n\t\\label{fig:big_baseline_diagram}\n\\end{figure*}\n\nDespite these considerations, we must be clear that this is not a test for \\textit{how well the layout would recover an underlying 21 cm signal}. \nSpecifically, two quantities are of interest for foreground mitigation: the total expected (foreground) power, and its covariance, $\\Sigma_P$.  \nWe have only addressed the expected power in this paper. \nAssuming a reliable model of this quantity can be subtracted from the observed data, the remaining $\\Sigma_P$ is the key factor in defining which power spectral modes are usable for averaging to a final one-dimensional power spectrum. \nTo a large extent, $\\Sigma_P$ is proportional to $P^2$, so the derivations in this paper are indicative of this quantity.\nNevertheless, both sample variance and thermal noise also play an independent role in the determination of $\\Sigma_P$, and these are dictated largely by the density of baselines (the former explicitly by the \\textit{angular} density).\n\nWhile the overall density of baselines should be similar in each of the layouts we have chosen -- due to our restriction of setting the antennas within a prescribed radius -- their angular density is decidedly \\textit{not}.\nWe thus expect those with lower angular coverage (eg. the various ``spokes'' layouts) to yield a greater value of $\\Sigma_P$, which could hamper 21 cm signal extraction.\nWe do not pursue a rigorous analysis of these considerations in this paper; our goal is to identify the general effects these layouts have on the establishment of the wedge feature -- not the prospects of 21 cm signal extraction.\nNevertheless, we suggest that such an analysis should be simple enough, by comparing a numerically-generated expectation of $\\Sigma_P$ in the presence of both sample variance and thermal noise for each layout.\n\n\\ifanalytic\n\tTo determine the expected PS for each layout, we numerically solve the triple integral given by the combination of Eqs. \\ref{eq:power_general} and \\ref{eq:general_var}.\n\tThis is a non-trivial task, requiring high-precision integration over $\\theta$ in order to yield reasonable estimates after dividing by the weight factor. \n\tDetails of the numerical method can be found in App. \\ref{app:numerical_integration}.\n\\else\n\tTo determine the expected PS for each layout, we evaluate the PS using the technique outlined in App.~\\ref{app:numerical} for the same set of 200 random skies for each layout, taking the mean and standard deviation. \n\\fi\n\nFigure \\ref{fig:big_layout_ps} shows the resulting expected 2D power spectra for each of these layouts.\nAll layouts show a strong wedge feature with similar shape, as expected, and each exhibits the same bandpass limits (the low-$\\omega$ ``brick''). \nTwo peculiar features require some explanation.\nFirst, due to our choice of using a gridding kernel which in principle has infinite extent (though in practice, we limit it to 50-$\\sigma_u$), grid points $\\vect{u}$ which are in extremely sparse UV-sampled locations will tend to evaluate to the same power, as the same distant baseline will be the dominant contributor for all grid points in the region. \nIf this is limited to a small arc of the full polar angle, its effect will be negligible, but some of these layouts are extremely sparse for all angles, especially at low-$u$. \nThis effect presents as a horizontal `smearing' of the power, and is most noticeable in the low-$u$ modes of the \\texttt{spokes\\_log\\_4} layout. \nA related effect produces the many thin vertical ``spikes'' witnessed at high-$u$ in many of the spectra. \nIn this case, however, it seems to be a combination of the irregularity of the baselines with this local sparsity that produces the effect. \nWe emphasize that the vertical features, though they appear `noisy', do not disappear as more realizations are averaged, and are therefore systematic. \n\nIn Figure \\ref{fig:big_layout_compare} we show the ratio of each expected 2D PS against the result of a delay spectrum.\nThis is precisely the result of \\S\\ref{sec:classic} (i.e. the limit of sparse baselines), except that each $u$ is assumed to be exactly obtainable.\nThus comparison to this spectrum is appropriate as the sparse limit of baseline density. \nWe reiterate that the physical layouts here can be non-local, so that the evaluated power is determined by a relatively distant baseline, whereas the reference `delay spectrum' is always exactly local in this comparison. \n\nWith this in mind, we note that most of the layouts produce less foreground power over most modes than a simple delay spectrum. \nThis is to be expected, as the averaging over baselines effectively lowers the amplitude of fluctuations. \nThe single exception to this seems to be the hexagonal layout. \nHowever, on closer inspection, most of the power here is exactly the same as the delay spectrum, as expected from its inherent sparsity. \nAt low-$u$, the \\texttt{hexagon} is in a region of extreme local sparsity, as discussed above, and therefore cannot be trusted (in the same way as the low-$u$ region of \\texttt{spokes\\_log\\_4}).\nThe region in and around the wedge does exhibit significantly more power than the delay spectrum, but this is common to all layouts, and we will discuss this momentarily.\n\nThe most significant reduction of power occurs in the EoR window for the \\texttt{spokes\\_log\\_6} and \\texttt{rlx\\_grid\\_log} layouts, at 2-3 orders of magnitude.\nThese layouts have strong logarithmic regularity at the most polar angles compared to other layouts in our sample. \nThough their radial density is not as high as the log-spoke layouts with fewer spokes, it appears that providing some regularity at more angles (and therefore decreasing overall irregularity) outweighs this deficit.\nHowever, these layouts, along with \\texttt{circle\\_filled\\_1}, also have the highest density of short baselines, so it is difficult to isolate the contribution of any single characteristic.\n\nThe most visually obvious feature of the ratio plots (figure \\ref{fig:big_layout_compare}) is the excess power appearing as irregular vertical stripes protruding from the wedge. \nThis power appears to arise due to sparsity of baselines at these scales, such that for a particular grid-point, baselines ``move through'' the grid-point with frequency and leave nulls in the effective spectrum before another baseline passes through. \nThis creates a ringing in the Fourier-space spectrum, which throws power outside the wedge. \nThis interpretation is supported by the fact that the two layouts which minimize this effect are those with the highest density of baselines at high $u$. \nConversely, the hexagonal layout, with its extremely sparse and regular baselines, maximizes this effect over much of the range.\nThis is a well-known key advantage of the delay spectrum, which in principle limits the foreground power exclusively to the theoretical ``horizon line'' for each baseline (notwithstanding other chromatic effects of the instrument and sky).\nNevertheless, it is unclear how this advantage balances against the reduction of window power offered by the dense regular baseline layouts.\nUltimately, these scales, where the density of baselines is low enough to cause this effect, should be ignored in any analysis.\n\nAnother interesting feature are the diagonal strips at high $u$ in the \\texttt{spokes\\_lin} layouts, which appear to be manifestations of the same effect illustrated in Fig. \\ref{fig:dense_linear_concentric}, i.e. dense linear radial regularity introducing scale-dependent harmonics in the sky response.\nNevertheless, these are muted compared to the purely regular theoretical arrays previously considered.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=1cm 2cm 0cm 2cm]{figures\/big_layout_straight}\n\t\\caption{Average 2D PS (over 200 realizations) for each of the layouts we consider (see Table \\ref{tab:baseline_layouts} for details of the layouts).}\n\t\\label{fig:big_layout_ps}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 2cm 0cm 2cm]{figures\/big_layout_ratio}\n\t\\caption{Ratio of 2D PS of each layout in Table \\ref{tab:baseline_layouts} to a delay spectrum evaluated with one baseline at each grid point (see \\S\\ref{sec:mitigation:arrays} for details).}\n\t\\label{fig:big_layout_compare}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 2cm 0cm 2cm]{figures\/big_layout_std}\n\t\\caption{Standard deviation of 2D PS over 200 realizations for each layout in Table \\ref{tab:baseline_layouts}.}\n\t\\label{fig:big_layout_std}\n\\end{figure*}\n\nTo verify that the results of this section are not subject to high statistical uncertainty, we show the ratio of the standard error of the mean (SEM) to the mean of each PS in Fig. \\ref{fig:big_layout_std}.\nThis illustrates that the mean is accurate to within $\\sim10$\\%, which means that statistical uncertainty is of minor concern, and that the conceptual results of this section can be trusted in this regard.\nInterestingly, the regions of excess power have a relatively high uncertainty compared to the rest of the spectrum, indicating that these regions are more sensitive to the exact positions of point-sources on the sky.\nThis supports our interpretation that this excess power arises from a dearth of baselines, which would increase the sensitivity of the measured power at a particular $\\vect{u}$ to the sky realization, and also increase the variance of measurements over polar angles. \n\nIn summary, with the number of antennae considered, the precise layout has only a minimal effect on the expected PS within the wedge and window. \nNevertheless, in accord with our semi-analytic considerations of previous sections, it appears that dense logarithmically regular layouts can improve the spectral smoothness of the array, and mitigate foreground power, at the level of 2-3 orders of magnitude. \nWe expect this to improve with a higher number of antennae, so that layout considerations will become relatively more important in future high-$N$ arrays.\nConversely, gridding baselines, as opposed to delay transforming on a per-baseline basis, produces artifacts at high-$u$ which can throw excess power out of the wedge. \nIt is beyond the scope of this paper to quantitatively assess which method is preferable for measuring the EoR.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\\subsection{Summary}\nUsing a simple formalism to describe the expected 2D power spectrum of point-source foregrounds (Eqs. \\ref{eq:simple_vis}, \\ref{eq:vis_gridded}, \\ref{eq:vis_full} and \\ref{eq:power_general}), we verified the standard schematic `wedge', which has been extensively discussed in the literature.\nWe then used this formalism, which includes the ability to utilise arbitrary $uv$-sampling functions, to examine the effects of such on the wedge, with the primary conclusion that dense, radially regular layouts can diminish the extent and amplitude of the wedge, but that this effect is small for physically achievable layouts.\n\nUsing a semi-analytic approach based on a discrete polar grid $uv$-sampling (largely focusing on logarithmically-separated radial baselines) we find, as suggested in previous works \\citep[eg.][]{Bowman2009,Morales2012,Parsons2012}, that increasing the radial density of baselines tends to decrease the amplitude of the wedge (defined as that part of the foreground signature which emerges from the low-$k_{||}$ ``brick''). \nIndeed, we find that for regular log-spaced baselines, a density may (in principle) be achieved at which the wedge effectively disappears. \nWe explain these results intuitively via radial baseline ``replacement'' with change of frequency (cf. \\S\\ref{sec:weighted:conceptual}).\n\n\nUsing this semi-analytic model, we explored some of the ramifications of these ideas.\nWe found that a characteristic separation can be determined which defines the threshold for wedge emergence.\nThis characteristic separation is proportional to the baseline magnitude $u$ and also to the bandwidth of the observation (cf. Eq. \\ref{eq:solution_for_concentric}).\nWe find that the physical baseline separation (in metres) is approximately $u\/\\tau$, where our default value for $\\tau$ is approximately 100, and in the regime of the wedge, $u > \\tau\/\\sqrt{2}\\pi\\sigma \\approx 150$. \nThe minimum baseline separation to mitigate the entire wedge (which occurs at $u=\\tau\/\\sqrt{2}\\pi\\sigma$, or an antenna separation of $\\sim 300$m) is $1\/3\\sigma \\approx 1.7{\\rm m}$.\nWe concluded that such a baseline density is physically impossible for the most efficient antenna layout corresponding to the polar grid baseline layout.\nWith a less compact array, the baseline density is technically achievable, but highly impractical.\n\nFurther, we found that randomising the radial distribution of baselines tends to re-instate the wedge, as the series of overlaid oscillations is smeared out over $\\omega \\propto \\eta$.\nThus the optimal array for wedge mitigation is both dense \\textit{and} regular.\n\nUpon examination of the expected 2D PS from some simple antenna layouts, we found that in practice both the window and wedge can be reduced by up to 3 orders of magnitude by employing antenna spokes which are regular in log-space and as dense as possible across many angles.\nWe noted that such a layout competes with the requirement of angular baseline density to mitigate sample variance. \n\n\n\\subsection{Future Considerations}\nThe work in this paper paints a rather bleak picture: \nit will be very difficult to combat the wedge via any array design.\nNevertheless, we have shown that in principle, layouts with a higher degree of radial alignment and regularity will serve to reduce the magnitude of the wedge, and therefore potentially yield some increase in the fidelity of future PS estimation.\n\nTo establish this rigorously, one needs to consider not only the expected 2D PS, but also its covariance.\nThese will compete with one another -- the more aligned the layout, the lower the expected wedge, but the higher the overall covariance of the estimate.\nA proper analysis of these quantities, and their effect on the signal-to-noise of a fiducial 21 cm signal, is the most pressing future consideration to arise from this work.\nAlong with this, consideration of non-Gaussian band-pass (or taper) shapes, non-co-planar arrays, non-zenith pointings, and non-flat SED's may be interesting realistic effects to add to the analysis.\n\n\n\n\\begin{acknowledgements}\n\tThe Centre for All-Sky Astrophysics (CAASTRO) is an Australian Research Council Centre of Excellence, funded by grant CE11E0090. \n\tParts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.\n\tThis research has made use of NASA's Astrophysics Data System.\n\tAll plots in this paper were generated using \\textsc{matplotlib}.\n\\end{acknowledgements}\n\n\\clearpage\n\n\\begin{appendix}\n\t\n\t\\section{Derivation of Analytic Examples}\n\t\\label{app:delay}\n\t\n\tIn this section we derive the solutions to the three analytic examples shown in Table \\ref{tab:simple_summary}.\n\tRecall that in this section, we have employed the following conditions:\n\t\\begin{enumerate}\n\t\t\\item Sparse discrete polar grid baseline layout\n\t\t\\item Artificially narrow gridding kernel (width $\\rightarrow 0$)\n\t\t\\item Evaluation at grid points co-located with reference baseline positions.\n\t\\end{enumerate}\n\n\tWe note that in this scheme, only one baseline may contribute to a given $\\vect{u}$ over all frequency. \n\tThus, Eq.~\\ref{eq:vis_gridded} is simplified to $V(\\nu, \\vect{u}) = V_i(\\nu,\\vect{u}_i)$, removing the sum over baselines.\n\t\n\tFinally, since the baselines are arranged symmetrically, the average over $\\theta$ is unweighted -- each baseline contributes to the same arc-length.\n\tThe power can thus be simply expressed as\n\t\\begin{equation}\n\t\\label{eq:power_ngp}\n\t\\langle P(\\omega, u_\\mu) \\rangle = \\frac{1}{N_\\theta} \\sum_{k=1}^{N_\\theta} \n\t{\\rm Var}(\\tilde{V}(\\omega, \\vect{u}_{\\mu, k})) + \\left|\\langle \\tilde{V}(\\omega, \\vect{u}_{\\mu, k})\\rangle \\right|^2,\n\t\\end{equation}\n\twhere the sum is over all baselines in a given ring, and the subscript $\\mu$ is meant to indicate that we evaluate the visibilities at $\\vect{u}_i$, rather than at an arbitrary location.\n\tNote that evaluation of the PS at $u \\neq u_i$ is simple, but will yield the same PS as that evaluated at the closest $u_i$. \n\tThese solutions therefore really represent a series of step-functions in $u$, where each step is centred on $u_i$. \n\t\n\t\n\t\\subsection{Sky Model: Single Source}\n\tWithin this subsection we will consider a single-source sky at $\\vect{l} = \\vect{l}_0$ with $S=S_0$.\n\tDue to the non-stochastic nature of the sky we only require the mean visibility, which is simply:\n\t\n\n\n\n\n\t%\n\t%\n\n\n\t\\begin{align}\n\t\\langle \\tilde{V}(\\omega, \\vect{u}_\\mu) \\rangle = \\nu_0 S_0 \\int df\\ \\phi_\\nu  B_\\nu(l_0) e^{-2\\pi i f(\\omega + \\vect{u}_\\mu \\cdot \\vect{l}_0)}.\n\t\\end{align}\n\t\n\t\n\t\\subsubsection{Static Beam} \n\tIf the beam is frequency-independent, it comes out of the integral and we are left with\n\t\\begin{equation}\n\t\\label{eq:tmp0}\n\t\\tilde{V}(\\omega, \\mathbf{u}_\\mu) = S_0 \\nu_0 B(l_0) \\int df e^{-\\tau^2 (f-1)^2} e^{-2\\pi if (\\omega +\\vect{u}_\\mu \\cdot\\vect{l}_0)}.\n\t\\end{equation}\n\t\n\tHere we make use of the following useful identity, and we shall repeatedly do so throughout this section:\n\t\\begin{equation}\n\t\\label{eq:int_of_exp}\n\t\\int_{-\\infty}^{+\\infty} e^{-ax^2 - bx + c}dx = \\sqrt{\\frac{\\pi}{a}}e^{b^2\/4a + c}.\n\t\\end{equation}\n\tRearranging Eq. \\ref{eq:tmp0}, we find \n\t\\begin{align}\n\ta &= \\tau^2,\\nonumber \\\\\n\tb &= 2\\pi i (\\omega + \\vect{u}\\cdot \\vect{l}_0) - 2  \\tau^2, \\nonumber \\\\\n\tc &= -\\frac{l_0^2}{2\\sigma^2} - \\tau^2.\n\t\\end{align}\n\tThis yields\n\t\\begin{align}\n\t\\label{eq:vis_ss_sb}\n\t\\tilde{V}(\\omega, \\mathbf{u}_\\mu) =  &\\frac{S_0 \\nu_0 \\sqrt{\\pi}}{\\tau} \\exp\\left(-\\frac{l_0^2}{2\\sigma^2}\\right) \\nonumber \\\\\n\t& \\times\\exp\\left( -\\frac{\\pi^2(\\omega + \\vect{u}_\\mu\\cdot \\vect{l}_0)^2}{\\tau^2}  - 2 i\\pi (\\omega + \\vect{u}_\\mu\\cdot \\vect{l}_0)\\right)\n\t\\end{align}\n\t\n\tFurthermore, the power can be written:\n\t\\begin{align}\n\t\\label{eq:ss_ngp_static}\n\tP(\\omega, u_\\mu) = \\frac{1}{N_\\theta}  & \\frac{S_0^2 \\nu_0^2 \\pi}{\\tau^2} \\exp\\left(-\\frac{l_0^2}{\\sigma^2}\\right) \\nonumber \\\\\n\t& \\times \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2(\\omega + u_\\mu l_0 \\cos (2\\pi k\/N_\\theta))^2}{\\tau^2} \\right).\n\t\\end{align}\n\tThis sum has no general closed form solution.\n\tNevertheless, it is not difficult to ascertain its general behaviour. \n\tThe two terms in the exponential will compete for dominance, and since cosine has a maximum of unity, we can determine a line of equal weight: $\\omega = ul_0$.\n\tWhen the $\\omega$ term is dominant, the integrand loses sensitivity to $\\theta$, and the power can be written\n\t\\begin{equation}\n\tP(\\omega \\gg ul_0, u) = \\frac{S_0^2 \\nu_0^2 \\pi}{\\tau^2} \\exp\\left(-\\frac{l_0^2}{\\sigma^2}\\right)  \\exp\\left( -\\frac{2\\pi^2 \\omega^2}{\\tau^2} \\right),\n\t\\end{equation}\n\tThus we expect that there will be a (sharp) exponential drop in the power for $\\omega \\gg ul_0$. \n\tWhen $ul_0$ is small, the entire function $P(\\omega)$ (i.e. a vertical line in the 2D PS) will obey this equation, and the cutoff will appear at a characteristic scale $\\omega \\sim \\tau\/\\pi\\sqrt{2}$. \n\tLarger $ul_0$ acts as a buffer, requiring $\\omega$ to overcome it before the exponential drop is realised (at a much sharper rate, due to the increased amplitude of the exponential). \n\tThe exact point at which $\\omega$ overcomes the $ul_0$ term is difficult to obtain in closed form (it can easily be obtained as a power series), but we merely state the empirical result that it is close to $ul_0$. \n\tThus we have a cutoff at $\\omega \\approx {\\rm max}(\\tau\/\\pi\\sqrt{2}, ul_0)$, where the first limit defines a ``brick'' at low $(\\omega,u)$, and the second defines a ``wedge'' at higher $u$.\n\t\n\t\n\t\n\t\n\t\n\t\\subsubsection{Chromatic Beam}\n\t\\citet{Liu2014} have pointed out that regardless of whether the beam is chromatic or not, it is a much broader function of frequency than the taper, and therefore it is a good approximation to bring it outside the integral, and evaluate it at $f=1$. \n\tThis would yield precisely the same result as the achromatic beam of the previous section.\n\tNevertheless, we wish to present an exact formula in this section.\n\t\n\tIn this case, the only aspect that changes from the previous section is that we have $a \\rightarrow \\tau^2 + l_0^2\/2\\sigma^2$ since the beam moves back inside the integral.\n\tThus we achieve\n\n\n\n\n\n\t\\begin{align}\n\tP(\\omega, u_\\mu) =  & \\frac{S_0^2 \\nu_0^2 \\pi}{N_\\theta p^2} \\exp\\left[-\\tau^2 \\left(1 - \\frac{\\tau^2}{p^2}\\right)\\right] \\nonumber \\\\\n\t& \\times \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2 (\\omega + u l_0 \\cos (2\\pi k\/N_\\theta))^2}{p^2}\\right),\n\t\\end{align}\n\twith $p^2 = \\tau^2+l_0^2\/2\\sigma^2$.\n\tThe behaviour of this equation is very similar to the previous static case, except that the effect of $\\tau$ is balanced by the effect of the beam-width. \n\tThat is, setting $\\tau$ arbitrarily small (i.e. very wide band-pass) will no longer yield an arbitrarily tight ``brick'', as the beam-width will have the effect of broadening it.\n\t\n\tIn practice, for instruments targeted at observing the EoR, $\\tau^2 \\gg 1\/2\\sigma^2$, so that the achromatic beam  is a reasonable approximation, as expected by the arguments from \\citet{Liu2014}.\n\t\n\t\\subsection{Sky Model: Stochastic Uniform}\n\t\t\n\tWe merely need to solve Eqs. \\ref{eq:su_meanvis} and \\ref{eq:su_covvis} for a Gaussian beam, and then integrate over frequency.\n\tWe evaluate only for a static beam in this case, as we have already seen that a chromatic beam is too broad (in frequency) to have a significant impact on the result.\n\t\n\tThe mean term is simply\n\t\\begin{align}\n\t\t\\langle V(\\omega, u_\\mu) \\rangle &= 2\\pi \\sigma^2 \\nu_0\\bar{S} \\int df\\ e^{-2\\pi i f\\omega} \\phi_\\nu  e^{-2\\pi^2 \\sigma_\\nu^2 f^2 u^2}.\n\t\\end{align}\n\n\tUsing the identity Eq.~\\ref{eq:int_of_exp}, and noting that the equation depends only on $u^2$ and therefore needs not be integrated around the annulus, we find\n\t\\begin{align}\n\t|\\langle \\tilde{V}\\rangle|^2 &= \\frac{4 \\bar{S}^2 \\nu_0^2 \\pi^3 \\sigma^4}{\\tau^2 + 2\\pi^2 \\sigma^2 u^2} \\exp\\left[ 2\\left(\\frac{\\tau^4 - \\pi^2\\omega^2}{\\tau^2 + 2\\pi^2\\sigma^2 u^2} - \\tau^2 \\right) \\right].\n\t\\end{align}\n\t\n\tTo begin the variance, we use Eq.~\\ref{eq:su_covvis}, along with the various assumptions we have thus far made, to obtain\n\t\\begin{align}\n\t\t\\label{eq:stoch_uniform}\n\t\t{\\rm Var}(\\tilde{V}) = \\mu_2 \\nu_0^2 & \\int d\\vect{l}e^{-l^2\/\\sigma^2} \\left| \\int df   e^{-2\\pi if(\\omega + \\vect{u}\\cdot \\vect{l})} \\phi_\\nu  \\right|^2\n\t\\end{align}\n\tWe use the result of (Eq. \\ref{eq:vis_ss_sb}) directly to obtain\n\t\\begin{align}\n\t\t{\\rm Var}(\\tilde{V}) = \\frac{\\pi \\mu_2 \\nu_0^2}{\\tau^2} \\int d\\vect{l}\\ \\  e^{-l^2\/\\sigma^2} \n\t\t\\exp\\left(-\\frac{2\\pi^2(\\omega+\\vect{u}_\\mu\\cdot\\vect{l})^2}{\\tau^2} \\right) .\n\t\\end{align}\n\tDue to statistical isotropy, we may without loss of generality evaluate the case $\\vect{u} = (u,0)$, and perform the integral over $\\vect{l}$ in 2D Cartesian space, to finally find\n\t\\begin{align}\n\t\\label{eq:sparse_solution}\n\t{\\rm Var}(\\tilde{V}) = \\frac{\\mu_2 \\nu_0^2 \\pi^3 \\sigma^2}{\\tau\\sqrt{\\tau^2 + 2\\pi^2\\sigma^2u^2}} \\exp\\left(-\\frac{2\\pi^2 \\omega^2}{2\\pi^2\\sigma^2u^2+\\tau^2}\\right).\n\t\\end{align}\n\t\n\tNow combining both terms of the power spectrum, we can make some simple observations.\n\tFirstly, for $\\pi u \\sigma \\ll \\tau$, we have\n\t\\begin{equation}\n\tP_{\\pi u \\sigma \\ll \\tau} = \\frac{\\nu_0^2 \\pi^2 \\sigma^2}{\\tau^2} e^{-2\\pi^2 \\omega^2\/\\tau^2} \\left(\\bar{S}^2 \\sigma^2 + \\mu_2 \\pi \\right).\n\t\\end{equation}\n\tThis has a sharp cut-off at $\\omega \\approx \\tau\/\\sqrt{2}\\pi$, creating the familiar lower-left ``brick'' in the 2D PS.\n\tConversely, we have\n\t\\begin{align}\n\tP_{\\pi u \\sigma \\gg \\tau} &= \\frac{\\nu_0^2 \\sigma}{u} e^{-\\omega^2\/2u^2\\sigma^2}\\left[\\frac{ 2\\pi  \\sigma \\bar{S}^2 e^{-\\tau^2}}{ u}  + \\frac{\\mu_2 \\pi e^{-\\omega^2\/2\\sigma^2 u^2}}{\\sqrt{2}\\tau}\\right] \\nonumber \\\\\n\t&\\approx \\frac{\\nu_0^2\\mu_2 \\pi \\sigma}{\\sqrt{2}\\tau u} e^{-\\omega^2\/u^2\\sigma^2}, \n\t\\end{align}\n\twhere the final line assumes that $\\omega < \\tau^2$, which covers all the reasonable values of $\\omega$.\n\n\tThis clearly has a sharp cut-off at $\\omega = u\\sigma$, creating the wedge (cf. rightmost panel of Fig. \\ref{fig:ss_ngp_static}). \n\t\n\t\n\t\n\n\n\\section{Radially Smooth Layout}\n\\label{app:radial}\nHere we consider a polar grid layout in which the radial spokes are no longer discrete but are of such high density that they may be considered smooth.\nThis will allow us to derive some constraints on how `smooth' the radial distribution of baselines must be to avoid a wedge.\n\nLet $\\rho = \\rho_\\theta \\rho_u$ be the density of baselines as a function of $u$ and $\\theta$.\nThen the sums over baselines in Eq. \\ref{eq:wg_master} reduce to integrals over $\\rho$: \n\\begin{align}\nI = \\int df \\frac{\\phi_\\nu}{W_\\nu} \\int d^2 \\vect{u}_i \\frac{\\rho_\\theta \\rho_u}{u_i} e^{-q^2(\\vect{u} - f\\vect{u}_i)^2} e^{-2if(\\omega +\\vect{l}\\cdot\\vect{u}_i)}.\n\\end{align}\n\nWe may calculate the total weight, performing the integration in polar co-ordinates, making the substitution $u'_i = fu_i$:\n\\begin{align}\nW_\\nu = &\\frac{e^{-2\\pi^2 \\sigma^2u^2}}{2\\pi f^2} \\int_0^{2\\pi} d\\theta\\  \\rho_\\theta \\nonumber \\\\\n&\\times \\int du'_i\\  \\rho_u(u'_i\/f^2) e^{-q^2({u'_j}^2 - 2uu'_i\\cos\\theta )}.\n\\end{align}\nIt is difficult to proceed further without specifying some form for $\\rho_u$. \nNevertheless, we note that $\\rho_u$ will only contribute to the $u'_i$ integral if it is sufficiently sharply peaked -- otherwise it can be treated as a constant and removed from the integral.\nWe let $\\rho_u$ be an arbitrary linear combination of Gaussians, centered around points $u_l$:\n\\begin{equation}\n\\rho_u \\propto \\sum_l a_l \\exp\\left(-\\frac{(u'_j - u_l)^2}{2 \\sigma_l^2}\\right),\n\\end{equation}\nwhere the normalisation constant is irrelevant as it cancels in the final visibility. \n\nThe equation for $W_\\nu$ may thus be re-written as\n\\begin{align}\nW^T_j = &\\frac{e^{-q^2 u^2}}{f^2} \\int d\\theta\\  \\rho_\\theta \\sum_l a_l e^\\frac{-u_l^2}{2f^4\\sigma_l^2} \\nonumber \\\\\n&\\times \\int du'_i\\  \\exp\\left(-{u'}_i^2(q^2 + \\frac{1}{2\\sigma_l^2})\\right. \\nonumber \\\\\n&+ \\left. 2u'_i(q^2 u\\cos \\theta + \\frac{u_l}{2f^2\\sigma_l^2})\\right).\n\\end{align}\nPerforming the $u'_i$ integral, each term in the sum becomes\n\\begin{equation}\na_l \\sqrt{\\frac{\\pi}{q^2 + \\frac{1}{2\\sigma_l^2}}} \\exp\\left(\\frac{q^2\\left[q^2 u^2\\cos\\theta\/2 + \\frac{uu_l\\cos\\theta}{2f^2\\sigma_l^2} - \\frac{u_l^2}{f^4 \\sigma_l^2}\\right]}{2q^2 + \\frac{1}{\\sigma_l^2}}\\right).\n\\end{equation}\nIf $\\sigma_l \\gg 1\/2q = 1\/2\\pi\\sigma$, then we can ignore the $\\sigma_l$ term in both the square root and the denominator of the exponential. \nIn fact, this same inequality also reduces the numerator to its first term (for $f\\sim 1$), such that the form for $W_\\nu$ is\n\\begin{align}\nW_\\nu = &\\sqrt{\\frac{\\pi}{q^2}} \\frac{e^{-q^2u^2}}{f^2}   \\sum_l a_l \\int d\\theta\\ \\rho_\\theta \\exp\\left(\\frac{q^2 u^2\\cos\\theta}{4}\\right).\n\\end{align}\n\nThe condition that $\\sigma_l \\gg 1\/2\\pi\\sigma$ for all terms $l$ is thus a well-specified ``smoothness\" bound, though we note that it is a conservative bound;\neven if a term is \\textit{more} peaked than permitted by the bound, if it has a small relative amplitude then its contribution may be ignored. \nThis is important for real arrays, in which the baselines form delta-functions in the UV plane. \nThough every point consists of a ``Gaussian'' which is more peaked than the bound, they may be spaced closely enough that each of them contributes negligible weight, thereby approximating a ``smooth\" array.\n\n\nThis smoothness bound, for a realistic array at $\\nu_0 \\approx 150$ MHz, corresponds to constraining $\\sigma_l \\gg 2D$, where $D$ is the diameter of an array tile.\nBreaking this condition would require quite a peaked baseline density indeed.\n\n\n\n\nWith this in mind, for an arbitrary radially smooth layout, the solution is of the form\n\\begin{equation}\nW_\\nu = g(q)\/f^2.\n\\end{equation}\nWe can use the same procedure to determine the final integral of $I$ (except that it has an extra factor of $2\\pi i \\vect{l}\\cdot\\vect{u}'_i$ in the exponent). \nThis implies that the factors of $f^2$ cancel, so that we have\n\\begin{equation}\nI =  \\frac{g'(\\vect{u}, \\vect{l})}{g(\\vect{u})} \\int df \\phi_\\nu e^{-2\\pi if\\omega},\n\\end{equation}\nand it is clear that the solution must be separable in $u$ and $\\omega$. \nThis clearly defines a ``brick'' structure valid for all $u$ (and which again has a cut-off at $\\tau\/\\sqrt{2}\\pi$).\nThus a wedge is precluded for any radially smooth layout.\n\nWe note that this was determined for arbitrary angular density $\\rho_\\theta$. \n\n\\ifanalytic\n\\section{Numerical Integration Algorithm}\n\\label{app:numerical_integration}\nTo determine the expected power spectrum for an arbitrary layout (as is done in \\S\\ref{sec:mitigation:arrays}) requires performing the triple-integral implicit in the combination of Eqs. \\ref{eq:power_general} and \\ref{eq:general_var}.\nThere are several difficulties in doing so, because all integrals must be performed numerically (in general).\n\nThe first difficulty is that the $\\theta$ integral must have zero absolute error tolerance. \nThis is due to the fact that it is normalised by a similar $\\theta$ integral over the weight function. \nSince the absolute value of either integral may be arbitrarily small, a small but constant absolute error will be magnified, and results in artificial vertical stripes in the 2D PS. \nConsequently, the efficiency of the procedure is highly reduced.\n\nThe second difficulty is that the $f$-integrals are highly oscillatory when $\\omega$ is large.\nThese then also must be performed with high precision, and often by breaking the integral into independent chunks. \nEven so, the authors have not been able to find a numerical integration scheme which yields acceptable results at high $\\omega$, and this can result in ``negative'' power at some grid-points. \nThis can be partially overcome by making a change of variable $x = f-f'$ so that only one integral contains the oscillations, rather than two. \nThis increases efficiency by tens of percent, but does not allow accurate computation to arbitrarily high $\\omega$.\n\nWith these considerations, the efficiency of the integration is very poor indeed --- for a layout of $\\sim10000$ baselines, some grid-points can take many hours to days to complete. \nClearly this becomes infeasible when multiple arrays with multiple grid-points are required. \n\nTo increase performance, we perform several optimizations. \nFirst, we quickly reduce the number of baselines required to be summed over for a given $(u, \\omega)$ by determining their weight, $W(u)$, and culling all baselines whose weight is smaller than some threshold by the mean. \nIf a single baseline remains, we simply return the sparse-layout solution, Eq.~\\ref{eq:sparse_solution} which shortcuts the process.\n\nSecond, we increase the \\textit{relative} error tolerance on the $\\theta$ to $10^{-3}$. Doing so will yield a final result which is also nominally accurate to 0.1\\%.\n\nA more aggressive solution to this performance issue is described in App.~\\ref{app:approx_variance}, involving analytic approximations of the integral itself.\n\n\n\\section{An Approximate Solution For The Variance}\n\\label{app:approx_variance}\nIn this appendix we derive an approximate analytical solution for the variance in our fiducial case of static beam and stochastic sky (cf. Eq. \\ref{eq:wg_master}).\nThe approximation we make is to replace the weighted gridding with a nearest-baseline gridding. \nThat is, instead of evaluating the visibility at a given grid point $\\vect{u}$ as the weighted average of visibilities at surrounding baselines, we assume that the \\textit{closest} baseline will contribute the dominant weight \\textit{for a given frequency}, and we neglect the rest of the terms in the sum.\nThis is a reasonable approximation to make, since if any other terms are co-dominant, they must be very close to the dominant point, and their visibility will be very similar anyway.\nIt is least accurate when two baselines are a similar distance from the grid-point, but in opposite directions. \nHowever, such a case will not occupy a large fraction of frequency space, and therefore its effect should be limited.\n\nAssumed in this setup is the fact that different baselines could be the dominant contributors at different frequencies. \nNeglecting this point results in the solutions of App. \\ref{app:delay}, which cannot avoid a wedge. \n\nThese assumptions lead to the frequency-space integral being split into a sum of terms containing a single baseline each, which is the closest to $\\vect{u}$ for that range of frequencies.\nThe frequency range for each term will be labeled $(f_i, f_{i+1})$, and the closest baseline will be labeled $\\vect{u}_i$.\n\n\\subsection{Variance at a grid point}\nUsing Eq. \\ref{eq:wg_master} as a starting point, we ignore the sum over baselines within the integral, and first perform the $\\vect{l}$-integral to achieve\n\\begin{align}\n\t{\\rm Var}(\\tilde{V}) = \\mu_2 \\nu_0^2 \\sum_{ij} &\\int_{f_i}^{f_{i+1}} \\int_{f_j}^{f_{j+1}}  df df' \\phi_\\nu \\phi_{\\nu'} \\\\ \\nonumber\n\t& \\times e^{-2\\pi i \\omega (f-f')} e^{-2\\pi^2 \\sigma^2 (f\\vect{u}_i - f'\\vect{u}_j)^2}.\n\\end{align}\nWe now use a change of variables: $x = f- f'$ to get\n\\begin{align}\n{\\rm Var}(\\tilde{V}) = \\mu_2 \\nu_0^2 \\sum_{ij} &\\int_{f_i - f'_j}^{f'_i - f_j}dx\\  e^{-2\\pi i \\omega x} \\\\ \\nonumber\n& \\int_{f_i - x}^{f'_i - x}  df' \\ e^{-\\tau^2 ((x+f'-1)^2 + (f'-1)^2)} \\\\ \\nonumber \n& \\times e^{-2\\pi^2 \\sigma^2 ((x+f')\\vect{u}_i - f'\\vect{u}_j)^2}.\n\\end{align}\nLetting\n\\begin{align}\n\tp_{ij}^2 &= 2\\tau^2 + \\pi^2 \\sigma^2 (\\vect{u}_i - \\vect{u}_j)^2 \\\\\n\tr_{ij} &= \\tau^2 + \\pi^2\\sigma^2 \\vect{u}_i (\\vect{u}_i - \\vect{u}_j),\n\\end{align}\nwe can perform the $f'$-integral ($I_i$) simply, resulting in\n\\begin{align}\n\tI^{(')}_{ij} = \\frac{\\sqrt{\\pi}}{2p} e^{r_{ij}^2 x^2\/p_{ij}^2} {\\rm erf}\\left[\\frac{r_{ij}-p_{ij}^2}{p}x + p_{ij}f^{(')}_i\\right]\n\\end{align}\nwhen evaluated at a given boundary.\nWe perform another change of variables:\n\\begin{align}\n\tz &= \\frac{r-p^2}{p}x + pf_i \\\\\n\tdx &= \\frac{p}{r-p^2} dz,\n\\end{align}\nto find the variance is\n\\begin{align}\n{\\rm Var}(\\tilde{V}) &= \\mu_2 \\nu_0^2 \\sum_{ij} \\mathbb{V}_{i'}^j - \\mathbb{V}_{i}^j, \\ \\ \\ {\\rm with} \\nonumber \\\\\n \\mathbb{V}_i^j &= \\frac{\\sqrt{\\pi}}{2(r-p^2)}\\int_{z_{ij}}^{z'_{ij}}dz\\  \\exp\\left[-2\\pi i \\omega\\left(\\frac{zp - p^2f_i}{r-p^2}\\right)\\right] \\nonumber \\\\\n & \\times \\exp\\left[-\\left(\\frac{zp - p^2f_i}{r-p^2}\\right)^2(\\tau^2 + \\pi^2\\sigma^2 u_i^2 - \\frac{r^2}{p^2})\\right] \\nonumber \\\\  &\\times \\exp\\left[ 2\\tau^2\\left(\\left(\\frac{zp-p^2f_i}{r-p^2}\\right) -1\\right)\\right] {\\rm erf}(z), \n\\end{align}\nnoting that the upper limit term $i'$ applies to all $f_i$ \\textit{within the integrand} and setting\n\\begin{align}\n\tz_{i,j} &= \\frac{r-p^2}{p}(f_i - f'_j) + pf_i, \\\\\n\tz'_{i,j} &= \\frac{r-p^2}{p}(f'_i - f_j) + pf_i.\n\\end{align}\n.\n\nWe now expand ${\\rm erf}(z)$ in its Maclaurin series, noting that this series always converges\\footnote{For large $z$, the number of terms required for convergence is high, however these terms should be adequately suppressed by other factors in the integral to make this a viable procedure.}.\nSetting the following variables:\n\\begin{align}\n\tt^2 &= \\tau^2 + \\pi^2 \\sigma^2 u_i^2 - r^2\/p^2 \\\\\n\ta^2 &= \\left(\\frac{pt}{r-p^2}\\right)^2 \\\\\n\tb_i &= \\frac{2p^3 f_i t^2}{(r-p^2)^2} + \\frac{2\\pi i \\omega p}{r-p^2} \\\\\n\tc_i &= 2\\tau^2 \\frac{p^2(1 - f_i) - r}{r-p^2} - \\frac{2\\pi i \\omega p^2 f_i}{r-p^2} - \\left(\\frac{p^2}{r-p^2}\\right)^2 f_i^2 t^2,\n\\end{align}\nwe have\n\\begin{align}\n\t\\mathbb{V}_i^j &= \\frac{1}{r-p^2} \\int_{z_0}^{z_1}dz\\  e^{-a^2 z^2 + b z + c } \\sum_{n=0}^\\infty \\frac{(-1)^n z^{2n+1}}{n!(2n+1)}.\n\\end{align}\n\nCompleting the square in the exponent, and shifting $z$ such that $z \\rightarrow z - b\/2a^2$,\nwe find\n\\begin{align}\n\\label{eq:full_variance_sblpf}\n\\mathbb{V}_i^j &= \\frac{e^{b^2\/4a^2 + c}}{r-p^2}  \\sum_{n=0}^\\infty \\frac{(-1)^n}{n!(2n+1)} \\sum_{k=0}^{2n+1}\\binom{2n+1}{k}  \\left(\\frac{b}{2a^2}\\right)^{2n+1-k} \\nonumber \\\\\n& \\times\\left[-\\frac{1}{2} z^{k+1} \\left(\\frac{1}{|az|}\\right)^{k+1} \\Gamma\\left(\\frac{k+1}{2}, a^2 z^2\\right)\\right|^{z'_{ij} - b\/2a^2}_{z_{ij} - b\/2a^2}.\n\\end{align}\nWe note that the solution is the sum of real parts of $\\mathbb{V}$, because the imaginary parts will cancel in the summation when swapping $i$ and $j$.\n\n\\subsection{Circular Average}\nThe solution of the previous subsection needs to be angularly averaged to yield the power at $u$. \nThe angular dependence enters both through the determination of contributing baselines, $\\vect{u}_i$, and the relative weight at each point.\n\nThe latter is computed as \n\\begin{equation}\n\tW(\\vect{u}) = \\int df \\phi_\\nu w_\\nu(\\vect{u} - f\\vect{u}_i).\n\\end{equation}\nAgain, we break the $f$ integral into independent sections, to yield\n\\begin{align}\n\t\tW(\\vect{u}) &= \\sum_{i} \\int_{f_i}^{f'_i}  df \\phi_\\nu w_\\nu(\\vect{u} - f \\vect{u}_i) \\\\ \\nonumber\n\t\t&= \\sum_i  \\frac{\\sqrt{\\pi}}{2p_i}\\exp\\left(-\\frac{2\\pi^2\\sigma^2\\tau^2d_i^2}{p_i^2}\\right) {\\rm erf}\\left[\\frac{p^2}{p_i^2} - f p_i \\right|^{f'_i}_{f_i},\n\\end{align}\nwith\n\\begin{align}\n\t\\vect{d} &= \\vect{u} - \\vect{u}_i \\\\\n\tp_i^2 &= \\tau^2 + 2\\pi^2 \\sigma^2 u_i^2.\n\\end{align}\nThe integration over $\\theta$ only occurs for each baseline as far as its contribution allows, and cannot be written fully analytically without reference to the layout of all baselines. \n\n\\subsection{Determination of contributing baselines}\nEvaluation of the power in this solution will require numerically summing the terms in the equations (though these sums should be significantly faster than performing a full 3D integration).\nTo accomplish this, precise integration limits must be derived for each term, along with the corresponding contributing baseline. \nThis can in principle be done both in frequency and angle, however we opt to limit ourselves to the frequency limits, as the angular limits do not offer a great deal in terms of computational performance.\n\nOur procedure for determination of the frequency limits in the general case is as follows:\n\\begin{enumerate}\n\t\\item Evaluate $W_i(u) = \\int d\\theta W_i(\\vect{u})$ for each baseline $\\vect{u}_i$ and retain only those whose contribution is greater than $10^{-t} \\bar{W}(u)$. \n\t\\item Determine at what frequency each pair of baselines is equidistant from $\\vect{u}$, saved as matrix $f^{ij}_{\\rm eq}$.\n\t\\item Determine the closest baseline to $\\vect{u}$ at $f = f_{\\rm min}$, with $f_{\\rm min}$ suitably low such that the bandpass\/taper renders it ignorable. Set $f_0 = f_{\\rm min}$ and $\\vect{u}_0$ to this baseline.\n\t\\item Until $f\\geq f_{\\rm max}$:\n\t\\begin{enumerate}\n\t\t\\item $j = {\\rm argmin}(f_{\\rm eq}^{ij} > f_{k-1})$\n\t\t\\item $f_k = f_{\\rm eq}^{ij}$\n\t\t\\item $\\vect{u}_k = \\vect{u}_j$\n\t\t\\item $i=j$\n\t\t\\item $k = k+1$\n\t\\end{enumerate}\n\\end{enumerate}\n\nThe equidistant frequencies are computed as \n\\begin{equation}\n\tf^{ij}_{\\rm eq} \\equiv \tf^{ji}_{\\rm eq} = \\left|\\frac{2(\\vect{u}\\cdot(\\vect{u}_i - \\vect{u}_j))}{u_i^2 - u_j^2}\\right|, \\ \\ i\\neq j.\n\\end{equation}\n\n\n\\section{Semi-analytic logarithmic spoke solution}\n\\label{app:logsolution}\nIn this appendix we derive a semi-analytic solution to the case of a logarithmic spoke layout in the high-$u$ limit. \nThis case is partially solved in \\S\\ref{sec:weighted}, for the limits in which the distances between baselines are either very large or very small. \nHowever, all realistic cases lie between these limits, and it is here that we focus in this subsection.\n\nThe logarithmic spoke layout ensures that the single-baseline-per-frequency limit is obtained for $u^2 \\gg 1\/\\pi^2\\sigma^2 \\Delta^2$, which exists for every array.\nThis is the limit explored in detail in App. \\ref{app:approx_variance}, and so we can use those results here.\n\nWe note further that at sufficiently high $u$, and for a sufficiently low number of radial spokes, we can ignore the angular component of the baselines and interpret Eq. \\ref{eq:full_variance_sblpf} directly as the power spectrum at $(u, \\omega)$.\n\nFor simplicity (and without sacrificing a great deal of accuracy), we will consider a point $u$ which is co-located with a baseline $u_i$ at $f=1$. \nFurthermore, we will consider an infinite array, such that the baselines grow arbitrarily small and large.\nDenoting the co-located baseline by the index 0, and larger baselines with negative indices (and vice versa), we can explicitly write the frequency limits within which a given baseline $i$ singularly contributes:\n\\begin{subequations}\n\t\\begin{align}\n\tf_i &= \\frac{1}{(1+\\Delta)^{i}(1+\\Delta\/2)}, \\\\\n\tf'_i &= \\frac{1}{(1+\\Delta)^{i-1}(1+\\Delta\/2)} \\equiv f_i(1+\\Delta).\n\t\\end{align}\n\\end{subequations}\n\nGiven that our primary point of focus is the wedge structure, we investigate the limit $\\pi\\sigma u \\gg \\tau$, which is where we expect the wedge to emerge.\nIn this case, several simplifications can be made. Firstly, we have \n\\begin{align}\n\tp^2 &\\approx \\pi^2\\sigma^2 u^2 \\left[(1+\\Delta)^i - (1+\\Delta)^j\\right] \\\\\n\tr &\\approx \\pi^2\\sigma^2 u^2 \\left[(1+\\Delta)^{2i} - (1+\\Delta)^{i+j}\\right] \\nonumber \\\\\n\t\\frac{r-p^2}{p} &\\approx \\pi \\sigma u \\frac{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}{(1+\\Delta)^i - (1+\\Delta)^j}.\n\\end{align}\n\nWe also have \n\\begin{align}\n\tt^2 &\\approx \\pi^2 \\sigma^2 u_i^2 \\\\\n\ta^2 &\\approx \\left(\\frac{u_i\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]}{u\\left[(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}\\right]}\\right)^2 \\\\\n\tb_i &\\approx 2\\frac{f_i t^2}{\\pi\\sigma u\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]} \\left(\\frac{(1+\\Delta)^i - (1+\\Delta)^j}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}\\right)^2  \\nonumber \\\\\n\t& + \\frac{2 i \\omega}{\\sigma u} \\left(\\frac{(1+\\Delta)^i - (1+\\Delta)^j}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}\\right) \\\\\n\tc_i &\\approx -2\\pi i \\omega f_i \\frac{\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]^2}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}} \\nonumber \\\\\n\t& - \\left[\\frac{\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]^2}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}\\right]^2 f_i^2 \\pi^2 \\sigma^2 u_i^2.\n\\end{align}\n\\fi\n\n\\section{Description of Numerical Techniques}\n\t\\label{app:numerical}\n\tThe gridding of visibility data and its transformation into an averaged power spectrum are non-trivial tasks that can require a significant computational effort.\n\tIn this appendix we describe the simple method we have taken in this work to accelerate this process and ensure its accuracy.\n\t\n\tNaively, the application of Gaussian weights from each baseline $\\vect{u}'_j$ to a particular point of interest $\\vect{u}_i$ is an order $N\\times M$ calculation (where $N$ is the number of baselines, and $M$ the number of grid-points at which to evaluate the power spectrum).\n\tSeveral standard algorithms can reduce this calculation to of order $N\\log M$.\n\tThe most popular is to use an FFT-backed convolution.\n\tHowever, we do not choose this route, as it requires the  $\\vect{u}_i$ to be arranged on a regular Cartesian grid, which has its own difficulties in terms of angular averages and dynamic range.\n\t\n\tInstead, we use a KD-tree algorithm (from the \\textsc{scikit-learn} Python package) to efficiently determine the baselines within a given radius of every $\\vect{u}_i$, and apply the weights\n\tfrom only these baselines. \n\tThe radius can be arbitrarily set, based on the beam width.\n\tThis allows the $\\vect{u}_i$ to be placed arbitrarily. \n\tSince we require an angular average, it is most convenient to choose the $\\vect{u}_i$ in a polar grid, so that the angular average is merely the average of a particular row in the array. \n\tThis has the dual benefits of simplicity and accuracy -- the average is specified at a particular magnitude of $q$, rather than an average over a complicated distribution of $q$ within an annulus. \n\t\n\tThis algorithm enables the numerical calculation of the 2D PS as an arbitrarily precise quantity.\n\tThat is, if the number of nodes in an angular ring is arbitrarily large, the operation exactly converges to the integral Eq. \\ref{eq:power_general}.\n\tIn practice then, if one simultaneously tests for convergence, this algorithm provides an exact non-gridding solution to the numerical calculation of the 2D PS.\n\tIn this paper we do not formally test for convergence, but rather simply use a number of angular nodes we deem to be sufficient to capture the integral adequately.\n\tIn real-world applications, the extension to formal convergence-monitoring is rather simple, and may provide for quite efficient accurate calculations of the 2D PS.\n\t\n\\end{appendix}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nRadio hotspots are bright and compact regions located at the end of\npowerful radio galaxies \n\\citep[FRIIs,][]{fr2} and\nconsidered to be the working surfaces of supersonic jets. In these\nregions,\nthe jet emitted by the active galactic nucleus\n(AGN) impacts on the surrounding ambient medium producing a shock that\nmay re-accelerate relativistic particles transported by the jet and\nenhance the radio emission. \nElectrons responsible for synchrotron emission in the\noptical band must be very energetic (Lorentz factor $\\gamma >\n10^{5}$),\nand therefore with short radiative lifetime.\nConsequently the detection of optical emission from hotspots\nsupports the scenario where the emitting electrons are accelerated\nat the hotspots, possibly by strong shocks generated by the impact of \nthe jet with the ambient medium \\citep{meise89,meise97,gb03}. \nThe detection of X-ray synchrotron counterparts of radio hotspots\n  would imply the\n  presence of electrons with even higher energies. \n  However the main radiation process\n  responsible for the X-ray emission seems to differ between high and\n  low luminosity hotspots \\citep{hardcastle04}. \n  In bright hotspots, like Cygnus A and 3C\\,295, the X-ray\n  emission is produced by synchrotron-self Compton (SSC) in the presence\n  of a magnetic field that is roughly in equipartition, while in \nlow-luminosity hotspots, like 3C\\,390.3, the emission at such high\n  energies is likely due to\n  synchrotron radiation \\citep{hardcastle07}.\\\\\nThe discovery of optical emission extended to kpc scale \nquestions the standard shock acceleration model, suggesting that \nother efficient\nmechanisms must take place across the hotspot region.\nAlthough it may seem an uncommon phenomenon due to the\ndifficulty to produce high-energy electrons on large scales, deep\noptical images showed that diffuse\noptical emission is present in a handful of hotspots: 3C\\,33, \n3C\\,111, 3C\\,303, 3C\\,351\n\\citep{valta99}, 3C\\,390.3 \\citep{aprieto97}, \n3C\\,275.1 \\citep{cheung05}, Pictor A \\citep{thomson95}, \nand 3C\\,445 \\citep{aprieto02}. \nA possible mechanism able to keep up the optical emission in the post-shock\nregion on kpc scale is a continuous, relatively efficient, stochastic \nmechanism\\footnote{More recently these stochastic\nmechanisms have been also proposed for\nthe acceleration of ultra-high energy cosmic-rays in the lobes of\nradiogalaxies \\citep{hardcastle09}.}.\\\\\nThe sample of low-power hotspots presented by \\citet{mack09} is\ncharacterized by low magnetic field strengths between 40 and 130\n$\\mu$G, a factor 2\nto 5 lower than that estimated in hotspots with optical counterparts\npreviously studied in the literature. A surprisingly high optical\ndetection rate ($\\geq$ 45\\%)\nof the hotspots in this sample was found, and in most cases \nthe optical counterpart extends on kpc scales. This\nis the case of 3C\\,445 South, 3C\\, 445 North, 3C\\,105 South and\n3C\\,227 West \\citep{mack09}.\\\\\nThis\npaper focuses on a multi-band, from radio to X-rays, \nhigh spatial resolution study of the two\nmost interesting cases among the low-luminosity hotspots from\n  \\citet{mack09}, 3C\\,105 South and 3C\\,445 South, in which the\nhotspot regions are resolved into multiple components. \n3C\\,105 is hosted by a narrow-line radio galaxy (NLRG) at redshift\n  $z=0.089$ \\citep{tadhunter93}. At this redshift 1$^{\\prime\\prime}$\n  corresponds to 1.642 kpc. The radio source 3C\\,105 is about\n  330$^{\\prime\\prime}$ (542 kpc) in size, and the hotspot complex 3C\\,105 South is\n  located about 168$^{\\prime\\prime}$ (276 kpc) from the core in the\n  south-east direction. 3C\\,445 is hosted by a broad-line radio galaxy\n  (BLRG) \nat redshift $z=0.05623$ \\citep{eracleous94}. At this redshift\n1$^{\\prime\\prime}$ corresponds to 1.077 kpc. The radio source 3C\\,445\nis about 562$^{\\prime\\prime}$ (608 kpc) in size, and the hotspot\ncomplex 3C\\,445 South is located 270$^{\\prime\\prime}$ (291 kpc)\nsouth of the core.\n\nThroughout this paper, we assume the following cosmology: $H_{0} =\n71\\; {\\rm km\/s\\, Mpc^{-1}}$, \n$\\Omega_{\\rm M} = 0.27$ and $\\Omega_{\\rm \\Lambda} = 0.73$,\nin a flat Universe. The spectral index\nis defined as \n$S {\\rm (\\nu)} \\propto \\nu^{- \\alpha}$.\\\\ \n\n\n\n\\section{Observations}\n\n\\subsection{Radio observations}\n\nVLA observations at 1.4, 4.8, and 8.4 GHz \nof the radio hotspots 3C\\,445 South and\n3C\\,105 South were carried out in July 2003 (project code AM772)\nwith the array in\nA-configuration. Each source was observed for about half an\nhour at each frequency, spread into a number of scans\ninterspersed with other source\/calibrator scans in order to improve\nthe $uv$-coverage. About 4 minutes were spent on the\nprimary calibrator 3C\\,286, while secondary phase calibrators\nwere observed for 1.5 min about every\n5 min. Data at 1.4 and 4.8 GHz were previously published by \\citet{mack09}.\nThe data reduction was carried out following the standard procedures\nfor the VLA implemented in the NRAO AIPS package.\nFinal images were produced after a few phase-only self-calibration\niterations. The r.m.s. noise level on the image plane is negligible if compared\nto the uncertainty of the flux density due to amplitude calibration\nerrors that, in this case, are estimated to be $\\sim$3\\%.\\\\\nBesides the {\\it full-resolution} images, we also produced {\\it\n  low-resolution} images at both 4.8 and 8.4 GHz, \nusing the same $uv$-range, image sampling and restoring beam of the\n1.4 GHz data. These new images were obtained with natural grid \nweighting in order to mitigate the differences in the sampling density \nat short spacing, and to perform a robust spectral analysis. \\\\\n\n\n\n\n\\subsection{Optical observations}\n\nFor both 3C\\,105 South and 3C\\,445 South, VLT high spatial resolution\nimages in standard filters taken with both ISAAC in J-, H-, K-, and FORS\nin I-, R-, B- and U- bands are used in this work. All the images have\nexcellent spatial resolutions in the range of 0.5$^{\\prime\\prime}$ $<$ FWHM $<$\n0.7$^{\\prime\\prime}$. Details on the observations and data reduction are given in\n\\citet{mack09}. The pixel scale of the ISAAC images is 0.14 arcsec\npixel$^{-1}$. In the case of the FORS images the pixel scale is 0.2\narcsec pixel$^{-1}$, with the exception of the I-band where it is\n  0.1 arcsec pixel$^{-1}$.\\\\ \nFurther HST observations on 3C\\,445 South only, were obtained with\nthe ACS\/HRC camera on 7th July, 2005  \nin the filters F814W (I-band, exposure time $\\sim$ 1.5 hr) and\nF475W (B-band, exposure time $\\sim$ 2.3 hr).\\\\  \nFor science analysis we used the ``*drz'' images delivered \nby the HST ACS pipeline. These final images are calibrated, \ncosmic-ray cleaned, geometrically corrected, and drizzle-combined, \nprovided in electrons per sec. The final pixel scale of the \ndrizzled images is 0.025$^{\\prime\\prime}$$\\times$0.025$^{\\prime\\prime}$ per pixel. \nThe flux calibration was done using the standard HST\/ACS procedure\nthat relies on the PHOTFLAM keyword in the respective image headers.\nThe quality of the pipeline-delivered images was adequate for the\npurposes of analyzing the hotspot region. \n\n\n\\begin{table*}\n\\caption{Radio flux density and angular size of the hotspot\n  components. \\newline {\\it Note 1}: deconvolved angular sizes from a Gaussian\n  fit. \\newline {\\it Note 2}:\nthe angular sizes are derived from the lowest contour on the image\nplane; \\newline\n{\\it Note 3}: The diffuse emission is estimated by subtracting\nthe flux density of SW and SE from the total flux density (see Section\n5.3).}\n\\begin{center}\n\\begin{tabular}{|c|r|r||r|r|r|r|r|r|}\n\\hline\nSource&Comp.&z&scale&S$_{1.4}$&S$_{4.8}$&S$_{8.4}$&$\\theta_{\\rm maj}$&$\\theta_{\\rm min}$\\\\\n & & &kpc\/$^{\\prime\\prime}$ \n&mJy&mJy&mJy&arcsec&arcsec\\\\\n\\hline\n&&&&&&&&\\\\\n3C\\,105&S1\\footnotemark[1]&0.089&1.642&130$\\pm$10&67$\\pm$5&45$\\pm$5&1.0&0.8\\\\\n &S2\\footnotemark[1]& & &1250$\\pm$40&620$\\pm$20&460$\\pm$15&1.30&1.0\\\\\n &S3\\footnotemark[1]& & &1180$\\pm$35&510$\\pm$15&320$\\pm$12&1.5&0.8\\\\\n &Ext& & &174$\\pm$10&75$\\pm$5&50$\\pm$3& &  \\\\\n3C\\,445&SE\\footnotemark[2]&0.0562&1.077&290$\\pm$30&98$\\pm$15&65$\\pm$10&3.5&1.0\\\\\n &SW\\footnotemark[2]& & &220$\\pm$25&51$\\pm$10&36$\\pm$6&1.5&0.5\\\\\n &Diff\\footnotemark[3]& & & & &13.0$\\pm$1.1& &  \\\\\n&&&&&&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center} \n\\label{tab_flux_rad}\n\\end{table*}\n\n\\subsection{X-ray observations}\n\n\nThe radio source \n3C\\,105 was observed by {\\it Chandra} on 2007 December 17 (Obs ID 9299)\nduring ``The {\\it Chandra} 3C Snapshot Survey for Sources with z$<$0.3''\n\\citep{massaro10}.  \nAn $\\sim$8 ksec \nexposure was obtained with the ACIS-S camera, operating in \nVERY FAINT mode.\nThe data analysis was performed following the standard\nprocedures described in the {\\it Chandra} Interactive Analysis of\nObservations (CIAO) threads and using the CIAO software package v4.2 \n(see Massaro et al. 2009 for more details). The {\\it Chandra} Calibration\nDatabase (CALDB) version 4.2.2 was used to process all files.  \nLevel 2 event files were generated using the $acis\\_process\\_events$ task,\nafter removing the hot pixels with $acis\\_run\\_hotpix$.  Events were\nfiltered for grades 0,2,3,4,6, and we removed pixel randomization.\\\\\n3C\\,445 South was observed by {\\it Chandra} on 2007 October 18\n\\citep{perlman10}, ACIS chip S3, with an exposure time of 45.6 ksec. \nThe data were retrieved from the archive and\nanalysed following the same procedure as for 3C\\,105 South. This\nre-analysis was necessary in order to achieve a proper alignment with\nthe radio data. \\\\\nWe created 3 different flux maps in the soft, medium, and hard X-ray bands\n(0.5 -- 1, 1 -- 2, and 2 -- 7 keV, respectively) by dividing the data with\nmonochromatic exposure maps with nominal energies = 0.8 keV (soft),\n1.4 keV (medium), and 4 keV (hard).\nBoth the exposure maps and the flux maps were regridded to a \npixel size of 0.25 the size of a native ACIS pixel\n(native=0.492$^{\\prime\\prime}\\times0.492^{\\prime\\prime}$).  To obtain\nmaps with brightness units of ergs~cm$^{-2}$~s$^{-1}$~pixel$^{-1}$, we\nmultiplied each event by the nominal energy of its respective band.\\\\\nFor 3C\\,445 South, we measured a flux density consistent with what reported by\n\\citet{perlman10}. The flux density was extracted from {\\it Chandra}\nACIS-S images in which the hotspot was placed on axis. \nBoth hotspots have been detected also by {\\it Swift} in the energy range\n0.3-10 keV (See Appendix A). This is remarkable given {\\it\n  Swift}'s survey operation mode and its poor spatial resolution. The\ndetection level is about 7$\\sigma$ and 12$\\sigma$ for 3C\\,105 South\nand 3C\\,445 South, respectively. However, given the large {\\it Swift}\nerrors in the counts-to-flux conversion and its low angular\nresolution, \nwe do not provide any further\nflux estimate.\\\\\n\n\\subsection{Image registration}\n\nThe alignment between radio and optical images was done by the\nsuperposition of the host galaxies with the nuclear component of the\nradio source using the AIPS task LGEOM. This results in a shift of\n3.5$^{\\prime\\prime}$. \nTo this purpose, the optical images\nwere previously brought on the same grid, orientation and coordinate system \nas the radio images by\nmeans of the AIPS task CONV and REGR \\citep[see\n  also][]{mack09}. \nThe final overlay of radio and optical images is\naccurate to 0.1$^{\\prime\\prime}$.\\\\\nFor 3C\\,105 South the X-ray image has been aligned with the radio one\nby comparing the core position. Then, the final\noverlay of X-ray contours on the VLT image \nis accurate to 0.1$^{\\prime\\prime}$. In the case of 3C\\,445 the\nshape of the nucleus of the galaxy is badly distorted in the {\\it Chandra} image\nbecause of its location far off axis of {\\it Chandra}.\nThe alignment was then\nperformed using three background sources visible both in X-ray and B\nband, and located around the hotspot. The achieved accuracy with \nthis registration is better than 0.15 arcsec, allowing us to confirm \na shift of about 2$^{\\prime\\prime}$ in declination between the X-rays and\nB-band emission centroids, the X-ray one being the closest to the core\n(Fig. \\ref{fig_3c445}).\\\\  \n\n\n\n\\begin{table*}\n\\caption{Near infrared, optical flux density and X-ray (0.5 - 7 keV) \nflux of hotspot components. In the case of 3C\\,445 the X-ray flux \nis not associated to any of the two\nmain components. The X-ray flux reported refers the total emission measured\non the whole hotspot region. \\newline {\\it Note 1}: units in\n10$^{-15}$ erg cm$^{-2}$ s$^{-1}$; \\newline\n{\\it Note 2}: the X-ray value, in $\\mu$Jy, is from\n\\citet{perlman10}; \\newline\n{\\it Note 3}: The diffuse emission is inclusive of the SC\ncomponent and it is estimated by subtracting from\nthe total flux density those arising from SW and SE (see Section 5.3).} \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nSource&Comp.&S$_{\\rm K}$&S$_{\\rm\n  H}$&S$_{\\rm J}$&S$_{\\rm I}$&S$_{\\rm R}$&S$_{\\rm B}$&S$_{\\rm\n  U}$&S$_{\\rm I}^{\\rm HST}$&S$_{\\rm B}^{\\rm HST}$&S$_{X}$\\\\\n & &$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&\\\\\n\\hline\n&&&&&&&&&&&\\\\\n3C\\,105&S1&4.6$\\pm$0.9&4.4$\\pm$1.1&$<$2.5&- &0.5$\\pm$0.1\n&0.2$\\pm$0.1&-&-&-& 7.5$\\pm$2.4\\footnotemark[1]\\\\\n &S2&18.4$\\pm$1.4&12.3$\\pm$1.1&3.4$\\pm$1.0&-&0.7$\\pm$0.1&0.2$\\pm$0.1&-&-&-&\n $<$2.0\\footnotemark[1]\\\\\n &S3&31.9$\\pm$2.8&25.7$\\pm$2.9&4.4$\\pm$1.8&-&0.9$\\pm$0.1&0.3$\\pm$0.1&-&-&-&\n 3.2$\\pm$1.6\\footnotemark[1]\\\\\n &Ext&15.4$\\pm$2.0&5.4$\\pm$2.0&-& -&0.4$\\pm$0.1&0.2$\\pm$0.1&-&-&-& -\\\\\n3C\\,445&SE&8.0$\\pm$1.0&5.6$\\pm$2.0&6.0$\\pm$1.5&2.0$\\pm$0.2&1.3$\\pm$0.2&0.7$\\pm$0.1&0.5$\\pm$0.3&1.7$\\pm$0.2&1.5$\\pm$0.3&9.38$\\times$10$^{-4}$\n\\footnotemark[2]\\\\\n &SW&4.6$\\pm$1.4&3.6$\\pm$1.5&3.0$\\pm$0.4&1.7$\\pm$0.3&1.4$\\pm$0.1&0.7$\\pm$0.1&\n0.5$\\pm$0.2&1.4$\\pm$0.1&0.3$\\pm$0.1&-\\\\\n &SC&- &- &- & - &0.8$\\pm$0.1&0.6$\\pm$0.1&0.4$\\pm$0.1&- & - & -\\\\\n &Diff\\footnotemark[3]& -\n&2.1$\\pm$0.6&3.2$\\pm$1.3&1.2$\\pm$0.2&1.0$\\pm$0.2&0.8$\\pm$0.2& - \\\\\n&&&&&&&&&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center} \n\\label{tab_flux_opt}\n\\end{table*}\n\n\n\\section{Photometry}\n\nTo construct the spectral energy distribution (SED) of individual hotspot\ncomponents, the flux density at the various wavelengths must be\naccurately measured in the same region, avoiding \ncontamination from unrelated features. \nTo this purpose, we produced a cube where each plane consists of radio\nand optical images regridded to the same size and smoothed to the same\nresolution. Then the flux\ndensity was derived by means of AIPS task BLSUM which performs an\naperture integration on a selected polygonal region common to all the\nimages. The values derived in this way were then used to construct the\nradio-to-optical SED, and they are reported in\nTables 1 and 2.\\\\\nIn addition to the low-resolution approach, \nwe derive the hotspot flux densities and\nangular sizes on the full resolution images, in order to better\ndescribe the source morphology.\n\nOn the radio images, we estimate the flux density of each component \nby means of TVSTAT, which is similar to BLSUM, but instead of working\non an image cube it works on a single image. The angular size was\nderived from the lowest contour on the image plane, and it\ncorresponds to roughly twice the size of the full width half maximum\n(FWHM) of a conventional Gaussian covering a similar area. \nIn the case of 3C\\,105 South, the hotspot components are unresolved\nat 1.4 GHz, and we derive the flux density at this frequency by means of\nAIPS task JMFIT, which performs a Gaussian fit in the image plane. \nThe angular size was measured on the images in\nwhich the components were resolved, i.e. in the case of 3C\\,105\nSouth we use the 4.8 and 8.4-GHz images, which provide the same value, while\nfor 3C\\,445 South the components could be reliably resolved in the\nimage at 8.4 GHz only (Table 1).\\\\\nFull-resolution infrared and optical flux densities of hotspot sub-components \nwere measured by means of the IDL-based task ATV using \na circular aperture centred on each component.\nSuch values were compared to those derived from the analysis of the\ncube and they were found to be within the expected uncertainties.\\\\\nFor the X-ray flux we constructed photometric apertures \nto accommodate the {\\it Chandra}\npoint spread function and to include the total extent of the\nradio structures.\nThe background regions, with a total area typically twice that of the\nsource region, have been selected close to the source, and\ncentred on a position where other sources or extended structures are\nnot present. The X-ray flux was measured\nin any aperture with only a small correction for the\nratio of the mean energy of the counts within the aperture to the\nnominal energy for the band. \nWe note that in 3C\\,105 South,\nthe hotspot components \nare well separated (2$^{\\prime\\prime}$), allowing us to accurately\nisolate the corresponding X-ray emission. In\n3C\\,445 South the X-ray emission is not associated with the two main\ncomponents clearly visible in the radio and optical bands, and \nflux was derived by using an aperture large enough to include all of the X-ray\nemission extending over the entire hotspot region. Our estimated value\nis in agreement with the one reported by \\citet{perlman10}.\nAll X-ray flux densities have been corrected for the\nGalactic absorption with the column density N$_H$ =\n1.15$\\cdot$10$^{21}$cm$^{-2}$ given by\n\\citet{kalberla05}. \nX-ray fluxes are reported in Table \\ref{tab_flux_opt}.\\\\  \n\n\\begin{figure*}\n\\begin{center}\n\\special{psfile=9pan_label1.ps voffset=-600 hoffset=-40\n  vscale=100 hscale=100 angle=0}\n\\vspace{16cm}\n\\caption{Multifrequency images of 3C\\,105 South. From the left to\n  right and top to bottom: Radio images at 1.4, 4.8, 8.4 GHz (VLA\n  A-array), NIR\/optical images in K, H, J, R, B bands (VLT), and X-ray 0.5-7\n  keV ({\\it Chandra}) contours. Each panel covers 9.5$^{\\prime\\prime}$\n  (15.6 kpc) in DEC and 14$^{\\prime\\prime}$ (23 kpc) in RA. In the\n  radio images the lowest contours are 0.9 mJy\/beam at 1.4 GHz, 0.20\n  mJy\/beam at 4.8 GHz, and 0.18 mJy\/beam at 8.4 GHz, and they\n  correspond to 3 times the off-source rms noise level measured on the\nimage plane. Contours increase by a factor of 4. The restoring beam is\n1.3$^{\\prime\\prime}$$\\times$1.1$^{\\prime\\prime}$ at 1.4 GHz,\n0.38$^{\\prime\\prime}$$\\times$0.36$^{\\prime\\prime}$ at 4.8 GHz, and  \n0.32$^{\\prime\\prime}$$\\times$0.22$^{\\prime\\prime}$ at 8.4 GHz. In the\noptical images the contour levels are in arbitrary units and increase\nby a factor of 2. The FWHM is about 0.4$^{\\prime\\prime}$,\n0.5$^{\\prime\\prime}$, 0.7$^{\\prime\\prime}$, 0.6$^{\\prime\\prime}$,\n0.7$^{\\prime\\prime}$ in K, H, J, R, and B band respectively.\nThe X-ray contours were generated from an 0.5-7 keV image, smoothed\nwith a Gaussian of FWHM=0.72$^{\\prime\\prime}$. Contour levels increase\nlinearly: 0.02, 0.04, 0.06,.. 0.14 counts per 0.123$^{\\prime\\prime}$ pixel. \nThe X-ray contours\nare superposed to the R band image, previously shifted as so to\nalign with X-ray.} \n\\label{fig_3c105}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\special{psfile=13pan445label_1.ps voffset=-350 hoffset=0\n  vscale=90 hscale=90}\n\\vspace{15cm}\n\\caption{Multifrequency images of 3C\\,445 South. From the left to\n  right and top to bottom: Radio images at 1.4, 4.8, 8.4 GHz (VLA\n  A-array), NIR\/optical images in K, H, J, I, R, B, U bands (VLT),\n  optical images in I and U bands (HST), \nand X-ray 0.5-7\n  keV ({\\it Chandra}) contours. Each panel covers 7.3$^{\\prime\\prime}$\n  (7.8 kpc) in DEC and 11.4$^{\\prime\\prime}$ (12.2 kpc) in RA. In the\n  radio images the lowest contours are 1.3 mJy\/beam at 1.4 GHz, 0.20\n  mJy\/beam at 4.8 GHz, and 0.10 mJy\/beam at 8.4 GHz, and they\n  correspond to 3 times the off-source rms noise level measured on the\nimage plane. Contours increase by a factor of 4. The restoring beam is\n1.43$^{\\prime\\prime}$$\\times$0.96$^{\\prime\\prime}$ at 1.4 GHz,\n0.45$^{\\prime\\prime}$$\\times$0.37$^{\\prime\\prime}$ at 4.8 GHz, and  \n0.24$^{\\prime\\prime}$$\\times$0.21$^{\\prime\\prime}$ at 8.4 GHz. In the\noptical images the contour levels are in arbitrary units and increase\nby a factor of 2. The VLT FWHM are 0.7$^{\\prime\\prime}$,\n0.6$^{\\prime\\prime}$, 0.5$^{\\prime\\prime}$, 0.7$^{\\prime\\prime}$,\n0.6$^{\\prime\\prime}$, 0.6$^{\\prime\\prime}$, 0.7$^{\\prime\\prime}$,\nin K, H, J, I, R, B, and U band respectively. In HST images\neach pixel is 0.025$^{\\prime\\prime}$.\nThe X-ray contours in the last panel are superposed on the B band\nimage. They come from an 0.5-7 keV image, smoothed with a Gaussian of\nFWHM=0.87$^{\\prime\\prime}$. Contour levels increase by a factor of 2;\nthe lowest contour is at a brightness of 0.01 counts per\n0.0615$^{\\prime\\prime}$ pixel.} \n\\label{fig_3c445}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Morphology}\n\n\\subsection{3C\\,105 South}\n\nThe southern hotspot complex of 3C\\,105 shows a curved\nstructure of about 8$^{\\prime\\prime}$$\\times$4.5$^{\\prime\\prime}$ \n($\\sim$13$\\times$7 kpc) in\nsize. It is dominated by three bright components, all resolved at\nradio frequencies,\nconnected by a low surface brightness emission also visible in\noptical and infrared (Fig. \\ref{fig_3c105}). \nThe central component, labeled S2 in Fig. \\ref{fig_3c105}, is the\nbrightest in radio and, when imaged with high spatial resolution, it\nis resolved in two different structures separated by about 1.2 kpc. \n\\citet{leahy97} interpreted \nthis as the true jet termination hotspot, while S1, with an elongated\nstructure of (1.6$\\times$1.3) kpc and located 5.7 kpc\nto the north of S2 is considered as jet emission. The southernmost\ncomponent S3,\nlocated about 4.1 kpc from S2, has a resolved structure of\n(2.4$\\times$1.3) kpc in size, and it is elongated in a direction\nperpendicular to the line leading to S2. Its morphology suggests that\nS3 is a secondary hotspot similar to 3C\\,20 East \\citep{cox91}.\\\\ \nAt 1.4 GHz, \nan extended tail\naccounting for $S_{\\rm 1.4} =$ 608 mJy \nand embedding the jet\nis present to the west of the hotspot complex, in agreement with the\nstructure previously found by \\citet{neff95}. \nAt higher frequencies the lack of the short spacings prevents\n  the detection of such an extended structure, and only a hint of the\njet,\naccounting for $S_{\\rm 4.8} \\sim 70$ mJy, is still\nvisible at 4.8 GHz. \\\\\nIn the optical and NIR the hotspot complex is characterized by\nthe three main components detected in radio. In NIR and optical, \nthe southernmost component S3\nis the brightest one, with a radio-to-optical spectral index\n  $\\alpha_{r-o}=$0.95$\\pm$0.10. It displays an elongated\nstructure rather similar in shape and size to that \nfound in radio. It is resolved in all bands with the only exception of \nB band, likely due to the lower spatial resolution \nachieved. Component S1 is resolved in all NIR\/optical\nbands, showing a tail extending towards S2. Its radio-to-optical\nspectral index is $\\alpha_{r-o}=$0.95$\\pm$0.10.\nOn the other hand, S2 appears unresolved in all bands, with the\nexception of K and H bands, i.e. those with the highest resolution\nachieved. In these NIR bands S2 is extended in the southern direction, \nresembling what is observed in radio. Its radio-to-optical\nspectral index is $\\alpha_{r-o}=$1.05$\\pm$0.10.\\\\\nDiffuse emission connecting the main hotspot\ncomponents and extending to the southwestern part of the hotspot\ncomplex is\ndetected in most of the NIR and optical images.\\\\\nIn the X-ray band, \nS1 is the brightest component, whereas the emission from \nS3 is very weak (formally detected at only 2$\\sigma$ level).\nFor this reason in the following we will use the\nnominal X-ray flux of S3 as a conservative upper limit. \nFor component S2 only an upper limit could be set. \n\n\n\\subsection{3C\\,445 South}\n\nThe hotspot 3C\\,445 South displays an extended east-west structure of\nabout 9.3$^{\\prime\\prime}$ $\\times$ 2.8$^{\\prime\\prime}$\n(10$\\times$3 kpc) in size in radio\n(Fig. \\ref{fig_3c445}). At 8.4 GHz, the hotspot complex is almost\ncompletely resolved out\nand the two main components, clearly visible in\nNIR\/optical images, are hardly distinguishable.\nWhen imaged with enough resolution, these components display an\narc-shaped structure both in radio and NIR\/optical bands,\nwith sizes of about (3.4$\\times$1.5) kpc and\n(2.1$\\times$1.1) kpc for SE and SW respectively.  \nComponent SE is elongated in a direction almost perpendicular to the\nline leading to the source core, while SW forms an angle of about\n-20$^{\\circ}$ with the same line.\\\\ \nIn radio and NIR, the \nSE component is the brightest one, with a flux density ratio\nSE\/SW $\\sim$ 1.6, while in the optical both\ncomponents have similar flux densities. Both components have a\nradio-to-optical spectral index $\\alpha_{r-o}=$  0.9$\\pm$0.10.\nIn the optical R-, B-, and U-band \nimages a third component (labelled SC in Fig. \\ref{fig_3c445}) \naligned with the jet direction becomes visible between SE\nand SW. \nDespite the good resolution and sensitivity of the radio and \nNIR images, SC is not present at such wavelengths.\nWhen imaged with the high\nresolution provided by HST, both SE and SW are clearly resolved, and\nno compact regions can be identified in the hotspot complex. Trace of\nthe SC component is seen in the B-band, in agreement\nwith the VLT images.\\\\\nIn the VLA and VLT images, \nthe two main components are enshrouded by a diffuse emission, visible\nin radio and NIR\/optical bands. \nThe flux densities of the SE and SW components measured on the HST images\nare consistent (within the errors) with those derived on the VLT images.\\\\ \nThe optical component W located about 2.8$^{\\prime\\prime}$ (3 kpc) \non the northwestern part of\nSW does not have a radio counterpart, as it is clearly shown by\n  the superposition of I-band HST and 8.4-GHz VLA images\n  (Fig. \\ref{hst_vla}), and thus it is considered an\nunrelated object, like a background galaxy. Another possibility\n  is that this is a synchrotron emitting region where the impact of the\n  jet produces very efficient particle acceleration. However, its steep\n  optical spectrum ($\\alpha \\sim 2$ between I and U\n  bands, see Section 5.3, Fig. \\ref{slope_3c445}) together with the\n  absence of detected radio emission disfavour this\n  possibility. Future spectroscopic information would further unveil \n  the nature of this optical region.\\\\\n{\\it Chandra} observations of 3C\\,445 South detected X-ray emission from a\nregion that extends over 6$^{\\prime\\prime}$ in the east-west direction\n(Fig. \\ref{fig_3c445}), and \nit peaks almost in the middle of the hotspot structure, suggesting a\nspatial displacement \nbetween X-ray and radio\/NIR\/optical emission \\citep{perlman10}.\\\\\n\\begin{figure}\n\\begin{center}\n\\special{psfile=hst_i-radio.eps voffset=-245 hoffset=0 vscale=33\n  hscale=33}\n\\vspace{8cm}\n\\caption{3C\\,445 South. 8.4-GHz VLA contours are superimposed on the I-band\nHST image.} \n\\label{hst_vla}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c105n_units_last.ps voffset=-230 hoffset=10 vscale=33 hscale=33}\n\\vspace{6.5cm}\n\\caption{The broad-band SED of the northern component, S1, of \n3C\\,105 South. The solid lines represent the synchrotron model where\n$\\nu_{\\rm b} =5 \\times 10^{12}$ Hz and $\\nu_{\\rm c} = 2 \\times\n10^{15}$ Hz, and the SSC\nmodels computed assuming a magnetic field of 50 and 150 $\\mu$G. The\nshort-dashed line represent a synchrotron model where $\\nu_{\\rm b} = 5\n\\times 10^{12}$ Hz, and $\\nu_{\\rm c} = \\infty$.\nThe long-dashed lines represent the IC-CMB models\ncomputed assuming B=16 (and B=32) $\\mu$G,\n$\\Gamma$=6 ($\\Gamma=4$), $\\theta$=0.1 ($\\theta=0.2$) rad,\nwith or without flattening in the observed synchrotron spectrum\nat $\\nu<$  60 MHz. The magnetic field is in the rest frame.}\n\\label{fig_spectra_105n}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c105c_units_last.ps voffset=-230 hoffset=10 vscale=33\n  hscale=33}\n\\vspace{6.5cm}\n\\caption{The broad-band SED of the central component, S2, of \n3C\\,105 South. The solid line represents the synchrotron model where\n$\\nu_{\\rm b} =7.5 \\times 10^{12}$ Hz and $\\nu_{\\rm c} = 3 \\times\n10^{14}$ Hz, and the SSC\nmodels computed assuming a magnetic field of 50 and 225 $\\mu$G. The\narrow indicates the X-ray upper limit.}\n\\label{fig_spectra_105c}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c105s_units_last.ps voffset=-230 hoffset=10 vscale=33\n  hscale=33}\n\\vspace{6.5cm}\n\\caption{The broad-band SED of the southern component, S3, of \n3C\\,105 South. The solid line represents the synchrotron model where\n$\\nu_{\\rm b} =1.5 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 3 \\times\n10^{14}$ Hz, and the SSC\nmodels computed assuming a magnetic field of 50 and 150 $\\mu$G. The\narrow indicates the X-ray upper limit.}\n\\label{fig_spectra_105s}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=Figura_445sw_new.ps voffset=-240 hoffset=10 vscale=33\n  hscale=33}\n\\vspace{7cm}\n\\caption{The broad-band SED of the western component, SW, of \n3C\\,445 South. The morphology from {\\it Chandra} image shows that\nX-rays are not associated with the western component.\nThe synchrotron models assume \n$\\nu_{\\rm b} =9.4 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 4.7 \\times\n10^{15}$ Hz ({\\it dotted line}), $\\nu_{\\rm b} =5.5 \\times 10^{13}$ Hz \nand $\\nu_{\\rm c} = 2.2 \\times10^{16}$ Hz ({\\it dashed line}), \n$\\nu_{\\rm b} =4.4 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 1.8 \\times\n10^{18}$ Hz ({\\it solid line}), $\\nu_{\\rm b} =4.4 \\times 10^{13}$ Hz\nand $\\nu_{\\rm c} = \\infty$ ({\\it thick solid line}). }\n\\label{fig_spectra_445w}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=Figura_445se_new.ps voffset=-240 hoffset=10 vscale=33\n  hscale=33}\n\\vspace{7cm}\n\\caption{The broad-band SED of the eastern component, SE, of \n3C\\,445 South. The morphology from {\\it Chandra} image shows that\nX-rays are not associated with the eastern component.\nThe synchrotron models assume \n$\\nu_{\\rm b} =5.2 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 2.6 \\times\n10^{15}$ Hz ({\\it dotted line}), $\\nu_{\\rm b} =2.4 \\times 10^{13}$ Hz \nand $\\nu_{\\rm c} = 9.4 \\times10^{15}$ Hz ({\\it dashed line}), \n$\\nu_{\\rm b} =1.2 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 4.7 \\times\n10^{17}$ Hz ({\\it solid line}), $\\nu_{\\rm b} =1.2 \\times 10^{13}$ Hz\nand $\\nu_{\\rm c} = \\infty$ ({\\it thick solid line}). }\n\\label{fig_spectra_445e}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c445_res_nuFnu_units.ps voffset=-240 hoffset=10 vscale=33\n  hscale=33}\n\\vspace{7cm}\n\\caption{The broad-band SED of the diffuse emission (see text) of \n3C\\,445 South. The morphology from the {\\it Chandra} image does not\nallow us to firmly exclude a connection between the X-rays and the\ndiffuse (including SC component) emission.\nThe synchrotron model assumes\n$\\nu_{\\rm b} =8 \\times 10^{16}$ Hz, $\\nu_{\\rm c} \\gg \\nu_{\\rm b}$ and $p$=2.7.}\n\\label{fig_spectra_445diff}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Spectral energy distribution}\n\n\\begin{table}\n\\caption{Synchrotron parameters. Column 1: Hotspot; Column 2:\n  component; Columns 3, 4: spectral index and break frequency as \nderived from the fit to the radio-to-optical SED (Section 5.1); \nColumn 5: equipartition magnetic field, computed following the\napproach presented in Brunetti et al. (2002); Column 6: radiative age\ncomputed using Eq. 2.} \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nSource&Comp.&$\\alpha$&$\\nu_{\\rm b}$&$B_{\\rm eq}$&$t_{\\rm rad}$\\\\\n & & &10$^{13}$ Hz&$\\mu$G&yr\\\\\n\\hline\n&&&&&\\\\\n3C\\,105&S1&0.8&0.50&150&12\\\\\n       &S2&0.8&0.75&290&4\\\\\n       &S3&0.8&1.5&270&3\\\\\n3C\\,445&SE&0.75&5.2&60&15\\\\\n       &SW&0.75&9.4&50&15\\\\\n&&&&&\\\\\n\\hline\n\\end{tabular}\n\\label{tab_fit}\n\\end{center}\n\\end{table}\n\n\\subsection{The broad-band energy distribution}\n\nWe model the broad band energy distribution, from radio to optical, of the\nhotspot regions in order to determine the mechanisms at the basis of\nthe emission. The comparison between the model expectation in the\nX-rays and {\\it Chandra} data sets additional\nconstraints.\nIn the adopted models, the\nhotspot components are described by homogeneous spheres with constant magnetic\nfield and constant properties of the relativistic electron\npopulations. \nThe spectral energy distributions of the emitting electrons are modelled\nassuming the formalism described in \\citet{gb02}. According to\n  this model a population of seed electrons (with $\\gamma \\leq\n  \\gamma_{*}$) is accelerated at the shock and is injected in the\n  downstream region with a spectrum dN($\\gamma$)\/dt $\\propto$\n  $\\gamma^{-p}$, for $\\gamma_{*} < \\gamma < \\gamma_{c}$, \n  $\\gamma_{c}$ being the maximum energy of the electrons accelerated at the\n  shock. Electrons accelerated at the shock are advected in the\n  downstream region and age due to radiative losses. Based on\n  \\citet{gb02}, the volume integrated spectrum of the electron\n  population in the downstream region of size $L \\sim T v_{\\rm adv}$\n  ($T$ and $v_{\\rm adv}$ being the age and the advection velocity of\n  the downstream region) is given by either a steep power-law $N$($\\gamma$)\n  $\\propto \\gamma^{-(p+1)}$ for $\\gamma_{b} < \\gamma < \\gamma_{c}$,\n  where $\\gamma_{b}$ is the maximum energy of the ``oldest'' electrons\nin the downstream region, or by $N$($\\gamma$)\n  $\\propto \\gamma^{-p}$ for $\\gamma_{*} < \\gamma < \\gamma_{b}$, or by\na flatter shape for $\\gamma_{\\rm low} < \\gamma < \\gamma_{*}$, where\n$\\gamma_{\\rm low}$ is the minimum energy of electrons accelerated at\nthe shock.\\\\\nAs the first step we fit the SED in the\nradio-NIR-optical regimes with a synchrotron model, and we derive the\nrelevant parameters of the synchrotron spectrum  \n(injection spectrum $\\alpha$, break frequency\n$\\nu_{\\rm b}$, cut-off frequency $\\nu_{\\rm c}$) \nand the slope of the energy distribution of the electron population\nas injected at the shock (p $= 2 \\alpha +1$). Since hotspots have\n  spectra with injection slope $\\alpha$ ranging between 0.5 and 1 (as\n  a reference, the classical value from the diffuse particle\n  acceleration at strong shocks is $\\alpha = 0.5$, e.g. Meisenheimer\n  et al. 1997), we decided to consider the injection spectral index as\n  a free parameter.\nSuch constraints allow us to determine the spectrum of the emitting\nelectrons (normalization, break and cut-off\nenergy), once the magnetic field strength has been assumed, and\nto calculate the emission from either synchrotron-self-Compton (SSC) \nor inverse-Compton scattering of the cosmic background radiation \n(IC-CMB) expected from the hotspot (or jet)\nregion \\citep[following][]{gb02}.\nModels described in \\citet{gb02} take also into account the boosting\neffects arising from a hotspot\/jet that is moving at relativistic\nspeeds and oriented at a given angle with respect to our line of\nsight.\\\\\n\n\\subsection{3C\\,105 South}\n\nIn Figures \\ref{fig_spectra_105n} to \\ref{fig_spectra_105s} \nwe show the SED \nfrom the radio band to high energy emission measured\nfor the hotspot components of 3C\\,105 South, together with the model fits.\nSynchrotron models with an injection spectral index $\\alpha$=0.8\nprovide an adequate representation of the SED\nof the central and southern components of\n3C\\,105 South, with break frequencies ranging from\n5$\\times$10$^{12}$ to 1.5$\\times$10$^{13}$ W\/Hz, while the cutoff\nfrequencies are between 3$\\times$10$^{14}$ and 2$\\times$10$^{15}$ W\/Hz.\nIn both components, the upper limit to the X-ray emission does\nnot allow us to constrain the validity of the SSC model\n(Figs. \\ref{fig_spectra_105c} and \\ref{fig_spectra_105s}).  \nOn the other hand, the northern component of 3C\\,105 shows a prominent\nX-ray emission.\nA synchrotron model (dashed line in Fig. \\ref{fig_spectra_105n}) may\nfit quite reasonably the radio, NIR and X-ray emission, but\nit completely fails in reproducing\nthe optical data. An additional contribution of the SSC \nis not a viable option since it requires a \nmagnetic field much smaller than that\nobtained assuming equipartition (see Section 5.4) (solid lines), and \nimplying an\nunreasonably large energy budget.\nOn the other hand, the high energy emission is well modelled by\nIC-CMB \\citep[e.g.][]{tavecchio00,celotti01} \nwhere the CMB photons are scattered by relativistic electrons with\nLorentz factor $\\Gamma \\sim 6$, and $\\theta$=5$^{\\circ}$ with a\nmagnetic field of 16 $\\mu$G. \nThis model\nimplies that boosting effects play an important role in the X-ray emission\nof this component, suggesting that S1 is more likely a relativistic\nknot in the jet, rather than a hotspot feature. The weakness of\n  this interpretation is that 3C\\,105 is a NLRG and its jets are\n  expected to form a large angle with our line of\n  sight.\nAlternatively,\n  the X-ray emission may be synchrotron from a different population of\nelectrons, as suggested in the case of the jet in 3C\\,273 (Jester et\nal. 2007).\\\\ \n\n\\subsection{3C\\,445 South}\n\nThe analysis of the southern hotspot of 3C\\,445 \nas a single unresolved component was carried out in previous work by\n\\citet{aprieto02,mack09,perlman10}. In this new analysis, \nthe high spatial resolution and\nmultiwavelength VLT and HST data of 3C\\,445 South allow us \nto study the SED of each\ncomponent separately in order to investigate in more detail the\nmechanisms at work across the hotspot region.\nIn Figures \\ref{fig_spectra_445w} to \\ref{fig_spectra_445diff} \nwe show the SED \nfrom the radio band to high energy emission measured\nfor the components of 3C\\,445 South, together with the model fits.\nWe must note that at 1.4 GHz the resolution is not\nsufficient to reliably separate the contribution from the two\nmain components. For this reason,\nwe do not consider the flux density at this frequency in constructing\nthe SED. The X-ray emission\n(Fig. \\ref{fig_3c445}) is misaligned with respect to the radio-NIR-optical\nposition. For this reason, on the SED of\nboth components (Figs. \\ref{fig_spectra_445w} and\n\\ref{fig_spectra_445e}) we plot the total X-ray flux which must be\nconsidered an upper limit. \nFor the components of 3C\\,445 South the synchrotron models with\n$\\alpha$=0.75 reasonably\nfit the data, providing break frequencies in the range of 10$^{13}$\nand $10^{14}$ W\/Hz, and cutoff frequencies from 10$^{15}$ Hz and\n10$^{18}$ Hz.\\\\\nBoth the morphology (Fig. \\ref{fig_3c445}) and the SED\n(Figs. \\ref{fig_spectra_445w} and \n\\ref{fig_spectra_445e}) indicate that the bulk of {\\it Chandra} X-ray\nemission detected in 3C\\,445 is not due to synchrotron emission from\nthe two components (Section 6).\\\\\nAs discussed in Section 4.2, diffuse IR and optical emission\n  surrounds the two components SE and SW of 3C\\,445 South, and a third\ncomponent, SC, becomes apparent in the optical. We attempt to evaluate the\nspectral properties of the diffuse emission (including component\nSC). When possible, depending on statistics, we subtract from the\ntotal flux density of the hotspot, the contribution \narising from the two main\ncomponents, obtaining in this way the SED of the diffuse emission\n(inclusive of SC component) of\n3C\\,445 South. In the image we also plot the total X-ray flux. \nAs expected the emission has a hard spectrum ($\\alpha\n\\sim 0.85$) without evidence of a break up to the optical band,\n10$^{15}$ Hz $<$ $\\nu_{b}$ $\\leq$ 8$\\times$10$^{16}$ Hz. We also note\nthat this hard component may represent a significant contribution of\nthe observed X-ray emission, although the X-ray peak appears shifted\n($\\sim$ 1$^{\\prime\\prime}$) from the SC component.\nDue to the extended nature of the emission in this hotspot, we\n  created a\npower-law spectral index map \nillustrating the change of the spectral index $\\alpha$\nacross the hotspot region (Fig. \\ref{slope_3c445}). \nThe spectral energy distributions presented in\nFigs. \\ref{fig_spectra_445w}, \\ref{fig_spectra_445e}, and\n\\ref{fig_spectra_445diff} show the\ncurvature of the integrated spectrum for the main\ncomponents and the diffuse emission (see Section 5.1). \nThe spectral map in Fig. \\ref{slope_3c445} attempts to provide\ncomplementary information on the spectral slope for the diffuse\ninter-knot emission. Extracting these maps using the largest \npossible frequency range is complicated as it implies combining images \nfrom different instruments with different scale sampling, \nnoise pattern, etc. These effects sum up to produce very \nlow contrast maps given the weakness of the hotspot signal. \nTo minimise these effects it was decided to extract the slope maps from\nthe optical and -IR images only.\\\\\nThe spectral index map between I- and U-band (Fig. \\ref{slope_3c445}) shows \ntwo sharp edges, at the SW and SE components, with the highest value \n$\\alpha \\sim  1.5 $. \nBetween these two main regions there \nis diffuse emission that is clearly seen\nin the I-\/U-band spectral index map. The slope of this \ncomponent is flatter than that of the two main regions \nand rather uniform all over the hotspot, with $\\alpha \\sim 1$.\\\\ \n\n\n\\subsection{Physical parameters}\n\nWe compute the magnetic field of each hotspot component by \nassuming minimum energy conditions,\ncorresponding to equipartition of energy between radiating\nparticles and magnetic field, \nand following the approach by \\citet{gb97}.\nWe assume for the hotspot components an ellipsoidal volume $V$ with a\nfilling factor $\\phi$=1 (i.e. the volume is fully and homogeneously\nfilled by relativistic plasma). \nThe volume $V$ is computed by means:\\\\\n\n\\begin{equation}\nV = \\frac{\\pi}{6} d_{\\min}^{2} d_{\\max}\n\\end{equation}\n\n\\noindent where d$_{\\min}$ and d$_{\\max}$ are the linear size of the\nminor and major axis, respectively.  \nWe consider $\\gamma_{\\rm min} =$100, \nand we assume that the energy densities\nof protons and electrons are equal. \nWe find equipartition \nmagnetic fields ranging from $\\sim$ 50 - 290 $\\mu$G (Table\n\\ref{tab_fit}) that is \nlower than those \ninferred in high-power radio hotspots \nwhich range from $\\sim$ 250 to 650 $\\mu$G\n\\citep{meise97, cheung05}. \nRemarkably, if we compare these results with those from \\citet{mack09},\nwe see that in 3C\\,445 South the value\ncomputed considering the entire source volume is similar to those obtained in\nits individual sub-components, suggesting that compact\nand well-separated emitting regions are not present in the hotspot volume. \nOn the other hand, the magnetic field\naveraged over the whole 3C\\,105 South hotspot complex is much smaller\nthan those derived in its sub-components.\\\\\nIn the presence of such low magnetic fields \nhigh-energy electrons may have longer radiative lifetime than\nin high-power radio hotspots. \nThe radiative age $t_{\\rm rad}$ is related to the\nmagnetic field and the break frequency by\\footnote{The magnetic field\n  energy density in these hotpots are at least an order of magnitude\n  higher than the energy density of the cosmic microwave background\n  (CMB) radiation. Inverse Compton\n  losses due to scattering of CMB photons\n  are negligible.}:\\\\\n\n\\begin{equation}\nt_{\\rm rad} = 1610 \\; B^{-3\/2} \\nu_{b}^{-1\/2} (1+z)^{-1\/2}\n\\label{eq_trad}\n\\end{equation}\n\n\\noindent where B is in $\\mu$G, $\\nu_{b}$ in GHz and $t_{\\rm rad}$ in\n10$^{3}$ yr. If in Eq. \\ref{eq_trad} we assume the equipartition\nmagnetic field \nwe find that the radiative ages are just a few years (Table 3). \nAs the hotspots\nextend over kpc distances, it is indicative that a very efficient\nre-acceleration mechanism is operating in a similar way over\nthe entire hotspot region.\\\\\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=alphaqiu.ps voffset=-270 hoffset=-18 hscale=48 vscale=48}\n\\vspace{6.5cm}\n\\caption{Power-law spectral index map for 3C\\,445 South determined from FORS\nI-band and FORS U-band. Contours are\n1, 1.3, 1.5, 1.6, 1.7. First contour is 3 sigma.}\n\\label{slope_3c445}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Discussion}\n\nThe detection of diffuse optical emission occurring well outside\nthe main shock region and distributed over a large fraction of the\nwhole kpc-scale hotspot structure is somewhat surprising. Deep optical\nobservations pointed out that this is a rather common phenomenon\ndetected in about a dozen hotspots \\citep[e.g.][]{mack09,cheung05,thomson95}.\nFirst-order Fermi\nacceleration alone cannot explain optical emission extending on kpc\nscale and additional efficient mechanisms taking place away from the\nmain shock region should be considered, \nunless projection effects play an important role in smearing compact\nregions where acceleration is still occurring.\\\\\nTheoretically, we can consider several scenarios that are able to reproduce\nthe observed extended structures.\n(1) One possibility is that a very wide jet, with a size comparable to the \nhotspot region, impacts simultaneously into various locations across the\nhotspot generating a complex shocked region that defines an arc-shaped\nstructure. This, combined with projection effects may explain a\nwide (projected) emitting region.\n(2) Another possibility is a narrow jet that impacts into the hotspot in a\nsmall region where electrons are accelerated at a strong shock. In this\ncase the accelerated particles are then transported upstream \nin the hotspot\nvolume where they are continuously re-accelerated by stochastic mechanisms,\nlikely due to turbulence generated by the jet and shock itself.\n(3) Finally, extended emission may be explained \nby the ``dentist's drill'' scenario, in which the jet impacts into\nthe hotspot region in different locations at different times.  \\\\\nThe peculiar morphology and the rather high NIR\/optical luminosity \nof 3C\\,105 South and 3C\\,445 South, makes \nthese hotspots ideal targets to investigate the\nnature of extended diffuse emission. \\\\\nIn 3C\\,105 South, the detection of optical emission in both \nprimary and\nsecondary hotspots implies that in these regions there is a continuous \nre-acceleration of particles. The secondary hotspot S3 could be interpreted\nas a splatter-spot from material accelerated in the primary one, S2\n\\citep{williams85}.  \nBoth the alignment and the distance between these\ncomponents exclude the jet drilling scenario: \nthe light time between the two components is more than 10$^{4}$ years,\ni.e. much longer than their radiative time (Table 3), suggesting that\nacceleration is taking place in both S2 and S3\nsimultaneously. The secondary hotspot S3 shows some\nelongation, always in the same direction, in all the radio and optical\nimages with adequate spatial resolution. \nThis elongation is expected in a splatter-spot and it\nfollows the structure of the shock generated by the impact of the \noutflow from the\nprimary upon the cocoon wall. \\\\\nThis scenario, able to explain the presence of optical emission from two\nbright and distant components, fails in reproducing the diffuse\noptical emission enshrouding the main features, and the extended\ntail. In this case, an additional contribution from stochastic mechanisms\ncaused by turbulence in the downstream region is necessary. \nAlthough this acceleration mechanism is in general less\nefficient than Fermi-I processes, the (radiative) energy losses of\nparticles are smaller in the \npresence of low magnetic fields, such as those in between S2 and S3, \n(potentially)\nallowing stochastic mechanisms to maintain electrons at high energies. \\\\\nIn 3C 445 South the observational picture is complex.\nThe optical images of 3C 445 South show a spectacular 10-kpc arc-shape\nstructure. High resolution HST images allow a further step since they \nresolve this structure in two elongated components enshrouded by diffuse \nemission. \nThese components may mark the regions where a ``dentist's drill'' jet\nimpacts on the ambient medium, representing the most recent episode\nof shock acceleration due to the jet impact. On the other hand, they\ncould simply trace the locations \nof higher particle-acceleration efficiency from a wide\/complex \ninteraction between the jet and the ambient medium.\nHowever, the transverse extension, about 1 kpc, of the two elongated \ncomponents is much larger than what is derived if the relativistic\nparticles, accelerated at the shock, age in the downstream region (provided \nthat the hotspot advances at typical speeds of 0.05-0.1$c$).\nFurthermore, the diffuse optical emission on larger scale\nsuggests the presence of additional, complex, acceleration mechanisms,\nsuch as stochastic processes, \nable to keep particle re-acceleration ongoing in the \nhotspot region.\nThe detection of X-ray emission with {\\it Chandra} \nadds a new grade of complexity. This emission and its displacement \nare interpreted by \\citet{perlman10} as due to IC-CMB originating in\nthe fast part of the decelerating flow. Their model requires that the\nangle between the jet velocity and the observer's line  \nof sight is small. \nHowever, 3C\\,445 is a\n  classical double radio galaxy and the jet should form a large angle\n  with the line of sight (see also Perlman et al. 2010).\nOn the other hand, \nwe suggest that the X-ray\/optical offset might be the outcome of \n  ongoing efficient particle acceleration occurring in the hotspot\n  region. An evidence supporting this interpretation\nmay reside on the faint and diffuse blob\nseen in U- and B-bands (labelled SC in Fig. \\ref{fig_3c445}) just\nabout 1$^{\\prime\\prime}$ downstream the X-ray peak. The surface\nbrightness of this\ncomponent decreases rapidly as the frequency decreases, as it is shown in\nFig. \\ref{fig_3c445}: well-detected in U- and\nB-bands, marginally visible in I-band, and absent at NIR and radio\nwavelengths. The SED of the diffuse hotspot emission (including SC\ncomponent and excluding SW and SE) is consistent with synchrotron\nemission with a break at high frequencies, 10$^{15}$ Hz $<$ $\\nu_{b}$\n$\\leq$ 8$\\times$10$^{16}$ Hz, and may significantly contribute to the\nobserved X-ray flux. \nSuch a hard spectrum is in agreement with (i) a\nvery recent episode of particle acceleration (the radiative cooling time of the\nemitting particles being 10$^{2}$-10$^{3}$ yr); (ii) efficient\nspatially-distributed acceleration processes,\nsimilar to the scenario proposed for the western hotspot of Pictor A \n(Tingay et al. 2008, see their Fig.5). \\\\\n\n\n\\section{Conclusions}\n\nWe presented a multi-band, high spatial resolution study of the\nhotspot regions in two nearby radio galaxies,\nnamely 3C\\,105 South and 3C\\,445 South, on the basis of \nradio VLA, NIR\/optical VLT and HST, and X-ray {\\it Chandra}\nobservations. At the sub-arcsec resolution achieved at radio and\noptical wavelengths, both hotspots display\nmultiple resolved components connected by diffuse emission detected\nalso in optical. The hotspot region in 3C\\,105 resolves\nin three major components: a primary hotspot, unresolved and aligned\nwith the jet direction, and a secondary hotspot, elongated in shape,\nand interpreted as a splatter-spot arising from continuous outflow of\nparticles from the primary. \nSuch a feature, together with the extremely short\nradiative ages of the electron populations emitting in the optical,\nindicates that the jet has been impacting\nalmost in the same position for a long period, making the\ndrilling jet scenario unrealistic. \nThe detection of an excess of X-ray\nemission from the northern component of 3C\\,105 South \nsuggests that this region is likely a relativistic knot in the jet\nrather than a genuine hotspot feature.\nThe optical diffuse emission enshrouding\nthe main components and extending towards the tail can\nbe explained possibly assuming additional stochastic mechanisms\ntaking place across the whole hotspot region.\\\\ \nIn the case of 3C\\,445 South the optical observations probe a scenario \nwhere the interaction between jet and the ambient medium is very complex.\nTwo optical components pinpointed by HST observations mark either the locations \nwhere particle acceleration is most efficient or the remnants of the most\nrecent episodes of acceleration.\nAlthough projection effects may play an important role, the morphology \nand the spatial extension of the diffuse optical emission suggest that\nparticle accelerations, such as stochastic mechanisms, \nadd to the standard shock acceleration\nin the hotspot region.\nThe X-rays detected by {\\it Chandra} cannot be the counterpart at higher\nenergies of the two main components. It might be due to\nIC-CMB from the fast part of a decelerating flow.\nAlternatively the X-rays could pinpoint synchrotron emission from\nrecent episodes of efficient particle acceleration occurring in the\nwhole hotspot region, similarly to what proposed in other hotspots, that\nwould make the scenario even more complex.\nA possible evidence supporting this scenario comes from the\n  hard spectrum of the diffuse hotspot emission and from the\n  appearance of a new component (SC) in the optical images.\n\n\n\n\n\\section*{Acknowledgment}\nWe thank the anonymous referee for the valuable suggestions that improved the manuscript. \nF.M. acknowledges the Foundation BLANCEFLOR Boncompagni-Ludovisi, n'ee\nBildt for the grant awarded him in 2010 to support his research.\nThe VLA is operated by the US \nNational Radio Astronomy Observatory which is a facility of the National\nScience Foundation operated under cooperative agreement by Associated\nUniversities, Inc. This work has made use of the NASA\/IPAC\nExtragalactic Database NED which is operated by the JPL, Californian\nInstitute of Technology, under contract with the National Aeronautics\nand Space Administration. This research has made used of SAOImage DS9,\ndeveloped by the Smithsonian Astrophysical Observatory (SAO). Part of\nthis work is based on archival data, software or on-line services\nprovided by ASI Science Data Center (ASDC). The work at SAO is\nsupported by supported by NASA-GRANT GO8-9114A. \nWe acknowledge the use of public data from\nthe Swift data archive. This research has made use of software\nprovided by the Chandra X-ray Center (CXC) in the application packages\nCIAO and ChIPS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\noindent \\textbf{Nodal profile controllability.}\nThe problem of \\emph{nodal profile controllability} of partial differential equations on networks refers to the task of steering the solution thereof to prescribed profiles on specific nodes. Formally speaking, this amounts to saying that said solution should be controlled to given time-dependent functions (called \\emph{nodal profiles}) over certain time intervals by means of controls actuating at one or several other nodes.\nThis is in contrast to the classical question of exact controllability, wherein one seeks to steer the state, at a certain time, to a given final state on the entire network.\nThe nodes with prescribed profiles are then called \\emph{charged nodes} \\cite{YWang2019partialNP} (or \\emph{object-nodes} \\cite{Zhuang2021}), while the nodes at which the controls are applied are the \\emph{controlled nodes} \\cite{YWang2019partialNP} (or \\emph{control nodes} \\cite{Zhuang2021}).\n\n\nThe notion of exact boundary controllability of nodal profiles was, to our knowledge, first introduced by Gugat, Herty and Schleper in \\cite{gugat10}, motivated by applications in the context of gas transport through pipelines networks. \nTherein, consumers are located at the endpoints of the network and the nodal profiles represent the consumer satisfaction, and the former are sought to be attained by the flow which is controlled by means of a number of compressors actuating at several nodes.\n\n\\medskip\n\n\\noindent Motivated by the abundant practical relevance of such control problems, Tatsien Li and coauthors generalized the aforementioned results to one-dimensional first-order quasilinear hyperbolic systems with nonlinear boundary conditions \\cite{gu2011, li2010nodal, li2016book}. \nFor results on the wave equation on a tree-shaped network with a general topology or the unsteady flow in open canals, we refer the reader to \\cite{kw2011, kw2014, YWang2019partialNP} and \\cite{gu2013}, respectively. \n\nWhilst the exact-controllability of the Saint-Venant equations on networks with cycles is not true in general \\cite{LLS, li2010no}, for certain networks with cycles, the exact nodal profile controllability can be shown by means of a so-called \\emph{cut-off} method \\cite{Zhuang2021, Zhuang2018}.\nIn this regard, in line with intuition, the concept of nodal profile controllability is weaker than that of exact controllability.\nHence, when considering a system defined on a network with cycles, a situation which is encountered in many practical applications, the nodal profile control problem is a rather meaningful and feasible goal to attain.\n\n\\medskip\n\n\\noindent\nThe method used by Li et al. to prove nodal profile controllability is \\emph{constructive} in nature, in the sense that it relies on solving the equation forward in time and sidewise, to build a specific solution which achieves the desired goal, before evaluating the trace of this solution to obtain the desired controls. \nAll this is done in the context of regular $C^1_{x,t}$ solutions for first-order systems, which are \\emph{semi-global} in time -- this means that for any time $T>0$, and for small enough initial and boundary data, a unique solution exists at least until time $T>0$ --, a solution concept originating from \\cite{LiJin2001_semiglob}.\nIn \\cite{LiRao2002_cam, LiRao2003_sicon}, this notion of solution is used for proving local exact boundary controllability of one-dimensional quasilinear hyperbolic systems. In these works, a general framework for a constructive method is proposed, from which all subsequent constructive methods derive.\nThe cornerstone of Li's method is thus the proof of semi-global existence and uniqueness, and, in the case of networks, a thorough study of the transmission conditions at multiple nodes. As solving a sidewise problem entails exchanging the role of the spatial and time variables, \nthis method fundamentally exploits the one-dimensional nature of the system (see also Remark \\ref{rem:controllability_thm} \\ref{subrem:sidewise}).\n\n\nVery recently, in the context of the one-dimensional linear wave equation, the controllability of nodal profiles has also been studied in the context of less regular states and controls spaces, by using the duality between controllability and observability and showing an observability inequality. For star-shaped networks, one may see \\cite{YWang2021_NP_HUM} where the sidewise and D'Alembert Formula is used, and for a single string one may see \\cite{Sarac2021} which relies on sidewise energy estimates.\n\n\n\\medskip\n\n\\noindent \\textbf{Geometrically exact beams.} \nMulti-link flexible structures such as large spacecraft structures, trusses, robot arms, solar panels, antennae \\cite{chen_serial_EBbeams,  flotow_spacecraft, LLS} have found many applications in civil, mechanical and aerospace engineering.\nThe behavior of such structures is generally modeled by networks of interconnected beams.\n\nIn this article, we will address the problem of nodal profile controllability for \\emph{networks of beams}, possibly with cycles, a problem which has not yet been considered in the literature. \nThe network in question consists of $N$ beams, indexed by $i \\in \\{1, \\ldots, N\\}$, evolving in $\\mathbb{R}^3$, which are mutually linked via rigid joints. \nThe beams are assumed to be freely vibrating, meaning that external forces and moments, such as gravity or aerodynamic forces, have been set to zero.\n\n\\medskip\n\n\\noindent \nNowadays, there is a growing interest in modern highly flexible light-weight structures -- for instance robotic arms \\cite{grazioso2018robot}, flexible aircraft wings \\cite{Palacios2010aero} or wind turbine blades \\cite{Munoz2020, wang2014windturbine} -- which exhibit motions of large magnitude, not negligible in comparison to the overall dimensions of the object.\nTo capture such a behavior, one has to consider a beam model which is \\emph{geometrically exact}, in the sense that the governing system presents nonlinearities in order to also represent large motions -- i.e., large displacements of the centerline and large rotations of the cross sections. \n\nThis beam model, similarly to the more well-known Euler-Bernoulli and Timoshenko systems, is one dimensional with respect to the spatial variable $x$ and accounts for linear elastic material laws, meaning that the strains (which are the local changes in the shape of the material) are assumed to be small.\nModels for geometrically exact beams account for shear deformation, similarly to the Timoshenko system. \nMoreover, the geometrical and material properties of the beam may vary along the beam (indeed, we will see that the coefficients of the system depend on $x$), and the material may be anisotropic.\nAs a matter of fact, the Euler-Bernoulli and Timoshenko systems can be derived from geometrically exact beam models under appropriate simplifying assumptions \\cite[Section IV]{Artola2021damping}.\n\n\\medskip\n\n\\noindent We will see, in Subsection \\ref{subsec:GEBmodels}, that the mathematical model for geometrically exact beams may be written in terms of the position of the centerline of the beam and the orientation of its cross sections, with respect to a fixed coordinate system. This is the commonly known \\emph{Geometrically Exact Beam model}, or GEB, which originates from the work of Reissner \\cite{reissner1981finite} and Simo \\cite{simo1985finite}. The governing system is quasilinear, consisting of six equations. One may draw a parallel with the wave equation as the GEB model is of second order both in space and time.\n\nOn the  other hand, the mathematical model can also be written in terms of so-called \\emph{intrinsic} variables -- namely, velocities and internal forces\/moments, or equivalently velocities and strains -- expressed in a moving coordinate system attached to the beam.\nThis yields the \\emph{Intrinsic Geometrically Exact Beam model}, or IGEB, which is due to Hodges \\cite{hodges1990, hodges2003geometrically}. The governing system then counts twelve equations.\nAn interesting feature of the IGEB model is that it falls into the class of one-dimensional first-order hyperbolic systems and is moreover only semilinear. Therefore, from a mathematical perspective, one gains access to the broad literature which has been developed on such system -- see notably by Li and Yu \\cite{Li_Duke85}, Bastin and Coron \\cite{BC2016} -- beyond the context of beam models.\n\nDue to its less compound nature, the IGEB formulation is used in aeroelastic modelling and engineering, notably in the context of very light-weight and slender aircraft aiming to remain airborne almost perpetually, and that consequently exhibit great flexibility \\cite{Palacios2017modes, Palacios2011intrinsic, Palacios2010aero}; see also \\cite{Artola2020aero, Artola2019mpc, Artola2021damping} where the authors additionally take into account structural damping.\n\n\\medskip\n\n\\noindent On another hand, as pointed out in \\cite[Sec. 2.3.2]{weiss99}, one may see the GEB model and IGEB model as being related by a \\emph{nonlinear transformation} (which we define in \\eqref{eq:transfo}). In this work, we will keep track of this link between both models, studying mathematically the latter, and then deducing corresponding results for the GEB model.\n\nAs commonly done in solid mechanics, both the GEB and IGEB models are \\emph{Lagrangian descriptions} of the beam (as opposed to the \\emph{Eulerian description}), in the sense that the independent variable $x$ is attached to matter ($x$ is a label sticking to the particles of the beam's centerline throughout the deformation history) rather than being attached to an inertial frame of reference.\n\nThe IGEB model can also be seen as the beam dynamics being formulated in the \\emph{Hamiltonian} framework in continuum mechanics (see notably \\cite[Sections 5, 6]{Simo1988}), while the GEB model corresponds to the \\emph{Lagrangian} framework.\nThen, taking into account the interactions of the beam with its environment, one may study the IGEB model from the perspective of \\emph{Port-Hamiltonian Systems} (see \\cite{Maschke1992} for the finite dimension setting and \\cite{Schaft2002} and \\cite[Chapter 7]{Zwart2012bluebook} for the infinite dimensions setting), as in  \\cite{Macchelli2007, Macchelli2009} and \\cite[Section 4.3.2]{Macchelli2009book}. See also the case of the Timoshenko model in \\cite{Macchelli2004Timo}.\n\n\n\\subsection{Our contributions}\nIn this article we consider the problem of nodal profile controllability in the context of a specific network of geometrically exact beams containing one cycle.\n\\textcolor{black}{Afterwards, the case of other networks, possibly containing several cycles, is discussed in Section \\ref{sec:conclusion}: we give a few typical examples, together with a brief  algorithm (Algorithm \\ref{algo:control}) to realize nodal profile controllability under some requirements.}\n\n\\textcolor{black}{Our main results will be given on IGEB networks (Theorem \\ref{th:controllability}) and GEB networks (Corollary \\ref{coro:controlGEB}) as follows.}\n\\begin{enumerate}\n\\item We first consider a general network of beams whose dynamics are given by the IGEB model (System \\eqref{eq:syst_physical} below). We show, in Theorem \\ref{th:existence}, that there exists a unique semi-global in time $C_{x,t}^1$ solution to \\eqref{eq:syst_physical}. \n\n\nThis theorem is also a necessary step to show Theorem \\ref{th:controllability}, namely, the local exact controllability of nodal profiles for System \\eqref{eq:syst_physical}, in the special case of an A-shaped network (see Fig. \\ref{subfig:AshapedNetwork}).\nMore precisely, we drive the solution to satisfy given profiles at one of the multiple nodes by controlling the internal forces and moments at the two simple nodes.\n\n\n\\item For a general network, via Theorem \\ref{thm:solGEB}, we make the link between the IGEB network (System \\eqref{eq:syst_physical}) and the corresponding system \\eqref{eq:GEB_netw} in which the beams dynamics are given by the GEB model. \nMore precisely, we show that the existence of a unique $C^1_{x,t}$ solution to \\eqref{eq:syst_physical} implies that of a unique $C^2_{x,t}$ solution to \\eqref{eq:GEB_netw}, provided that the data of both systems fulfill some compatibility conditions.\n\nIn particular, Theorem \\ref{thm:solGEB}, permits to translate Theorems \\ref{th:existence} and \\ref{th:controllability} to corresponding results in terms of the GEB model \\eqref{eq:GEB_netw}, which are Corollaries \\ref{coro:wellposedGEB} and \\ref{coro:controlGEB}, respectively.\n\\end{enumerate}\n\n\n\n\n\n\\subsection{Notation}\n\\label{subsec:notation}\n\nLet $m, n\\in \\mathbb{N}$. Here, the identity and null matrices are denoted by $\\mathbf{I}_n \\in \\mathbb{R}^{n \\times n}$ and $\\mathbf{0}_{n, m} \\in \\mathbb{R}^{n \\times m}$, and we use the abbreviation $\\mathbf{0}_{n} = \\mathbf{0}_{n, n}$. The transpose of $M\\in \\mathbb{R}^{m\\times n}$ is denoted by $M^\\intercal$.\nThe symbol $\\mathrm{diag}(\\, \\cdot \\, , \\ldots, \\, \\cdot \\, )$ denotes a (block-)diagonal matrix composed of the arguments.\nWe denote by $\\mathcal{S}_{++}^n$ the set of positive definite symmetric matrices in $\\mathbb{R}^{n \\times n}$.\nThe cross product between any $u, \\zeta \\in \\mathbb{R}^3$ is denoted $u \\times \\zeta$, and we shall also write $\\widehat{u} \\,\\zeta = u \\times \\zeta$, meaning that $\\widehat{u}$ is the skew-symmetric matrix \n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{equation*}\n\\widehat{u} = \\begin{bmatrix}\n0 & -u_3 & u_2 \\\\\nu_3 & 0 & -u_1 \\\\\n-u_2 & u_1 & 0\n\\end{bmatrix}, \n\\end{equation*}\n\\end{linenomath}\n\\endgroup\nand for any skew-symmetric $\\mathbf{u} \\in \\mathbb{R}^{3 \\times 3}$, the vector $\\mathrm{vec}(\\mathbf{u}) \\in \\mathbb{R}^3$ is such that $\\mathbf{u} = \\widehat{\\mathrm{vec}(\\mathbf{u})}$. Finally, $\\{e_\\alpha\\}_{\\alpha=1}^3 = \\{(1, 0, 0)^\\intercal, (0, 1, 0)^\\intercal, (0, 0, 1)^\\intercal\\}$ denotes the standard basis of $\\mathbb{R}^3$. \n\n\n\n\n\n\\subsection{Outline}\n\nIn Section \\ref{sec:model_results}, we present in more detail the GEB and IGEB models (Subsection \\ref{subsec:GEBmodels}) before introducing the corresponding systems which give the dynamics of the beam network (Subsection \\ref{subsec:network_systems}). Then, in Subsection \\ref{subsec:main_results} we presents the main results of this article.\n\n\nSection \\ref{sec:exist} is concerned with the well-posedness of the network system \\eqref{eq:syst_physical}: \nin Subsections \\ref{subsec:hyperbolic} and \\ref{subsec:change_var} we show that the system \\eqref{eq:syst_physical} is hyperbolic and write it in Riemann invariants, we then study the transmission conditions for the diagonalized system in Subsection \\ref{subsec:out_in_info}, and finally, we prove Theorem \\ref{th:existence} in Subsection \\ref{subsec:proof_exist}.\n\n\nIn Sections \\ref{sec:controllability} and \\ref{sec:invert_transfo}, we give the proofs of Theorems \\ref{th:controllability} and \\ref{thm:solGEB}, respectively.\n\n\n\\textcolor{black}{Then, in Section 6, we give generalized considerations on more involved networks, namely, with more than one cycles, or with prescribed profiles on several nodes.}\n\n\n\n\n\n\n\n\n\\section{The model and main results}\n\\label{sec:model_results}\n\n\nAs mentioned in the introduction, the beams' dynamics may be given from different points of view, that we specify in the following subsection.\n\n\n\\subsection{Dynamics of a geometrically exact beam}\n\\label{subsec:GEBmodels}\n\n\n\\begin{figure} \\centering\n\\includegraphics[scale=0.7]{beam_netwC}\n\\caption{Beam $i$ in a straight reference configuration, before deformation and at time $t$. Here, $\\{b_i^\\alpha\\}_{\\alpha=1}^3$ denote the columns of $R_i$.}\n\\label{fig:beam_netw}\n\\end{figure}\n \nLet $i$ be the index of any beam of the network.\nFirst, we consider the mathematical model written in terms of the position $\\mathbf{p}_i$ of the centerline and of a rotation matrix $\\mathbf{R}_i$ whose columns $\\{\\mathbf{b}_i^\\alpha\\}_{\\alpha=1}^3$ give the orientation of the cross sections. Both $\\mathbf{p}_i$ and $\\mathbf{R}_i$ depend on $x$ and $t$, with $x\\in [0, \\ell_i]$ where $\\ell_i>0$ is the length of the beam, and both are expressed in the fixed basis $\\{e_\\alpha\\}_{\\alpha=1}^3$.\nThe former has values in $\\mathbb{R}^3$, while the latter has values in the special orthogonal group $\\mathrm{SO}(3)$.\\footnote{$\\mathrm{SO}(3)$ is the set of unitary real matrices of size $3$ and with a determinant equal to $1$, also called \\emph{rotation} matrices.} \n\n\nThe columns of $\\mathbf{R}_i$ may also be seen as a moving basis of $\\mathbb{R}^3$, attached to the beam, and with origin $\\mathbf{p}_i$; we call it \\emph{body-attached basis} as opposed to the fixed basis $\\{e_\\alpha\\}_{\\alpha=1}^3$. We refer to Fig. \\ref{fig:beam_netw} for visualization.\n\n\nThe corresponding model is called the \\emph{Geometrically Exact Beam} model (GEB) and, for a freely vibrating beam, is set in $(0, \\ell_i)\\times(0, T)$ and reads\n\\begin{linenomath}\n\\begin{equation}\n\\label{eq:GEB_pres}\n\\partial_t \\left( \\begin{bmatrix}\n\\mathbf{R}_i & \\mathbf{0}_{3}\\\\ \\mathbf{0}_{3} & \\mathbf{R}_i\n\\end{bmatrix} \\mathbf{M}_i\n\\begin{bmatrix}\nV_i \\\\ W_i\n\\end{bmatrix}\n\\right) = \\partial_x \\begin{bmatrix}\n\\phi_i \\\\ \\psi_i \\end{bmatrix} + \\begin{bmatrix}\n\\mathbf{0}_{3, 1} \\\\ (\\partial_x \\mathbf{p}_i) \\times \\phi_i\n\\end{bmatrix},\n\\end{equation}\n\\end{linenomath}\nwhere $V_i, W_i, \\phi_i, \\psi_i$ are functions of the unknowns $\\mathbf{p}_i, \\mathbf{R}_i$. More precisely, we introduce the linear velocity $V_i$, angular velocity $W_i$, internal forces $\\Phi_i$ and internal moments $\\Psi_i$ of the beam $i$, all having values in $\\mathbb{R}^3$ and being expressed in the body-attached basis. They are defined by (see Subsection \\ref{subsec:notation})\n\\begin{linenomath}\n\\begin{equation} \\label{eq:single_beam_VWPhiPsi}\n\\begin{bmatrix}\nV_i \\\\ W_i\n\\end{bmatrix}\n= \\begin{bmatrix}\n\\mathbf{R}_i^\\intercal \\partial_t \\mathbf{p}_i\\\\ \\mathrm{vec}\\left( \\mathbf{R}_i^\\intercal \\partial_t \\mathbf{R}_i \\right)\n\\end{bmatrix}, \\quad \n\\begin{bmatrix}\n\\Phi_i \\\\ \\Psi_i\n\\end{bmatrix}= \\mathbf{C}_i^{-1} \\begin{bmatrix}\n\\mathbf{R}_i ^\\intercal \\partial_x \\mathbf{p}_i  - e_1 \\\\ \n\\mathrm{vec}\\left(\\mathbf{R}_i^\\intercal \\partial_x \\mathbf{R}_i - R_i^\\intercal \\tfrac{\\mathrm{d}}{\\mathrm{d}x} R_i\\right)\n\\end{bmatrix},\n\\end{equation}\n\\end{linenomath}\nwhile the variables $\\phi_i, \\psi_i$ just correspond to $\\Phi_i, \\Psi_i$ when expressed in the fixed basis instead of the body-attached basis; in other words \n\\begin{linenomath}\n\\begin{equation} \\label{eq:def_smallphipsii}\n\\phi_i = \\mathbf{R}_i \\Phi_i, \\quad \\psi_i = \\mathbf{R}_i \\Psi_i.\n\\end{equation}\n\\end{linenomath}\nIn the above governing system and definitions, \n\\begin{linenomath}\n\\begin{equation} \\label{eq:reg_beampara}\n\\mathbf{M}_i, \\mathbf{C}_i \\in C^1([0, \\ell_i]; \\mathcal{S}_{++}^6), \\quad R_i \\in C^2([0, \\ell_i]; \\mathrm{SO}(3))\n\\end{equation}\n\\end{linenomath}\nare the so-called \\emph{mass matrix} $\\mathbf{M}_i$ and \\emph{flexibility matrix} $\\mathbf{C}_i$ which characterize the material and geometry of the beam $i$, while $R_i$ characterizes the initial form of this beam, as it may be pre-curved and twisted before deformation (at rest). All three are given parameters of the beam.\n\n\n\n\\begin{remark}\nConsider a single beam $i$ described by \\eqref{eq:GEB_pres}, with homogeneous Neumann boundary conditions at each end -- i.e., both $\\phi_i$ and $\\psi_i$ are identically equal to zero on $\\{0\\}\\times(0, T)$ and $\\{\\ell\\}\\times (0, T)$.\nWith appropriate initial conditions, rigid body motions such as defined below are solutions to the GEB model:\n\\begin{linenomath}\n\\begin{align} \\label{eq:rigid_body_motion}\n\\mathbf{p}_i(x,t) = f(t) + \\int_0^x R_i(s)e_1 ds, \\qquad \\mathbf{R}_i(x,t) = K(t)R_i(x)\n\\end{align}\n\\end{linenomath}\nfor all $(x,t) \\in [0, \\ell_i]\\times[0, T]$, where $(f, K) \\in C^2([0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$ \nare such that $\\frac{\\mathrm{d}}{\\mathrm{d}t}f \\equiv f_\\circ$ and $\\mathrm{vec}(K^\\intercal \\frac{\\mathrm{d}}{\\mathrm{d}t}K) \\equiv k_\\circ$ for some fixed $f_\\circ, k_\\circ \\in \\mathbb{R}^3$.\n\\end{remark}\n\n\n\n\\noindent The mathematical model may also be written in terms of intrinsic variables expressed in the body-attached basis, namely, linear\/angular velocities and internal forces\/moments $v_i, z_i \\colon [0, \\ell_i]\\times[0, T] \\rightarrow \\mathbb{R}^6$, respectively. In this case, one considers the unknown state $y_i \\colon [0, \\ell_i]\\times[0, T] \\rightarrow \\mathbb{R}^{12}$ of the form\n\\begin{linenomath}\n\\begin{align} \\label{eq:form_yi}\ny_i = \\begin{bmatrix}\nv_i \\\\ z_i\n\\end{bmatrix}, \\quad \\text{where} \\quad v_i = \\begin{bmatrix}\nV_i \\\\ W_i\n\\end{bmatrix}, \\ z_i = \\begin{bmatrix}\n\\Phi_i \\\\ \\Psi_i\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\nWe call the corresponding model the \\emph{Intrinsic Geometrically Exact Beam} model (IGEB), and it reads\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:IGEB_pres}\n\\partial_t y_i + A_i(x) \\partial_x y_i + \\overline{B}_i(x) y_i = \\overline{g}_i(x, y_i),\n\\end{align}\n\\end{linenomath}\nwhere the coefficients $A_i,\\overline{B}_i$ and the source $\\overline{g}_i$ depend on $\\mathbf{M}_i, \\mathbf{C}_i$ and $R_i$. \nMore precisely, $A_i \\in C^1([0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$ is defined by (see \\eqref{eq:reg_beampara})\n\\begin{linenomath}\n\\begin{align}\\label{eq:def_Ai}\nA_i = - \\begin{bmatrix}\n\\mathbf{0}_6 & \\mathbf{M}_i^{-1}\\\\\n\\mathbf{C}_i^{-1} & \\mathbf{0}_6\n\\end{bmatrix},\n\\end{align}\n\\end{linenomath}\nand we will see, in Subsection \\ref{subsec:hyperbolic}, that the matrix $A_i(x)$ is hyperbolic for all $x \\in [0, \\ell_i]$ (i.e., it has real eigenvalues only, with twelve associated independent eigenvectors).\n\n\nThe matrix $\\overline{B}_i(x)$ is indefinite and, up to the best of our knowledge, may not be assumed arbitrarily small\nimplying not only that the linearized system \\eqref{eq:IGEB_pres} is not homogeneous, but also that \\eqref{eq:IGEB_pres} cannot be seen as the perturbation of a system of conservation laws. The function $\\overline{B}_i \\in C^1([0, \\ell_i];\\mathbb{R}^{12 \\times 12})$ which depends, just as $A_i$, on the mass and flexibility matrices, also depends on the curvature $\\Upsilon_c^i \\colon [0, \\ell_i] \\rightarrow \\mathbb{R}^3$ of the beam before deformation, and is defined by\n\\begin{linenomath}\n\\begin{align*}\n\\overline{B}_i = \\begin{bmatrix}\n\\mathbf{0}_6 & - \\mathbf{M}^{-1}_i\\mathbf{E}_i\\\\\n\\mathbf{C}_i^{-1}\\mathbf{E}_i^\\intercal & \\mathbf{0}_6\n\\end{bmatrix}, \\quad \\text{with} \\ \\ \\mathbf{E}_i = \\begin{bmatrix}\n\\widehat{\\Upsilon}_c^i &  \\mathbf{0}_3\\\\\n\\widehat{e}_1 & \\widehat{\\Upsilon}_c^i\n\\end{bmatrix}, \\quad \\Upsilon_c^i = \\mathrm{vec}\\big(R_i^\\intercal \\tfrac{\\mathrm{d}}{\\mathrm{d}x} R_i \\big).\n\\end{align*}\n\\end{linenomath}\n\nThe function $\\overline{g}_i \\colon [0, \\ell_i]\\times \\mathbb{R}^{12} \\rightarrow \\mathbb{R}^{12}$ is defined by $\\overline{g}_i(x, u) = \\overline{\\mathcal{G}}_i(x, u)u$ for all $x \\in [0, \\ell_i]$ and $u=(u_1^\\intercal, u_2^\\intercal, u_3^\\intercal, u_4^\\intercal)^\\intercal \\in \\mathbb{R}^{12}$ with each $u_j \\in \\mathbb{R}^3$, where the map  $\\overline{\\mathcal{G}}_i$ is defined by (see Subsection \\ref{subsec:notation})\n\\begin{linenomath}\n\\begin{align*}\n\\overline{\\mathcal{G}}_i(x,u) = - \n\\begin{bmatrix}\n\\mathbf{M}_i(x)^{-1} & \\mathbf{0}_6\\\\\n\\mathbf{0}_6 & \\mathbf{C}_i(x)^{-1}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\widehat{u}_2 & \\mathbf{0}_3 & \\mathbf{0}_3 & \\widehat{u}_3\\\\\n\\widehat{u}_1 & \\widehat{u}_2 & \\widehat{u}_3 & \\widehat{u}_4 \\\\\n\\mathbf{0}_3 & \\mathbf{0}_3 & \\widehat{u}_2 & \\widehat{u}_1\\\\\n\\mathbf{0}_3 & \\mathbf{0}_3 & \\mathbf{0}_3 & \\widehat{u}_2\n\\end{bmatrix} \n\\begin{bmatrix}\n\\mathbf{M}_i(x) & \\mathbf{0}_6\\\\\n\\mathbf{0}_6 & \\mathbf{C}_i(x)\n\\end{bmatrix}.\n\\end{align*}\n\\end{linenomath}\nOne sees that $\\overline{g}_i$ is a quadratic nonlinearity (in the sense that its components are quadratic forms on $\\mathbb{R}^{12}$ with respect to the second argument), and that it has the same regularity as the mass and flexibility matrices $\\mathbf{M}_i, \\mathbf{C}_i$ with respect to its first argument, and is $C^\\infty$ with respect to its second argument. Moreover, $\\overline{g}_i(x, \\cdot)$ is locally Lipschitz in $\\mathbb{R}^{12}$ for any $x \\in [0, \\ell_i]$, and $\\overline{g}_i$ is locally Lipschitz in $H^1(0, \\ell_i; \\mathbb{R}^{12})$, but no global Lipschitz property is available.\n\n\n\\medskip\n\n\n\\noindent Finally, as mentionned in the introduction, one may see \\eqref{eq:GEB_pres} and \\eqref{eq:IGEB_pres} as being related by the nonlinear transformation $\\mathcal{T}$ defined by (see \\eqref{eq:single_beam_VWPhiPsi})\n\\begingroup\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{equation} \\label{eq:transfo}\n\\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}) = (\\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i))_{i \\in \\mathcal{I}}, \\quad \\text{where} \\quad\n\\mathcal{T}_i (\\mathbf{p}_i, \\mathbf{R}_i) = \n\\begin{bmatrix} V_i\\\\ W_i\\\\ \\Phi_i\\\\ \\Psi_i\\end{bmatrix}.\n\\end{equation}\n\\end{linenomath}\n\\endgroup\n\n\n\n\n\n\n\n\\subsection{Dynamics of the network of beams}\n\\label{subsec:network_systems}\n\n\n\\begin{figure}\n  \\begin{subfigure}{0.2\\textwidth}\n  \\centering\n    \\includegraphics[height=3.25cm]{star_blackC.pdf}\n    \\caption{Star-shaped}\n  \\end{subfigure}%\n  \\begin{subfigure}{0.24\\textwidth}\n    \\centering\n    \\includegraphics[height=3.25cm]{tree_blackC.pdf}\n    \\caption{Tree-shaped} \n  \\end{subfigure}%\n  \\begin{subfigure}{0.28\\textwidth}\n    \\centering\n    \\includegraphics[height=3.25cm]{A_pres_numC.pdf}\n    \\caption{A-shaped} \n    \\label{subfig:AshapedNetwork}\n  \\end{subfigure}%\n    \\hspace*{\\fill}  \n  \\begin{subfigure}{0.26\\textwidth}\n    \\centering\n    \\includegraphics[width=2.75cm]{xinC.pdf}\n\\caption{Orientation of an edge $i$ starting and ending at the nodes $k$ and $n$, respectively.}\n\\label{fig:xin}\n  \\end{subfigure}\n\n\\caption{Some oriented graphs representing beam networks, and orientation of the edges.}\n\\label{fig:exple_networks}\n\\end{figure}\n\nLet us now give the systems describing the entire beam network.\n\n\\subsubsection{Network notation}\nTo represent a collection of $N$ beams attached in a certain manner to each other at their tips, we use an oriented graph containing $N$ edges. Any edge $i$ is identified with the interval $[0, \\ell_i]$, which is the spatial domain for the beam model in question (GEB or IGEB). Hence, just as for the beams, the \\emph{edges} are indexed by $i \\in \\mathcal{I} = \\{1, \\ldots, N\\}$, while the \\textit{nodes} are indexed by $n \\in \\mathcal{N} = \\{1, \\ldots, \\#\\mathcal{N}\\}$, where $\\#$ denotes the set cardinality. The set of nodes is partitioned as $\\mathcal{N} = \\mathcal{N}_S \\cup \\mathcal{N}_M$, where $\\mathcal{N}_S$ is the set of indexes of \\emph{simple nodes}, while $\\mathcal{N}_M$ is the set of indexes \\emph{multiple nodes}. \n\nThe former set is in addition partitioned as $\\mathcal{N}_S = \\mathcal{N}_S^D \\cup \\mathcal{N}_S^N$, where $\\mathcal{N}_S^D$ contains the simple nodes with prescribed \\emph{Dirichlet} boundary conditions (i.e., the centerline's position and the cross section's orientation in the case of the GEB model, or the velocities in the case of the IGEB model, are prescribed), while $\\mathcal{N}_S^N$ contains the simple nodes with prescribed \\emph{Neumann} boundary conditions (i.e., the internal forces and moments are prescribed).\n\n\\medskip\n\n\\noindent For any $n\\in\\mathcal{N}$, we denote by $\\mathcal{I}^n$ the set of indexes of edges incident to the node $n$, by $k_n = \\# \\mathcal{I}^n $ the \\emph{degree} of the node $n$, and by $i^n$ the index\\footnote{Defining $i^n$ as the \\emph{smallest} element of $\\mathcal{I}^n$, and not the \\emph{largest} for example, is an arbitrary choice and is of no influence here.}\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_in}\ni^n = \\min_{i\\in \\mathcal{I}^n} i.\n\\end{align}\n\\end{linenomath}\nNote that in the case of a simple node, $\\mathcal{I}^n = \\{i^n\\}$.\n\n\n\n\nThe orientation of each beam is given by the variables $\\mathbf{x}_i^n$ and $\\tau_i^n$ defined as follows.\nFor any $i \\in \\mathcal{I}^n$, we denote by $\\mathbf{x}_i^n$ the end of the interval $[0, \\ell_i]$ which corresponds to the node $n$, while $\\tau_i^n$ is the outward pointing normal at $\\mathbf{x}_i^n$: \n\\begin{linenomath}\n\\begin{align*}\n\\tau_i^n = \n\\left\\{ \n\\begin{aligned}\n-1 \\qquad & \\text{if } \\mathbf{x}_i^n = 0,\\\\\n+1 \\qquad &\\text{if } \\mathbf{x}_i^n = \\ell_i.\n\\end{aligned}\n\\right.\n\\end{align*}\n\\end{linenomath}\nAs described in Fig. \\ref{fig:exple_networks}, each edge $i$ is represented by an arrow and each node $n$ by a circle. The arrowhead is at the ending point $x=\\ell_i$; see Fig. \\ref{fig:xin}.\n\n\n\n\n\n\n\\subsubsection{The network model}\n\n\nLet $T > 0$.\nIf all beams are described by the GEB model \\eqref{eq:GEB_pres}, then the overall network is described by System \\eqref{eq:GEB_netw} below, which gives the dynamics of the unknown state $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$:\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:GEB_netw}}\n\\nonumber\n\\partial_t \\left( \n\\left[\\begin{smallmatrix}\n\\mathbf{R}_i & \\mathbf{0}_3\\\\\n\\mathbf{0}_3 & \\mathbf{R}_i\n\\end{smallmatrix}\\right]\n\\mathbf{M}_i\n\\left[\\begin{smallmatrix}\nV_i \\\\ W_i\n\\end{smallmatrix}\\right]\n\\right) & \\\\\n\\label{eq:GEB_gov}\n\\hspace{1cm}= \\partial_x \\left[\\begin{smallmatrix}\n\\phi_i \\\\ \\psi_i \\end{smallmatrix} \\right] + \\left[\\begin{smallmatrix}\n\\mathbf{0}_{3, 1} \\\\ (\\partial_x \\mathbf{p}_i) \\times \\phi_i\n\\end{smallmatrix} \\right] \n&$\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:GEB_continuity_pi}\n\\mathbf{p}_i(\\mathbf{x}_i^n, t)  = \\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, t) &$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:GEB_rigid_angles}\n(\\mathbf{R}_i R_{i}^\\intercal)(\\mathbf{x}_i^n, t) = (\\mathbf{R}_{i^n} R_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n, t) &$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:GEB_Kirchhoff}\n{\\textstyle \\sum_{i\\in\\mathcal{I}^n}} \\tau_i^n \\left[ \\begin{smallmatrix} \n\\phi_i \\\\ \\psi_i\n\\end{smallmatrix} \\right] (\\mathbf{x}_i^n, t) = f_n (t)\n&$t \\in (0, T), \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:GEB_condNSz}\n\\tau_{i^n}^n \\left[ \\begin{smallmatrix} \n\\phi_{i^n} \\\\ \\psi_{i^n}\n\\end{smallmatrix} \\right] (\\mathbf{x}_{i^n}^n, t) = f_n (t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^N$\\\\\n\\label{eq:GEB_condNSv_p_R}\n(\\mathbf{p}_{i^n}, \\mathbf{R}_{i^n})(\\mathbf{x}_{i^n}^n, t) = (f_n^\\mathbf{p}, f_n^\\mathbf{R})(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^D$\\\\\n\\label{eq:GEB_IC_0ord}\n(\\mathbf{p}_i, \\mathbf{R}_i)(x, 0) = (\\mathbf{p}_i^0, \\mathbf{R}_i^0)(x) &$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:GEB_IC_1ord}\n(\\partial_t \\mathbf{p}_i, \\mathbf{R}_i W_i)(x, 0) = (\\mathbf{p}_i^1, w_i^0)(x) &$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$,\n\\end{subnumcases}\n\\end{linenomath}\nwhere we recall that $V_i, W_i, \\phi_i, \\psi_i$ are defined in \\eqref{eq:single_beam_VWPhiPsi}-\\eqref{eq:def_smallphipsii}.\nIn this system, \\eqref{eq:GEB_IC_0ord}-\\eqref{eq:GEB_IC_1ord} describe the initial conditions, with data\n\\begin{linenomath}\n\\begin{align} \\label{eq:reg_Idata_GEB}\n(\\mathbf{p}_i^0, \\mathbf{R}_i^0) \\in C^2([0, \\ell_i]; \\mathbb{R}^3 \\times \\mathrm{SO}(3)), \\quad \\mathbf{p}_i^1, w_i^0 \\in C^1([0, \\ell_i]; \\mathbb{R}^3), \\quad i \\in\\mathcal{I}.\n\\end{align}\n\\end{linenomath}\nThen, \\eqref{eq:GEB_continuity_pi}-\\eqref{eq:GEB_rigid_angles}-\\eqref{eq:GEB_Kirchhoff}\nare the so-called \\emph{transmission} (or \\emph{interface}) conditions for multiple nodes, while the conditions \\eqref{eq:GEB_condNSz}-\\eqref{eq:GEB_condNSv_p_R} are enforced at simple nodes. The nodal data is\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:reg_Ndata_GEB_N}\nf_n \\in C^1([0, T]; \\mathbb{R}^{6}), \\quad &n \\in \\mathcal{N}_M \\cup \\mathcal{N}_S^N\\\\\n\\label{eq:reg_Ndata_GEB_D}\n(f_n^\\mathbf{p}, f_n^\\mathbf{R}) \\in C^2([0, T]; \\mathbb{R}^3\\times \\mathrm{SO}(3)), \\quad &n \\in\\mathcal{N}_S^D.\n\\end{align}\n\\end{linenomath}\n\n\n\\medskip\n\n\n\\noindent On the other hand, if all beams are described by the IGEB model \\eqref{eq:IGEB_pres}, then for the overall network, the unknown state $(y_i)_{i \\in \\mathcal{I}}$ is described by System \\eqref{eq:syst_physical}, which reads\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:syst_physical}}\n\\label{eq:IGEB_gov}\n\\partial_t y_i + A_i \\partial_x y_i + \\overline{B}_i y_i = \\overline{g}_i(\\cdot,y_i) &$\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:IGEB_cont_velo}\n(\\overline{R}_i v_i)(\\mathbf{x}_i^n, t) = (\\overline{R}_{i^n} v_{i^n})(\\mathbf{x}_{i^n}^n, t) &$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:IGEB_Kirchhoff}\n\\sum_{i\\in\\mathcal{I}^n} \\tau_i^n (\\overline{R}_i z_i)(\\mathbf{x}_i^n, t) = q_n(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:IGEB_condNSz}\n\\tau_{i^n}^n z_{i^n} (\\mathbf{x}_{i^n}^n, t) = q_n(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^N$\\\\\n\\label{eq:IGEB_condNSv}\nv_{i^n}(\\mathbf{x}_{i^n}^n, t) = q_n(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^D$\\\\\n\\label{eq:IGEB_ini_cond}\ny_i(x, 0) = y_i^0(x) &$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$,\n\\end{subnumcases}\n\\end{linenomath}\n$v_i,z_i$ representing the first and last six components of $y_i$, respectively (see \\eqref{eq:form_yi}), and where $\\overline{R}_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{6 \\times 6})$ is defined by $\\overline{R}_i = \\mathrm{diag}(R_i, R_i)$ (see \\eqref{eq:reg_beampara}).\nHere, \\eqref{eq:IGEB_ini_cond} gives the initial conditions, with data \n\\begin{linenomath}\n\\begin{align} \\label{eq:reg_Idata_IGEB}\ny_i^0 \\in C^1([0, \\ell_i]; \\mathbb{R}^{12}), \\quad i \\in \\mathcal{I},\n\\end{align}\n\\end{linenomath}\nthe transmission conditions are \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff}, while the conditions \\eqref{eq:IGEB_condNSz}-\\eqref{eq:IGEB_condNSv} are imposed at simple nodes, with data\n\\begin{linenomath}\n\\begin{align}\\label{eq:reg_Ndata_IGEB}\nq_n \\in C^1([0, T]; \\mathbb{R}^{6}), \\quad n \\in \\mathcal{N}.\n\\end{align}\n\\end{linenomath}\n\n\n\n\n\n\n\\subsubsection{Origin of the nodal conditions}\n\nAs the form of transmission conditions is an essential aspect in the proof of nodal profile controllability of hyperbolic systems on networks, let us now explain the origin of these conditions for System \\eqref{eq:GEB_netw} and especially those of System \\eqref{eq:syst_physical}. See also \\cite{R2020} for a more detailed presentation, and for the meaning of the states and coefficients of \\eqref{eq:GEB_netw} and \\eqref{eq:syst_physical}.\n\n\n\\medskip\n\n\n\\noindent Let $n$ be the index of some multiple node. \nIn this work, we assume that, at all times, the beams incident with this node remain attached to each other. In other words, as imposed by \\eqref{eq:GEB_continuity_pi}, the position of their centerlines must coincide. \nMoreover, we work under the \\emph{rigid joint} assumption, namely, at any node, there is no relative motion between the incident beams. As the orientation of the cross sections before deformation is specified by the (given) function $R_i$, the rigid joint assumption is enforced by the condition \\eqref{eq:GEB_rigid_angles} which states that the change of orientation $\\mathbf{R}_iR_i^\\intercal$ (from the undeformed state of the beam network to its state at time $t$) is the same for all incident beams. See also \\cite[Subsection 2.4]{strohm_dissert}.\n\n\nFor the IGEB model, the condition corresponding to the continuity of the centerline's position and of the change of the cross section's orientation, is the \\emph{continuity} of velocities \\eqref{eq:IGEB_cont_velo}. Indeed, one may differentiate \\eqref{eq:GEB_continuity_pi} and \\eqref{eq:GEB_rigid_angles} with respect to time, and then left-multiply each of the obtained equations by $(R_j\\mathbf{R}_j^\\intercal)(\\mathbf{x}_j^n, t)$ for the corresponding beam index $j$ (thereby using the rigid joint assumption), to obtain\n\\begin{linenomath}\n\\begin{align*}\n(R_i\\mathbf{R}_i^\\intercal \\partial_t \\mathbf{p}_i)(\\mathbf{x}_i^n, t) &= (R_{i^n}\\mathbf{R}_{i^n}^\\intercal \\partial_t \\mathbf{p}_{i^n})(\\mathbf{x}_{i^n}^n, t), \\\\\n(R_i\\mathbf{R}_i^\\intercal \\partial_t \\mathbf{R}_i R_i^\\intercal)(\\mathbf{x}_i^n, t) &= (R_{i^n}\\mathbf{R}_{i^n}^\\intercal \\partial_t \\mathbf{R}_{i^n} R_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n, t),\n\\end{align*}\n\\end{linenomath}\nrespectively. The above equations turn out to equate to \\eqref{eq:IGEB_cont_velo}, by the definition of $V_i$ and $W_i$ (see \\eqref{eq:single_beam_VWPhiPsi}), and by using that the invariance of the cross product in $\\mathbb{R}^3$ under rotation provides the identity $R_i \\widehat{W}_iR_i^\\intercal = \\widehat{R_iW_i}$.\n\n\n\\medskip\n\n\n\\noindent Furthermore, at this multiple node $n$, we require the internal forces $\\phi_i$ and moments $\\psi_i$ exerted by incident beams $i\\in\\mathcal{I}_n$ to be balanced with the external load $f_n$ applied at this node, which reads as \\eqref{eq:GEB_Kirchhoff}, and is also called the \\emph{Kirchhoff} condition. \n\n\nThe corresponding Kirchhoff condition \\eqref{eq:IGEB_Kirchhoff} for the IGEB model is then obtained by left-multiplying each term in the right-hand side of \\eqref{eq:GEB_Kirchhoff} by $(R_i \\mathbf{R}_i^\\intercal)(\\mathbf{x}_i^n, t)$ for the corresponding index $i$ (once again using the rigid joint assumption), left-multiplying $f_n$ by $(R_i \\mathbf{R}_i^\\intercal)(\\mathbf{x}_i^n, t)$ for some $i \\in \\mathcal{I}_n$ (for instance as $i^n$), and recalling the relationship between $\\phi_i, \\psi_i$ and $\\Phi_i, \\Psi_i$ (see \\eqref{eq:def_smallphipsii}).\n\n\n\\medskip\n\n\n\\noindent Similar considerations hold for simple nodes. Here, either $n \\in \\mathcal{N}_S^N$ and an external load $f_n$ is applied at this node, yielding the condition \\eqref{eq:GEB_condNSz}, or $n \\in \\mathcal{N}_S^D$ and the centerline's position and cross section's orientation are prescribed as $f_n^\\mathbf{p}$ and $f_n^\\mathbf{R}$, respectively, for the beam $i^n$ incident with this node, yielding the condition \\eqref{eq:GEB_condNSv_p_R}. \n\n\nFor the IGEB model, this translates to \\eqref{eq:IGEB_condNSz} and \\eqref{eq:IGEB_condNSv}, respectively, when one left-multiplies \\eqref{eq:GEB_condNSz} and \\eqref{eq:GEB_condNSv_p_R} by $(R_{i^n}\\mathbf{R}_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n, t)$ and $\\mathbf{R}_{i^n}^\\intercal(\\mathbf{x}_{i^n}^n, t)$, respectively.\n\n\n\n\n\n\n\\subsubsection{Relationship between the data of both systems}\n\nAs mentioned earlier, the unknowns of the GEB and IGEB models are related by the transformation $\\mathcal{T}$, defined in \\eqref{eq:transfo}.\nThus, the initial data of both models are related as follows: for given $\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1$ and $w_i^0$, one has \n\\begin{linenomath}\n\\begin{align} \\label{eq:rel_inidata}\ny_i^0 = \\begin{bmatrix}\nv_i^0 \\\\ z_i^0\n\\end{bmatrix}, \\quad\nv_i^0 = \\begin{bmatrix}\n(\\mathbf{R}_i^0)^{\\intercal} \\mathbf{p}_i^1 \\\\\n(\\mathbf{R}_i^0 )^{\\intercal} w_i^0\n\\end{bmatrix}\n, \\quad\nz_i^0 = \\mathbf{C}_i^{-1} \n\\begin{bmatrix}\n(\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{p}_i^0 - e_1\\\\\n\\mathrm{vec}\\left( (\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{R}_i^0 - R_i^{\\intercal}\\frac{\\mathrm{d}}{\\mathrm{d}x} R_i \\right)\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\nSimilarly, the nodal conditions of \\eqref{eq:GEB_netw} and \\eqref{eq:syst_physical} are connected via $\\mathcal{T}$, and with the help of the above considerations on the nodal conditions, one can observe the following relationships between the nodal data of both systems.\nFor any $n \\in \\mathcal{N}_S^D$, for given $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$ of regularity \\eqref{eq:reg_Ndata_GEB_D}, one has\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_qn_D}\nq_n = \\begin{bmatrix}\n(f_n^\\mathbf{R})^\\intercal \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{p}\\\\\n(f_n^\\mathbf{R})^\\intercal \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{R}\n\\end{bmatrix},\n\\end{align}\n\\end{linenomath}\nwhile for any $n \\in \\mathcal{N}_M \\cup \\mathcal{N}_S^N$,\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_fn}\nf_n  = \n\\left\\{\n\\begin{aligned}\n&\\mathrm{diag}\\left((\\mathbf{R}_{i^n} R_{i^n}^\\intercal )(\\mathbf{x}_{i^n}^n, \\cdot), (\\mathbf{R}_{i^n} R_{i^n}^\\intercal) (\\mathbf{x}_{i^n}^n, \\cdot)\\right) q_n  &&n \\in \\mathcal{N}_M\\\\\n&\\mathrm{diag}\\big(\\mathbf{R}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot), \\mathbf{R}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot)\\big)  q_n  &&n \\in \\mathcal{N}_S^N.\n\\end{aligned}\n\\right.\n\\end{align}\n\\end{linenomath}\n\n\n\n\n\n\n\\subsection{Main results}\n\\label{subsec:main_results}\n\nWe may now present our main results, which are divided in two parts: one is concerned with the well-posedness and controllability of the IGEB network, and the other with showing that the transformation from the GEB to the IGEB network is invertible, by means of which one can deduce corresponding results for the former model. \n\n\\subsubsection{Study of the IGEB model}\n\nLet us define compatibility conditions for System \\eqref{eq:syst_physical}. As for the unknown, we write the initial data $(y_i^0)_{i \\in \\mathcal{I}}$ as\n\\begin{linenomath}\n\\begin{equation*}\ny_i^0 = \\begin{bmatrix}\nv_i^0 \\\\ z_i^0\n\\end{bmatrix} , \\qquad \\text{with } v_i^0, z_i^0 \\colon [0, \\ell_i] \\rightarrow \\mathbb{R}^6.\n\\end{equation*}\n\\end{linenomath}\n\n\\begin{definition}\nWe say that the initial data $y_i^0 \\in C^1([0, \\ell_i]; \\mathbb{R}^{12})$, for all $i\\in \\mathcal{I}$, and boundary data $q_n\\in C^0([0, T]; \\mathbb{R}^6)$, for all $n \\in \\mathcal{N}$, fulfill the first-order compatibility conditions of \\eqref{eq:syst_physical} if\n\\begin{linenomath}\n\\begin{equation} \\label{eq:compat_0} \n\\begin{aligned}\n&(\\overline{R}_i v_i^0)(\\mathbf{x}_i^n) = (\\overline{R}_j v_j^0)(\\mathbf{x}_j^n) \\qquad && i,j \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M\\\\\n&{\\textstyle \\sum_{i\\in\\mathcal{I}^n}} \\tau_i^n (\\overline{R}_i z_i^0)(\\mathbf{x}_i^n) = q_n(0) && n \\in \\mathcal{N}_M\\\\\n& \\tau_{i^n}^n z_{i^n}^0(\\mathbf{x}_{i^n}^n) = q_n(0) && n \\in \\mathcal{N}_S^N\\\\\n& v_{i^n}^0 (\\mathbf{x}_{i^n}^n) = q_n(0) && n \\in \\mathcal{N}_S^D,\n\\end{aligned}\n\\end{equation} \n\\end{linenomath}\nholds and  $y_i^1 \\in C^0([0, \\ell_i]; \\mathbb{R}^{12})$, for all $i\\in \\mathcal{I}$, defined by \n\\begin{linenomath}\n\\begin{equation*}\ny_i^1 = -  A_i \\frac{\\mathrm{d}y_i^0}{\\mathrm{d}x} - \\overline{B}_i y_i^0 + \\overline{g}_i(\\cdot, y_i^0) = \\begin{bmatrix}\nv_i^1 \\\\z_i^1\n\\end{bmatrix},\n\\end{equation*}\n\\end{linenomath}\nalso fulfills \\eqref{eq:compat_0}, where $v_i^0, z_i^0$ are replaced by $v_i^1, z_i^1$ respectively. \n\\end{definition}\n\n\nIn order to ensure a certain regularity of the eigenvalues and eigenvectors of $A_i$, we will later on make the following assumption.\n\n\n\\begin{assumption} \\label{as:mass_flex}\nFor all $i \\in \\mathcal{I}$, we suppose that\n\\begin{enumerate}\n\\item \\label{eq:assump1_1} $\\mathbf{C}_i, \\mathbf{M}_i \\in C^2([0, \\ell_i]; \\mathcal{S}_{++}^6)$;\n\\item \\label{eq:assump1_2} the function $\\Theta_i \\in C^2([0, \\ell_i]; \\mathcal{S}_{++}^6)$ defined by $\\Theta_i = (\\mathbf{C}_i^{\\sfrac{1}{2}} \\mathbf{M}_i\\mathbf{C}_i^{\\sfrac{1}{2}})^{-1}$, is such that there exists $U_i, D_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{6 \\times 6})$ for which\n\\begin{linenomath}\n\\begin{align*}\n\\Theta_i = U_i^\\intercal D_i^2 U_i, \\quad \\text{in }[0, \\ell_i],\n\\end{align*}\n\\end{linenomath}\nwhere $D_i(x)$ is a positive definite diagonal matrix containing the square roots of the eigenvalues of $\\Theta_i(x)$ as diagonal entries, while $U_i(x)$ is unitary.\n\\end{enumerate}\n\\end{assumption}\n\n\nOne may note that, in Assumption \\ref{as:mass_flex}, if \\ref{eq:assump1_1} holds, then \\ref{eq:assump1_2} is readily verified if $\\mathbf{M}_i, \\mathbf{C}_i$ have values in the set of diagonal matrices, or if the eigenvalues of $\\Theta_i(x)$ are distinct for all $x \\in [0, \\ell_i]$ (one may adapt \\cite[Th. 2, Sec. 11.1]{evans2}). Clearly, \\ref{eq:assump1_2} is also satisfied if $\\mathbf{M}_i, \\mathbf{C}_i$ are constant, entailing that the material and geometrical properties of the beam do not vary along its centerline.\n\n\\medskip\n\n\\noindent Our first task is to obtain the existence and uniqueness of semi-global in time solutions to \\eqref{eq:syst_physical} for any network. Henceforth, in the norms' subscripts, when there is no ambiguity, we use the abbreviations $C_x^1 = C^1([0, \\ell_i]; \\mathbb{R}^d)$, $C_t^1 = C^1(I; \\mathbb{R}^d)$ and $C_{x,t}^1 = C^1([0, \\ell_i]\\times I; \\mathbb{R}^d)$ for the appropriate time interval $I$ and dimension $d \\in \\{1, 2, \\ldots\\}$.\n\n\n\\begin{theorem} \\label{th:existence}\nConsider a general network, suppose that $R_i$ has the regularity \\eqref{eq:reg_beampara} and that Assumption \\ref{as:mass_flex} is fulfilled.\nThen, for any $T>0$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon \\in (0, \\varepsilon_0)$ and for some $\\delta>0$, and all initial and boundary data $y_i^0, q_n$ of regularity \\eqref{eq:reg_Idata_IGEB}-\\eqref{eq:reg_Ndata_IGEB}, and satisfying $\\|y_i^0\\|_{C_x^1} +\\|q_n\\|_{C_t^1} \\leq \\delta$ and the first-order compatibility conditions of \\eqref{eq:syst_physical}, there exists a unique solution $(y_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i] \\times [0, T]; \\mathbb{R}^{12})$ to \\eqref{eq:syst_physical}, with $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon$.\n\\end{theorem}\n\n\n\nThe proof of Theorem \\ref{th:existence}, given in Section \\ref{sec:exist}, consists in rewriting \\eqref{eq:syst_physical} as a single hyperbolic system and applying general well-posedness results \\cite{li2010controllability, wang2006exact}. To do so, one has to write \\eqref{eq:syst_physical} in Riemann invariants, the new unknown state being denoted $(r_i)_{i \\in \\mathcal{I}}$, and verify that the nodal conditions fulfill the following rule: at any node, the components of $r_i$ corresponding to characteristics \\emph{entering} the domain $[0, \\ell_i]\\times [0, +\\infty)$ at this node is expressed explicitly as a function of the components of $r_i$ corresponding to characteristics \\emph{leaving} the domain $[0, \\ell_i]\\times [0, +\\infty)$ at this node (more detail is given in Subsection \\ref{subsec:out_in_info}).\n\n\n\\begin{remark} \nAssuming that $R_i \\in C^2([0, \\ell_i]; \\mathrm{SO}(3))$ guaranties that $\\overline{B}_i \\in C^1([0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$. On the other hand, in Assumption \\ref{as:mass_flex}, the extra regularity for $\\mathbf{M}_i, \\mathbf{C}_i$ ($C^2$, instead of $C^1$ as in \\eqref{eq:reg_beampara}) permits us to ensure that the coefficients of the System \\eqref{eq:syst_physical} written in Riemann invariants, in particular $B_i$ (see Subsection \\ref{subsec:change_var}), are sufficiently regular.\n\\end{remark}\n\n\n\nWe now consider a problem of local exact boundary controllability of nodal profiles, for the specific case of the A-shaped network illustrated in Fig. \\ref{subfig:AshapedNetwork}, consisting of five nodes and five edges and having one cycle. More precisely, we consider the network defined by\n\\begin{linenomath}\n\\begin{align} \\label{eq:A_netw}\n\\begin{aligned}\n&\\mathcal{N}_S = \\mathcal{N}_S^N = \\{4, 5\\}, \\ \\mathcal{N}_M = \\{1, 2, 3\\}, \\ \\mathcal{I} = \\{1, \\ldots, 5\\}\\\\\n&\\mathbf{x}_1^1 = 0, \\ \\ \\mathbf{x}_2^1 = 0, \\ \\ \\mathbf{x}_3^2 = 0, \\ \\ \\mathbf{x}_4^2 = 0, \\ \\ \\mathbf{x}_5^3 = 0\\\\\n&\\mathbf{x}_1^2 = \\ell_1, \\ \\mathbf{x}_2^3 = \\ell_2, \\ \\mathbf{x}_3^3 = \\ell_3, \\ \\mathbf{x}_4^4 = \\ell_4, \\ \\mathbf{x}_5^5 = \\ell_5.\n\\end{aligned}\n\\end{align}\n\\end{linenomath}\nLet us first introduce some notation concerning the eigenvalues $\\{\\lambda_i^k (x)\\}_{k=1}^{12}$ of $A_i(x)$ for $i \\in \\mathcal{I}$ and $x \\in [0, \\ell_i]$, which, as we will see in in Subsection \\ref{subsec:hyperbolic}, are such that $\\{\\lambda_i^k\\}_{k=1}^{12} \\subset C^2([0, \\ell_i])$ under Assumption \\ref{as:mass_flex}, and\n\\begin{linenomath}\n\\begin{align} \\label{eq:sign_eigval}\n\\lambda_i^k(x) <0 \\ \\text{ if } \\ k\\leq 6, \\qquad \\lambda_i^k(x) >0 \\ \\text{ if } \\ k\\geq 7.\n\\end{align}\n\\end{linenomath}\nAlso under Assumption \\ref{as:mass_flex}, and for any $i\\in\\mathcal{I}$, we define $\\Lambda_i \\in C^0([0, \\ell_i]; (0, +\\infty))$ and $T_i>0$ by\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_Lambdai_Ti}\n    \\Lambda_i(x) = \\left( \\min_{k \\in \\{1, \\ldots, 6\\}} \\left| \\lambda_i^k(x) \\right| \\right)^{-1} \\quad \\text{and} \\quad T_i = \\int_0^{\\ell_i} \\Lambda_i(x) dx;\n\\end{align}\n\\end{linenomath}\nnote that the minimum ranges over the \\textit{negative} eigenvalues of $A_i(x)$. \nThe latter, $T_i$, corresponds to the transmission (or travelling) time from one end of the beam $i$ to its other end (see Section \\ref{sec:controllability}).\n\n\n\n\\begin{theorem} \\label{th:controllability}\nConsider the A-shaped network defined by \\eqref{eq:A_netw}.\nSuppose that $R_i$ has the regularity \\eqref{eq:reg_beampara} and that Assumption \\ref{as:mass_flex} is fulfilled.\nLet $\\overline{T}>0$ be defined by (see \\eqref{eq:def_Lambdai_Ti})\n\\begin{linenomath}\n\\begin{align} \\label{eq:minT}\n\\overline{T} = \\max \\left\\{T_1, T_2 \\right\\} + \\max \\left\\{T_4, T_5 \\right\\}. \n\\end{align}\n\\end{linenomath}\nThen, for any $T> T^*>\\overline{T}$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon \\in (0, \\varepsilon_0)$, for some $\\delta, \\gamma>0$, and\n\\begin{enumerate}[label=(\\roman*)]\n\\item for all initial data $(y_i^0)_{i \\in \\mathcal{I}}$ and boundary data $(q_n)_{n \\in \\{1, 2, 3\\}}$ of regularity \\eqref{eq:reg_Idata_IGEB}-\\eqref{eq:reg_Ndata_IGEB}, satisfying $\\|y_i^0\\|_{C_x^1} + \\|q_n\\|_{C_t^1} \\leq \\delta$ and the first-order compatibility conditions of \\eqref{eq:syst_physical}, and\n\\item for all nodal profiles $\\overline{y}_1, \\overline{y}_2 \\in C^1([T^*, T]; \\mathbb{R}^{12})$, satisfying $\\|\\overline{y}_i\\|_{C_t^1} \\leq \\gamma$ and the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} at the node $n=1$,\n\\end{enumerate}\nthere exist controls $q_4, q_5 \\in C^1([0, T]; \\mathbb{R}^6)$ with $\\|q_i\\|_{C_t^1}\\leq \\varepsilon$, such that \\eqref{eq:syst_physical} admits a unique solution $(y_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i] \\times [0, T]; \\mathbb{R}^{12})$, which fulfills $\\|y_i\\|_{C_x^1} \\leq \\varepsilon$ and\n\\begin{linenomath}\n\\begin{align}\\label{eq:aim}\ny_i(0, t) = \\overline{y}_i(t) \\quad \\text{for all }i \\in \\{1, 2\\}, \\, t \\in [T^*, T].\n\\end{align}\n\\end{linenomath}\n\\end{theorem}\n\nAs mentionned in Section \\ref{sec:intro}, the proof of Theorem \\ref{th:controllability}, given in Section \\ref{sec:controllability}, relies upon the existence and uniqueness theory of \\emph{semi-global} classical solutions to the network problem (here, Theorem \\ref{th:existence}), the form of the transmission condition of the network, and on a \\emph{constructive method}. The idea of the proof is to construct a solution $(y_i)_{i\\in \\mathcal{I}}$ to \\eqref{eq:syst_physical}, such that it satisfies the initial condition, the nodal conditions, and the given nodal profiles. Substituting this solution into the nodal conditions at the nodes $n \\in \\{4, 5\\}$, one then obtains the desired controls $q_4, q_5$. \\textcolor{black}{Our proof follows the lines of \\cite{Zhuang2018}, where the authors develop a methodology for proving the nodal profile controllability for A-shaped networks of canals governed by the Saint-Venant equations.}\n\n\\begin{remark} \\label{rem:controllability_thm}\nA few remarks are in order.\n\\begin{enumerate}\n\n\\item \\label{subrem:global}\n{\\color{black}\nThe smallness of the initial and nodal data and of the nodal profiles in (i) and (ii) is used to ensure the well-posedness of the mixed initial-boundary value problem for beams described by the IGEB model. This limitation leads to the local nature of the controllability result: in a sufficiently small $C^1$-neighborhood of the zero-steady state, we can construct continuously differentiable controls, which then generate a piecewise continuously differentiable solution on the whole network.  \nFurthermore, the study of other equilibrium solutions for the IGEB network is relevant to achieve further objectives. For instance, in the spirit of \\cite{gugatLeugering2003}, supposing that the set of equilibria is connected, one might look to use the result of local exact controllability of nodal profiles as a basis to then prove more global results.\n}\n\n\\item {\\color{black}\nFor the system linearized around the zero-steady state, a global nodal profile controllability result is also achieved, though without any limitation on the size of the of the data and nodal profiles, as it rests on an existence and uniqueness result that does not impose such limitations. Moreover, the `optimal' estimate for the controllability time $T^*$ remains that of in Theorem \\ref{th:controllability}, given in terms of the transmission times \\eqref{eq:def_Lambdai_Ti}.\n}\n\n\\item \nThe \\emph{controllability time} $T^*$ from which one can prescribe nodal profiles, has to be large enough, depending on the lengths of the beams and the eigenvalues of $(A_i)_{i \\in \\mathcal{I}}$ (and thus, it depends on the geometrical and material properties of the beam). As we will see in Section \\ref{sec:controllability}, $\\overline{T}$ is the transmission time from the controlled nodes to the charged node. One may note that $T_i \\leq \\frac{\\ell_i}{|\\lambda_i^*|}$, where the constant $\\lambda_i^*<0$ denotes the maximum over $x$ of the largest \\emph{negative} eigenvalue of $A_i(x)$.\n\n\\item \nOne will observe in the proof of Theorem \\ref{th:controllability} that the controls $q_4, q_5$ are not unique due to the use of interpolation and arbitrary nodal conditions throughout the proof. \n\n\\item \\label{subrem:sidewise}\nIn the proof of Theorem \\ref{th:controllability}, to construct the solution $(y_i)_{i\\in\\mathcal{I}}$, one is led to solve a series of forward and sidewise problems for \\eqref{eq:IGEB_gov} for the different beams $i \\in \\mathcal{I}$ of the network. Solving a sidewise problem  for \\eqref{eq:IGEB_gov} entails changing the role of $x$ and $t$, considering a governing system of the form \n\\begin{linenomath}\n\\begin{align*}\n\\partial_x y_i + A_i^{-1}\\partial_t y_i + A_i^{-1}\\overline{B}_i y_i = A_i^{-1}\\overline{g}_i(\\cdot, y_i)\n\\end{align*}\n\\end{linenomath}\nand providing ``boundary conditions'' at $t=0$ and $t = T$, and ``initial conditions'' at $x=0$ (rightward problem) or $x = \\ell_i$ (leftward problem). It is consequently important here that $A_i$ does not have any zero eigenvalue.\n\n\\end{enumerate}\n\\end{remark}\n\n\n\\subsubsection{Study of the GEB model}\n\nIn order to translate Theorems \\ref{th:existence} and \\ref{th:controllability} in terms of the GEB model, we prove the Theorem \\ref{thm:solGEB} below, which yields the existence of a unique classical solution to \\eqref{eq:GEB_netw}, provided that a unique classical solution exists for \\eqref{eq:syst_physical} and that the data of both models fulfill some compatibility conditions. \n\nLet us first introduce the compatibility conditions on the initial and boundary data of the GEB network \\eqref{eq:GEB_netw}, that will be of use in the theorem and corollaries that follow\n\\begin{linenomath}\n\\begin{subequations}\\label{eq:compat_GEB_-1_GEBtransmi}\n\\begin{align} \\label{eq:compat_GEB_-1}\n&(f_n^\\mathbf{p}, f_n^\\mathbf{R})(0) = (\\mathbf{p}_{i^n}^0, \\mathbf{R}_{i^n}^0)(\\mathbf{x}_{i^n}^n), \\quad n\\in \\mathcal{N}_S^D,\\\\\n\\label{eq:compat_GEB_transmi}\n&\\mathbf{p}_i^0(\\mathbf{x}_i^n)  = \\mathbf{p}_{i^n}^0(\\mathbf{x}_{i^n}^n), \\quad (\\mathbf{R}_i^0 R_i^\\intercal)(\\mathbf{x}_i^n)  = (\\mathbf{R}_{i^n}^0 R_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n), \\quad i\\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M,\n\\end{align}\n\\end{subequations}\n\\end{linenomath}\nand\n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{subequations}\\label{eq:compat_GEB_01}\n\\begin{align}\n\\label{eq:compat_GEB_01_1}\n&\\mathbf{p}_i^1(\\mathbf{x}_i^n) = \\mathbf{p}_{i^n}^1(\\mathbf{x}_{i^n}^n), \\quad w_i^0(\\mathbf{x}_i^n) = w_{i^n}^0(\\mathbf{x}_{i^n}^n), \\quad i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M\\\\\n\\label{eq:compat_GEB_01_2}\n&\\sum_{i \\in \\mathcal{I}^n} \\tau_i^n \\left(\\overline{R}_i \\mathbf{C}_i^{-1} \\begin{bmatrix}\n(\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{p}_i^0 - e_1\\\\\n\\mathrm{vec}\\left( (\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{R}_i^0 - R_i^{\\intercal}\\frac{\\mathrm{d}}{\\mathrm{d}x} R_i \\right)\n\\end{bmatrix}\\right)(\\mathbf{x}_i^n) = q_n(0), \\quad n \\in \\mathcal{N}_M,\\\\\n\\label{eq:compat_GEB_01_3}\n&\\tau_{i^n}^n \\left(\\overline{R}_{i^n} \\mathbf{C}_{i^n}^{-1} \\begin{bmatrix}\n(\\mathbf{R}_{i^n}^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{p}_{i^n}^0 - e_1\\\\\n\\mathrm{vec}\\left( (\\mathbf{R}_{i^n}^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{R}_{i^n}^0 - R_{i^n}^{\\intercal}\\frac{\\mathrm{d}}{\\mathrm{d}x} R_{i^n} \\right)\n\\end{bmatrix} \\right)(\\mathbf{x}_{i^n}^n) = q_n(0), \\quad n \\in \\mathcal{N}_S^N,\\\\\n\\label{eq:compat_GEB_01_4}\n&\\mathbf{p}_{i^n}^1(\\mathbf{x}_{i^n}^n) = \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{p}(0), \\quad w_{i^n}^0(\\mathbf{x}_{i^n}^n) = \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{R}(0), \\quad n \\in \\mathcal{N}_S^D.\n\\end{align}\n\\end{subequations}\n\\end{linenomath}\n\\endgroup\n\n\\begin{theorem} \\label{thm:solGEB}\nConsider a general network, and assume that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{thm:solGEB_c1} the beam parameters $(\\mathbf{M}_i, \\mathbf{C}_i, R_i)$ and initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ have the regularity \\eqref{eq:reg_beampara} and \\eqref{eq:reg_Idata_GEB}, and $y_i^0$ is the associated function defined by \\eqref{eq:rel_inidata},\n\n\\item \\label{thm:solGEB_c2} the Neumann data $f_n = f_n(t, \\mathbf{R}_{i^n})$ are of the form \\eqref{eq:def_fn}, for given functions $q_n$ of regularity \\eqref{eq:reg_Ndata_IGEB},\n\n\\item \\label{thm:solGEB_c3} the Dirichlet data $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$  are of regularity \\eqref{eq:reg_Ndata_GEB_D}, and $q_n$ are the associated functions defined by \\eqref{eq:def_qn_D},\n\n\\item \\label{thm:solGEB_c4} the compatibility conditions \\eqref{eq:compat_GEB_-1_GEBtransmi} hold.\n\\end{enumerate}\nThen, if there exists a unique solution $(y_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i]\\times [0, T]; \\mathbb{R}^{12})$ to \\eqref{eq:syst_physical} with initial and nodal data $y_i^0$ and $q_n$ (for some $T>0$), there exists a unique solution\n$(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in  \\prod_{i=1}^N C^2([0, \\ell_i]\\times [0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$\nto \\eqref{eq:GEB_netw} with initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ and nodal data $f_n$, $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$, and $(y_i)_{i\\in \\mathcal{I}} = \\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}})$.\n\\end{theorem}\n\n\\begin{remark}\nWe have the following restriction on the form of the Neumann data $f_n$: it must be possible to express it as a function $q_n = q_n(t)$ in the body-attached basis (see Subsection \\ref{subsec:GEBmodels}).\n\\end{remark}\n\n\nThe proof of Theorem \\ref{thm:solGEB}, given in Section \\ref{sec:invert_transfo}, consists in using the last six equations of \\eqref{eq:IGEB_gov} as compatibility conditions to prove that the transformation $\\mathcal{T}$, defined in \\eqref{eq:transfo}, is bijective on some spaces (see Lemma \\ref{lem:invert_transfo}); this relies on the use of quaternions \\cite{chou1992} to parametrize the rotations matrices, and existence and uniqueness results for (seemingly overdetermined) first-order linear PDE systems. Once that this property of the transformation is established, one recovers notably the governing system \\eqref{eq:GEB_gov} by using the first six equations of \\eqref{eq:IGEB_gov}. The transmission conditions are recovered by first showing that the rigid joint assumption \\eqref{eq:GEB_rigid_angles} is fulfilled and then deducing \\eqref{eq:GEB_continuity_pi}-\\eqref{eq:GEB_Kirchhoff} from \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff}.\n\n\nCorollary \\ref{coro:wellposedGEB} below follows from Theorem \\ref{th:existence} and Theorem \\ref{thm:solGEB}.\n\n\n\n\\begin{corollary}\n\\label{coro:wellposedGEB}\nConsider a general network and suppose that the conditions \\ref{thm:solGEB_c1}-\\ref{thm:solGEB_c2}-\\ref{thm:solGEB_c3}-\\ref{thm:solGEB_c4} of Theorem \\ref{thm:solGEB} are fulfilled, suppose that the beam parameters $(\\mathbf{M}_i, \\mathbf{C}_i)$ satisfy Assumption \\ref{as:mass_flex}, and that the compatibility conditions \\eqref{eq:compat_GEB_01} hold.\nThen, for any $T>0$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon \\in (0, \\varepsilon_0)$, and for some $\\delta>0$, if moreover $\\|y_i^0\\|_{C_x^1}+ \\|q_n\\|_{C_t^1}\\leq \\delta$, then there exists a unique solution $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^2([0, \\ell_i]\\times[0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$ to \\eqref{eq:GEB_netw} with initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ and nodal data $f_n$, $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$.\n\\end{corollary}\n\n\\begin{remark}\nUnder \\eqref{eq:compat_GEB_-1_GEBtransmi}, the conditions \\eqref{eq:compat_GEB_01} are just an equivalent way of imposing that $y_i^0$ fulfills the first-order compatibility conditions of \\eqref{eq:syst_physical}, but expressed in terms of the data of the GEB model.\n\\end{remark}\n\n\nFinally, from Theorems \\ref{th:controllability} and \\ref{thm:solGEB}, one obtains Corollary \\ref{coro:controlGEB} below.\n\n\n\n\\begin{corollary} \\label{coro:controlGEB}\nConsider the A-shaped network defined by \\eqref{eq:A_netw}, and assume that\n\n\\begin{enumerate}[label=(\\roman*)]\n\\item the beam parameters $(\\mathbf{M}_i, \\mathbf{C}_i, R_i)$ and initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ have the regularity \\eqref{eq:reg_beampara} and \\eqref{eq:reg_Idata_GEB}, the former satisfy Assumption \\ref{as:mass_flex} and the latter fulfill \\eqref{eq:compat_GEB_transmi}, and $y_i^0$ is the associated function defined by \\eqref{eq:rel_inidata},\n\n\\item the Neumann data $f_n = f_n(t, \\mathbf{R}_{i^n})$, for $n \\in \\{1, 2, 3\\}$ are of the form \\eqref{eq:def_fn}, for given functions $q_n$ of regularity \\eqref{eq:reg_Ndata_IGEB},\n\n\n\\item the compatibility conditions \\eqref{eq:compat_GEB_01_1}-\\eqref{eq:compat_GEB_01_2} for all $n \\in \\{1, 2, 3\\}$ hold.\n\\end{enumerate}\nLet $\\overline{T}>0$ be defined by \\eqref{eq:minT}. Then, for any $T>T^*>\\overline{T}$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon\\in (0, \\varepsilon_0)$, for some $\\delta, \\gamma>0$, and for any nodal profiles $\\overline{y}_1, \\overline{y}_2 \\in C^1([T^*, T]; \\mathbb{R}^{12})$ satisfying $\\|\\overline{y}_i\\|_{C_t^1}\\leq \\gamma$ and the  transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} at the node $n=1$, if additionally $\\|y_i^0\\|_{C_x^1} + \\|f_n\\|_{C_t^1} \\leq \\delta$ ($i \\in \\mathcal{I}, \\ n \\in \\{1, 2, 3\\}$), then there exist controls $f_4, f_5 \\in C^1([0, T]; \\mathbb{R}^6)$ with $\\|f_n\\|_{C_t^1}\\leq \\varepsilon$ such that System \\eqref{eq:GEB_netw} with initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ and boundary data $(f_n)_{n \\in \\{1, 2, 3\\}}$, admits a unique solution $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^2([0, \\ell_i]\\times[0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$, and $(y_i)_{i\\in\\mathcal{I}} := \\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}})$ fulfills $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon$ and the nodal profiles \\eqref{eq:aim}.\n\\end{corollary}\n\n\\begin{remark}\nIn Corollary \\ref{coro:controlGEB},\n\\begin{enumerate}\n\\item the profiles given at the node $n=1$ affect the intrinsic variables $\\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i)$, for $i \\in \\{1, 2\\}$, and not directly the displacements and rotations $(\\mathbf{p}_i, \\mathbf{R}_i)$;\n\n\\item for $i \\in \\{4,5\\}$ the control $f_i$ is given by \\eqref{eq:def_fn} where $q_i$ is the control provided by Theorem \\ref{th:controllability} for System \\eqref{eq:syst_physical}. The smallness of the $C^1$ norm of $f_i$ comes from a combination of the fact that $q_i$ and $y_i$ (and thus, as can be seen in \\eqref{eq:form_yi}, also the angular velocity $W_i$) have small $C^1$ norms, and that the expression of $f_i$ and $\\frac{\\mathrm{d}}{\\mathrm{d}t}f_i$ involves only the functions $q_i, W_i$ and the unitary matrices $\\mathbf{R}_i, R_i$. Indeed, $f_i = \\mathrm{diag}\\big(\\mathbf{R}_i(\\ell_i, \\cdot), \\mathbf{R}_i(\\ell_i, \\cdot)\\big)  q_i$ and one may compute that\n\\begin{linenomath}\n\\begin{align*}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} f_i = \\mathrm{diag}\\big(\\mathbf{R}_i(\\ell_i, \\cdot), \\mathbf{R}_i(\\ell_i, \\cdot)\\big) \\frac{\\mathrm{d}}{\\mathrm{d}t} q_i +  \\mathrm{diag}\\big((\\mathbf{R}_i \\widehat{W}_i)(\\ell_i, \\cdot), (\\mathbf{R}_i \\widehat{W}_i)(\\ell_i, \\cdot)\\big)  q_i.\n\\end{align*}\n\\end{linenomath}\n\\end{enumerate}\n\\end{remark}\n\n\n\n\n\n\\section{Existence and uniqueness for the IGEB network}\n\\label{sec:exist}\n\n\nWe now turn to the proof of Theorem \\ref{th:existence}.\n\n\\subsection{Hyperbolicity of the system}\n\\label{subsec:hyperbolic}\n\nLet $T >0$, $i\\in \\mathcal{I}$ and $x \\in [0, \\ell_i]$. One may quickly verify that the matrix $A_i(x)$, defined in \\eqref{eq:def_Ai}, has only real eigenvalues: six positive ones which are the square roots of the eigenvalues of $\\Theta_i(x)$ (defined in Assumption \\ref{as:mass_flex}), and six negative ones which are equal to the former but with a minus sign.\nFurthermore, some computations yield the following lemma whose proof is given in \\cite[Section 4]{R2020}. \n\n\n\\begin{lemma}\nSuppose that Assumption \\ref{as:mass_flex} is fulfilled and, for any $i \\in \\mathcal{I}$, let $U_i$, $D_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{6\\times 6})$ be the functions introduced in Assumption \\ref{as:mass_flex}.\nThen, $A_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{12\\times 12})$ may be diagonalized as follows. One has $A_i = L_i^{-1} \\mathbf{D}_i L_i$ in $[0, \\ell_i]$, where $\\mathbf{D}_{i}$, $L_i \\in C^2( [0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$ are defined by\n\\begin{linenomath}\n\\begin{equation} \\label{eq:def_bfDi_Li}\n\\mathbf{D}_i = \\mathrm{diag}(-D_{i}, D_{i}), \\qquad L_i = \\begin{bmatrix}\nU_i \\mathbf{C}_i^{-\\sfrac{1}{2}} & D_{i}U_i \\mathbf{C}_i^{\\sfrac{1}{2}} \\\\\nU_i \\mathbf{C}_i^{-\\sfrac{1}{2}} & - D_{i}U_i \\mathbf{C}_i^{\\sfrac{1}{2}}\n\\end{bmatrix},\n\\end{equation}\n\\end{linenomath}\nand the inverse $L_i^{-1} \\in C^2( [0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$ is given by\n\\begin{linenomath}\n\\begin{align} \\label{eq:inverseLi}\nL_i^{-1} = \\frac{1}{2} \\begin{bmatrix}\n\\mathbf{C}_i^{\\sfrac{1}{2}} U_i^\\intercal & \\mathbf{C}_i^{\\sfrac{1}{2}} U_i^\\intercal \\\\\n\\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1} & - \\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1}\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\n\\end{lemma}\n\n\n\n\\subsection{Change of variable to Riemann invariants} \n\\label{subsec:change_var}\n\nNow, we can write \\eqref{eq:syst_physical} in diagonal form by applying the change of variable \n\\begin{linenomath}\n\\begin{align} \\label{eq:change_var_Li}\nr_i(x,t) = L_i(x) y_i(x,t), \\qquad \\text{for all }x \\in [0, \\ell_i], \\ t \\in [0, T], \\ i \\in \\mathcal{I}.\n\\end{align}\n\\end{linenomath}\nThe first (resp. last) six components of $r_i$ correspond to the negative (resp. positive) eigenvalues of $A_i$, thus, for all $i \\in \\mathcal{I}$, we denote\n\\begin{linenomath}\n\\begin{align*}\nr_i = \\begin{bmatrix}\nr_i^-\\\\\nr_i^+\n\\end{bmatrix}, \\qquad r_i^-,\\, r_i^+ \\colon [0, \\ell_i]\\times [0, T] \\rightarrow \\mathbb{R}^6.\n\\end{align*}\n\\end{linenomath}\nIn addition, in order to write the transmission conditions concisely, we introduce the invertible matrix $\\gamma_i^n$ and positive definite symmetric matrix $\\sigma_i^n$\n\\begin{linenomath}\n\\begin{align*}\n\\gamma_i^n &= (\\overline{R}_i  \\mathbf{C}_i^{\\sfrac{1}{2}} U_i^\\intercal)(\\mathbf{x}_i^n), \\qquad\n\\sigma_i^n = (\\overline{R}_i  \\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1} U_i \\mathbf{C}_i^{-\\sfrac{1}{2}} \\overline{R}_i^\\intercal)(\\mathbf{x}_i^n)\n\\end{align*}\n\\end{linenomath}\nfor all $n \\in \\mathcal{N}$ and $i \\in \\mathcal{I}^n$.\nNotice that $\\sigma_i^n \\gamma_i^n = \\overline{R}_i(\\mathbf{x}_i^n) \\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1}$.\n\n\\medskip\n\n\\noindent Then, taking \\eqref{eq:def_bfDi_Li}-\\eqref{eq:inverseLi} into account, the system obtained by applying the change of variable \\eqref{eq:change_var_Li} to System \\eqref{eq:syst_physical} reads\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:syst_diagonal}}\n\\label{eq:r_IGEB_gov}\n\\partial_t r_i + \\mathbf{D}_i \\partial_x r_i + B_i r_i = g_i(\\cdot, r_i), &\\hspace{-0.45cm}$\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:r_IGEB_cont_velo}\n\\gamma_i^n (r_i^- + r_i^+)(\\mathbf{x}_i^n, t) &\\nonumber \\vspace{-0.2cm}\\\\\n\\qquad \\quad = \\gamma_{i^n}^n (r_{i^n}^- + r_{i^n}^+)(\\mathbf{x}_{i^n}^n, t), &\\hspace{-0.45cm}$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:r_IGEB_Kirchhoff}\n\\sum_{i\\in\\mathcal{I}^n} \\frac{\\tau_i^n}{2} \\sigma_i^n \\gamma_i^n (r_i^- -  r_i^+)(\\mathbf{x}_i^n, t) = q_n(t), &\\hspace{-0.45cm}$t \\in (0, T), \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:r_IGEB_condNSz}\n(r_{i^n}^- - r_{i^n}^+)(\\mathbf{x}_{i^n}^n, t) &\\vspace{-0.2cm} \\nonumber\\\\\n\\qquad \\quad = 2 \\tau_{i^n}^n (D_{i^n} U_{i^n}\\mathbf{C}_{i^n}^{\\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) q_n(t), &\\hspace{-0.45cm}$t \\in (0, T), \\, n \\in \\mathcal{N}_S^N$\\\\\n\\label{eq:r_IGEB_condNSv}\n(r_{i^n}^- + r_{i^n}^+)(\\mathbf{x}_{i^n}^n, t) &\\vspace{-0.2cm} \\nonumber\\\\\n\\qquad \\quad = 2 (U_{i^n} \\mathbf{C}_{i^n}^{- \\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) q_n(t), &\\hspace{-0.45cm}$t \\in (0, T), \\, n \\in \\mathcal{N}_S^D$\\\\\n\\label{eq:r_IGEB_ini_cond}\nr_i(x, 0) = r_i^0(x), &\\hspace{-0.45cm}$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$.\n\\end{subnumcases}\n\\end{linenomath}\nIn the governing system \\eqref{eq:r_IGEB_gov}, the coefficient $B_i \\in C^1([0, \\ell_i]; \\mathbb{R}^{12\\times 12})$ is defined by $B_i(x) = L_i(x) \\overline{B}_i(x) L_i(x)^{-1} + L_i(x) A_i(x) \\frac{\\mathrm{d}}{\\mathrm{d}x}L_i^{-1}(x)$, while the source is defined by $g_i(x,u) = L_i(x) \\overline{g}_i(x,L_i(x)^{-1} u)$ for all $i \\in \\mathcal{I}$, $x \\in [0, \\ell_i]$ and $u \\in \\mathbb{R}^{12}$. The corresponding initial data in \\eqref{eq:r_IGEB_ini_cond} for this system is $r_i^0 = L_i y_i^0$.\n\n\n\n\\subsection{Outgoing and incoming information}\n\\label{subsec:out_in_info}\n\n\n\\begin{figure}\n  \\begin{subfigure}{0.6\\textwidth}\n  \\centering\n    \\includegraphics[scale=0.75]{enter-leave-charactC}\n\\caption{Characteristic curves $(\\mathbf{x}(t), t)$ with $\\frac{\\mathrm{d}\\mathbf{x}}{\\mathrm{d}t}(t) = \\lambda(\\mathbf{x}(t))$, where either $\\lambda(s)>0$ or $\\lambda(s)<0$ for all $s \\in [0, \\ell_i]$.}\n\\label{fig:charac}\n  \\end{subfigure}%\n  \\hspace*{\\fill}  \n  \\begin{subfigure}{0.4\\textwidth}\n    \\centering\n\\includegraphics[scale=0.7]{NM_notationC}\n    \\caption{Form of $\\mathcal{I}_n$ at a node $n$.}\n    \\label{fig:NM_notation}\n  \\end{subfigure}\n\n\\caption{Outgoing and incoming information.}\n\\label{fig:out_in_info}\n\\end{figure}\n\n\n\nFor any $n \\in \\mathcal{N}$, let us denote by $s_n \\in \\{0, \\ldots, k_n\\}$  (resp. by $k_n-s_n$) the number of beams ending (resp. starting) at the node $n$; see Fig. \\ref{fig:NM_notation}. More precisely, we suppose that \n\\begin{linenomath}\n\\begin{align*}\n\\mathcal{I}^n = \\{i_1, \\ldots, i_{k_n}\\} \\quad \\text{with} \\quad i_1 < i_2 < \\ldots < i_{s_n} \\quad \\text{and} \\quad i_{s_n+1} < i_{s_n+2} < \\ldots < i_{k_n},\n\\end{align*}\n\\end{linenomath}\nand that $\\tau_{i_\\alpha}^n = -1$ for all $\\alpha \\in \\{1, \\ldots, s_n\\}$, while $\\tau_{i_\\alpha}^n = +1$ for all $\\alpha \\in \\{s_n+1, \\ldots, k_n\\}$. This is not to be confused with the notation $i^n$ introduced in \\eqref{eq:def_in}.\n\n\n\\medskip\n\n\n\\noindent For any node $n$ and any incident edge $i \\in \\mathcal{I}^n$, we call \\emph{outgoing} (resp. \\emph{incoming}) \\emph{information}, the components of $r_i$ which correspond to characteristics entering (resp. leaving) the domain $[0, \\ell_i]\\times[0, +\\infty)$ at this node (see Fig. \\eqref{fig:charac}). \n\n\nNamely, here, the outgoing (resp. incoming) information at the node $n$ is $r_{i_\\alpha}^-(\\ell_{i_\\alpha}, t)$ (resp. $r_{i_\\alpha}^+(\\ell_{i_\\alpha}, t)$) for all $\\alpha \\in \\{1, \\ldots, s_n\\}$, and $r_{i_k}^+(0, t)$ (resp. $r_{i_k}^-(0, t)$) for all $k \\in \\{s_n+1, \\ldots, k_n\\}$.\nWe then define the functions $r_n^\\mathrm{out}, r_n^\\mathrm{in} \\colon [0, T] \\rightarrow \\mathbb{R}^{6k_n}$ by\n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{align*}\nr_n^\\mathrm{out}(t) = \\begin{bmatrix}\nr_{i_1}^-(\\ell_{i_1}, t)\\\\\n\\vdots\\\\\nr_{i_{s_n}}^-(\\ell_{i_{s_n}}, t)\\\\\nr_{i_{s_n+1}}^+(0, t) \\\\\n\\vdots\\\\\nr_{i_{k_n}}^+(0, t)\n\\end{bmatrix}, \\qquad r_n^\\mathrm{in}(t) = \\begin{bmatrix}\nr_{i_1}^+(\\ell_{i_1}, t)\\\\\n\\vdots\\\\\nr_{i_{s_n}}^+(\\ell_{i_{s_n}}, t)\\\\\nr_{i_{s_n+1}}^-(0, t) \\\\\n\\vdots\\\\\nr_{i_{k_n}}^-(0, t)\n\\end{bmatrix}.\n\\end{align*}\n\\end{linenomath}\n\\endgroup\nWe also denote $r_n^\\mathrm{out} = ((r_{n,1}^\\mathrm{out})^\\intercal, \\ldots, (r_{n,k_n}^\\mathrm{out})^\\intercal)$, where $r_{n,\\alpha}^\\mathrm{out}(t) \\in \\mathbb{R}^6$ for all $\\alpha \\in \\{1, \\ldots, k_n\\}$; a similar notation is used for $r_n^\\mathrm{in}$.\n\n\n\\medskip\n\n\n\\noindent Taking into account this notation, and the sign of $\\tau_i^n$, we observe that the Kirchhoff condition \\eqref{eq:r_IGEB_Kirchhoff} is equivalent to\n\\begin{linenomath}\n\\begin{align*}\n-\\sum_{\\alpha =1}^{s_n} \\sigma_{i_\\alpha}^n \\gamma_{i_\\alpha}^n (r_{i_\\alpha}^- -  r_{i_\\alpha}^+)(0, t) + \\sum_{k=s_n+1}^{k_n} \\sigma_{i_k}^n \\gamma_{i_k}^n (r_{i_k}^- -  r_{i_k}^+)(\\ell_{i_k}, t) = 2q_n(t),\n\\end{align*}\n\\end{linenomath}\nwhich can also be written in the form\n\\begin{linenomath} \n\\begin{align*}\n\\sum_{\\alpha=1}^{k_n} \\sigma_{i_\\alpha}^n \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{out}(t) = \\sum_{\\alpha=1}^{k_n} \\sigma_{i_\\alpha}^n \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{in}(t) + 2q_n(t).\n\\end{align*}\n\\end{linenomath}\nThe continuity condition \\eqref{eq:r_IGEB_cont_velo} is equivalent to\n\\begin{linenomath}\n\\begin{align*}\n\\gamma_{i_1}^n (r_{i_1}^- + r_{i_1}^+)(\\mathbf{x}_{i_1}^n, t) = \\gamma_{i_\\alpha}^n (r_{i_\\alpha}^- + r_{i_\\alpha}^+)(\\mathbf{x}_{i_\\alpha}^n, t) \\quad \\text{for all }\\alpha \\in \\{2, \\ldots, k_n\\}\n\\end{align*}\n\\end{linenomath}\nwhich can be seen to also write as \n\\begin{linenomath}\n\\begin{align*}\n\\gamma_{i_1}^n r_{n,1}^\\mathrm{out}(t) - \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{out}(t) = - \\gamma_{i_1}^n r_{n,1}^\\mathrm{in}(t) + \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{in}(t) \\quad \\text{for all }\\alpha \\in \\{2, \\ldots, k_n\\}.\n\\end{align*}\n\\end{linenomath}\nHence, at any multiple node $n$, the transmission conditions \\eqref{eq:r_IGEB_Kirchhoff}-\\eqref{eq:r_IGEB_cont_velo} are equivalent to the following system:\n\\begin{linenomath}\n\\begin{align*}\n\\mathbf{A}_n \\mathbf{G}_n r_n^\\mathrm{out}(t) = \\mathbf{B}_n \\mathbf{G}_n r_n^\\mathrm{in}(t) + \\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.8}\n\\begin{bmatrix}\n2q_n(t) \\\\ \\mathbf{0}_{6k_n-6, 1}\n\\end{bmatrix},\n\\endgroup\n\\end{align*}\n\\end{linenomath}\nwhere $\\mathbf{A}_n, \\mathbf{B}_n, \\mathbf{G}_n \\in \\mathbb{R}^{6k_n \\times 6k_n}$ are defined by\n\\begin{linenomath}\n\\begin{align*}\n\\mathbf{A}_n = \\begin{bmatrix}\n\\mathbf{a}_n & \\mathbf{b}_n\\\\\n\\mathbf{c}_n & \\mathbf{I}_{6(k_n-1)}\n\\end{bmatrix}, \\quad\n\\mathbf{B}_n = \\begin{bmatrix}\n\\mathbf{a}_n & \\mathbf{b}_n\\\\\n-\\mathbf{c}_n & -\\mathbf{I}_{6(k_n-1)}\n\\end{bmatrix}, \\quad \\mathbf{G}_n = \\mathrm{diag}(\\gamma_{i_1}^n, \\ldots, \\gamma_{i_{k_n}}^n),\n\\end{align*}\n\\end{linenomath}\nthe sub-matrices $\\mathbf{a}_n \\in \\mathbb{R}^{6\\times 6}$, $\\mathbf{b}_n \\in \\mathbb{R}^{6\\times 6(k_n-1)}$ and $\\mathbf{c}_n \\in \\mathbb{R}^{6(k_n-1) \\times 6}$ being defined by $\\mathbf{a}_n = \\sigma_{i_1}^n$, $\\mathbf{b}_n = \\big[ \\sigma^n_{i_2} \\ \\sigma^n_{i_3} \\ \\ldots \\   \\sigma^n_{i_{k_n}}\\big]$ and $\\mathbf{c}_n = - \\big[\\mathbf{I}_6 \\ \\mathbf{I}_6 \\ \\ldots \\ \\mathbf{I}_6 \\big]^\\intercal$.\n\n\nThe matrix $\\mathbf{G}_n$ is clearly invertible and one can check that $\\mathbf{A}_n$ is also invertible (using the same reasoning as \\cite[Lemma 4.4]{R2020}).\nFor any $n \\in \\mathcal{N}$, let us define $\\mathcal{B}_n  \\in \\mathbb{R}^{6k_n \\times 6 k_n}$ by\n\\begin{linenomath}\n\\begin{align*}\n\\mathcal{B}_n = \\left\\{\\begin{aligned}\n&\\mathbf{G}_n^{-1}  \\mathbf{A}_n^{-1} \\mathbf{B}_n \\mathbf{G}_n && n \\in \\mathcal{N}_M\\\\\n&\\mathbf{I}_6 && n \\in \\mathcal{N}_S^N\\\\\n&-\\mathbf{I}_6 && n \\in \\mathcal{N}_S^D,\n\\end{aligned}\\right.\n\\end{align*}\n\\end{linenomath}\nas well as $\\mathcal{Q}_n \\in \\mathbb{R}^{6k_n \\times 6 k_n}$ and $\\mathbf{q}_n \\in C^1([0, T]; \\mathbb{R}^{6k_n})$ by\n\\begin{linenomath}\n\\begin{align*}\n\\mathcal{Q}_n = \\left\\{\\begin{aligned}\n&2 \\mathbf{G}_n^{-1}  \\mathbf{A}_n^{-1} && n \\in \\mathcal{N}_M\\\\\n&2 \\tau_{i^n}^n (D_{i^n} U_{i^n}\\mathbf{C}_{i^n}^{\\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) && n \\in \\mathcal{N}_S^N\\\\\n&2 (U_{i^n} \\mathbf{C}_{i^n}^{- \\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n)  && n \\in \\mathcal{N}_S^D,\n\\end{aligned}\\right. \\quad\n\\mathbf{q}_n(t) = \\begin{cases}\n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{bmatrix}\nq_n(t) \\\\ \\mathbf{0}_{6k_n-6, 1}\n\\end{bmatrix}\\endgroup & n \\in \\mathcal{N}_M\\\\\nq_n(t) & n \\in \\mathcal{N}_S.\n\\end{cases}\n\\end{align*}\n\\end{linenomath}\nThen, System \\eqref{eq:syst_diagonal} also reads\n\\begin{linenomath}\n\\begin{align*}\n\\begin{dcases}\n\\partial_t r_i + \\mathbf{D}_i(x) \\partial_x r_i + B_i(x) r_i = g_i(x, r_i) &\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}\\\\\nr^\\mathrm{out}_n(t) = \\mathcal{B}_n r^\\mathrm{in}_n(t) + \\mathcal{Q}_n \\mathbf{q}_n(t) & t \\in (0, T), \\, n \\in \\mathcal{N}\\\\\nr_i(x, 0) = r_i^0(x) & x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}.\n\\end{dcases}\n\\end{align*}\n\\end{linenomath}\n\n\n\n\\subsection{Proof of Theorem \\ref{th:existence}}\n\\label{subsec:proof_exist}\n\nRelying upon Subsections \\ref{subsec:hyperbolic}, \\ref{subsec:change_var} and \\ref{subsec:out_in_info}, and \\cite{li2010controllability, wang2006exact}, we now prove Theorem \\ref{th:existence}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:existence}]\n\nThe local and semi-global existence and uniqueness of $C_{x,t}^1$ solutions to general one-dimensional quasilinear hyperbolic systems have been addressed in \\cite[Lem. 2.3, Th. 2.1]{wang2006exact}, which is an extension of \\cite[Lem. 2.3, Th. 2.5]{li2010controllability} to nonautonomous systems.\n\n\nSuch results may be applied to the network system \\eqref{eq:syst_physical}, since it can be written as a single larger hyperbolic system. One needs only to apply the change of variable $\\widetilde{r}_i(\\xi, t) = r_i(\\ell_i \\ell^{-1} \\xi, t)$ for all $i \\in \\mathcal{I}$, $\\xi \\in [0, \\ell]$ and $t \\in [0, T]$ for some $\\ell>0$, in order to make the spatial domain identical for all beams, and consider the larger $\\mathbb{R}^{12N}$-valued unknown $\\widetilde{r} = (\\widetilde{r}_1^{\\,\\intercal}, \\ldots, \\widetilde{r}_N^{\\, \\intercal})^\\intercal$. Then, $\\widetilde{r}$ is governed by\n\\begin{linenomath}\n\\begin{align} \\label{eq:single_hyperb_syst}\n\\begin{cases}\n\\partial_t \\widetilde{r} + \\widetilde{\\mathbf{D}}(\\xi) \\partial_\\xi \\widetilde{r} + \\widetilde{B}(\\xi) \\widetilde{r} = \\widetilde{g}(\\widetilde{r}) & \\text{in }(0, \\ell)\\times(0, T)\\\\\n\\widetilde{r}^\\mathrm{\\, out}(t) = \\widetilde{\\mathcal{B}} \\, \\widetilde{r}^\\mathrm{\\, in}(t) + \\widetilde{Q}\\widetilde{\\mathbf{q}}(t) & t \\in (0, T)\\\\\n\\widetilde{r}(\\xi, 0) = \\widetilde{r}^0(\\xi) & \\xi \\in (0, \\ell),\n\\end{cases}\n\\end{align}\n\\end{linenomath}\nwhere $\\widetilde{\\mathbf{D}}, \\widetilde{B}, \\widetilde{\\mathcal{B}}, \\widetilde{\\mathcal{Q}}, \\widetilde{\\mathbf{q}}, \\widetilde{r}^\\mathrm{out}, \\widetilde{r}^\\mathrm{in}, \\widetilde{r}^0$ and $\\widetilde{g}$ are defined by\n\\begin{linenomath}\n\\begin{align*}\n&\\widetilde{\\mathbf{D}}(\\cdot) = \\ell \\mathrm{diag}\\left(\\ell_1^{-1}\\mathbf{D}_1(\\ell_1 \\ell^{-1} \\cdot), \\ldots, \\ell_N^{-1} \\mathbf{D}_N(\\ell_N \\ell^{-1} \\cdot) \\right),\\\\\n&\\widetilde{B}(\\cdot) = \\mathrm{diag}\\left(B_1(\\ell_1 \\ell^{-1} \\cdot), \\ldots, B_N(\\ell_N \\ell^{-1} \\cdot)\\right),\\\\\n&\\widetilde{\\mathcal{B}} = \\mathrm{diag}\\left(\\mathcal{B}_1, \\ldots, \\mathcal{B}_{\\#\\mathcal{N}}\\right), \\quad \\widetilde{\\mathcal{Q}} = \\mathrm{diag}\\left(\\mathcal{Q}_1, \\ldots, \\mathcal{Q}_{\\#\\mathcal{N}}\\right), \\quad \\widetilde{\\mathbf{q}} = (\\mathbf{q}_1^\\intercal, \\ldots, \\mathbf{q}_{\\# \\mathcal{N}}^\\intercal)^\\intercal,\\\\\n&\\widetilde{r}^\\mathrm{out} = \\left((r_1^\\mathrm{out})^\\intercal, \\ldots, (r_{\\#\\mathcal{N}}^\\mathrm{out})^\\intercal\\right)^\\intercal, \\quad \\widetilde{r}^\\mathrm{in} = \\left((r_1^\\mathrm{in})^\\intercal, \\ldots, (r_{\\#\\mathcal{N}}^\\mathrm{in})^\\intercal\\right)^\\intercal,\\\\\n&\\widetilde{r}^0(\\cdot) = \\left(r^0(\\ell_1 \\ell^{-1} \\cdot)^\\intercal, \\ldots, r^0(\\ell_N, \\ell^{-1} \\cdot)^\\intercal \\right)^\\intercal\\\\\n&\\widetilde{g}(\\cdot, \\mathbf{u}) = \\left(\\widetilde{g}_1(\\ell_1 \\ell^{-1} \\cdot, \\mathbf{u}_1)^\\intercal, \\ldots, \\widetilde{g}_N(\\ell_N \\ell^{-1} \\cdot, \\mathbf{u}_N)^\\intercal \\right)^\\intercal,\n\\end{align*}\n\\end{linenomath}\nwhere we denoted $\\mathbf{u} = (\\mathbf{u}_1^\\intercal, \\ldots, \\mathbf{u}_N^\\intercal)^\\intercal$ with $\\mathbf{u}_i \\in \\mathbb{R}^{12}$ for all $i \\in \\mathcal{I}$.\n\nDue to Subsection \\ref{subsec:out_in_info}, the boundary conditions of \\eqref{eq:single_hyperb_syst} are directly written in such a way that the outgoing information for System \\eqref{eq:single_hyperb_syst} is a function of the incoming information, a sufficient criteria in \\cite{li2010controllability, wang2006exact} to deduce well-posedness of the system.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Controllability of nodal profiles for the IGEB network}\n\\label{sec:controllability}\n\n\nWe now consider the A-shaped network defined by \\eqref{eq:A_netw} and our aim is to prove Theorem \\ref{th:controllability}. As pointed out in Section \\ref{sec:model_results}, we will solve several forward and sidewise problems for \\eqref{eq:IGEB_gov} (see Steps 1.3, 1.4, 1.5). The existence and uniqueness of semi-global in time solutions to these problems is provided by \\cite{li2010controllability, wang2006exact}, as in Section \\ref{sec:exist} for the overall network.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:controllability}]\nThe proof is divided in three steps.\nWe start by constructing a solution satisfying all transmission conditions and the nodal profiles.\nThe choice of $\\overline{T}$ (see \\eqref{eq:minT}), and thus $T^*$, is explained in Step 2. \n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.1 (see Fig. \\ref{fig:A} top-left).}\nConsider the forward problem for the entire network until time $\\overline{T}$, where at the simple nodes $n \\in \\{4,5\\}$, the controls $q_4, q_5$ are replaced by any functions $\\overline{q}_4, \\overline{q}_5 \\in C^1([0; \\overline{T}]; \\mathbb{R}^6)$ satisfying the first-order compatibility conditions of \\eqref{eq:syst_physical}. By Theorem \\ref{th:existence}, for any $\\gamma>0$ small enough, there exists $\\delta>0$ such that \\eqref{eq:syst_physical} admits a unique solution $(y_i^f)_{i\\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i]\\times[0, \\overline{T}]; \\mathbb{R}^{12})$ with $\\|y_i^f\\|_{C_{x,t}^1}\\leq \\gamma$, provided that $\\|y_i^0\\|_{C_x^1}+ \\|q_n\\|_{C_t^1}+\\|\\overline{q}_k\\|_{C_t^1} \\leq \\delta$ for all $i \\in \\mathcal{I}, n \\in \\{1, 2, 3\\}$ and $k \\in \\{4, 5\\}$.\n\n\nSimilarly to the state $y_i$ (see \\eqref{eq:form_yi}), we denote $y_i^f = ((v_i^f)^\\intercal, (z_i^f)^\\intercal)^\\intercal$, and later on, we will also use such a notation for $\\overline{y}_i$, $\\overline{\\overline{y}}_i$, $\\widetilde{y}_i$ and $\\mathbf{y}_i$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.2.}\nAt the node $n=1$, to obtain ``data'' $\\overline{\\overline{y}}_1, \\overline{\\overline{y}}_2 \\in C^1([0, T])$ for the entire time interval with small $C^1$ norm and fulfilling the transmission conditions at this node, we connect $y_i^f$ (from Step 1.1), which is defined on $[0, \\overline{T}]$, to the nodal profiles $\\overline{y}_i$ defined on $[T^*, T]$ (see \\eqref{eq:aim}).\n\n\nWe first find functions $\\overline{\\overline{v}}_1, \\overline{\\overline{z}}_1 \\in C^1([0, T]; \\mathbb{R}^6)$ \\textcolor{black}{with $\\|\\overline{\\overline{v}}_1\\|_{C_t^1} + \\|\\overline{\\overline{z}}_1\\|_{C_t^1} \\leq \\gamma$} and such that\n\\begin{linenomath}\n\\begin{align} \\label{eq:barbar_v1_z1}\n\\overline{\\overline{v}}_1(t) = \\left\\{ \\begin{aligned}\n&v_1^f(0, t) && t \\in [0, \\overline{T}]\\\\\n&\\overline{v}_1(t) && t \\in [T^*, T]\n\\end{aligned}\\right., \\quad \\overline{\\overline{z}}_1(t) = \\left\\{\\begin{aligned}\n&z_1^f(0, t) && t \\in [0, \\overline{T}]\\\\\n&\\overline{z}_1(t) && t \\in [T^*, T]\n\\end{aligned}\\right.,\n\\end{align}\n\\end{linenomath}\ncompleting the gap between via, for example, cubic Hermite splines fulfilling the values and first derivatives prescribed by \\eqref{eq:barbar_v1_z1} at $t = \\overline{T}$ and $t = T^*$. The $C^1$ norm of such functions is bounded by that of $v_i^f, \\overline{v}_i$ and $z_i^f,\\overline{z}_i$, respectively.\n\n\nThen, we define $\\overline{\\overline{v}}_2, \\overline{\\overline{z}}_2 \\in C^1([0, T]; \\mathbb{R}^6)$ by $\\overline{\\overline{v}}_2(t) = (\\overline{R}_2^\\intercal \\overline{R}_1)(0) \\overline{\\overline{v}}_1(t)$ and $\\overline{\\overline{z}}_2(t) = - (\\overline{R}_2^\\intercal \\overline{R}_1)(0) \\overline{\\overline{z}}_1(t)$, so that both the continuity and Kirchhoff conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} are fulfilled.\nSince $\\overline{R}_i$ ($i\\in\\mathcal{I}$) is unitary and independent of time, one has $|\\overline{\\overline{v}}_1| = |\\overline{\\overline{v}}_2|$ and $|\\overline{\\overline{z}}_1| = |\\overline{\\overline{z}}_2|$, as well as $|\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{v}}_1| = |\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{v}}_2|$ and $|\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{z}}_1| = |\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{z}}_2|$, \\textcolor{black}{implying that $\\|\\overline{\\overline{v}}_2\\|_{C_t^1} + \\|\\overline{\\overline{z}}_2\\|_{C_t^1}\\leq \\gamma$.}\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.3 (see Fig. \\ref{fig:A} top-right).}\nNow that we have $\\overline{\\overline{y}}_i$, we consider the sidewise (rightward) problem on $[0, \\ell_i] \\times [0, T]$ for the edges $i\\in \\{1, 2\\}$ (see Remark \\ref{rem:controllability_thm} \\ref{subrem:sidewise}), where at $x=0$ the ``initial data'' is $\\overline{\\overline{y}}_i$, at $t=0$ the ``boundary condition'' prescribes the velocities as $v_i(x,0) = v_i^0(x)$ (thus using a part of the initial conditions of System \\eqref{eq:syst_physical}), and at $t=T$ we set the artificial ``boundary condition'' $z_i(x,T) = \\overline{q}_i(x)$ for any function $\\overline{q}_i \\in C^1([0, \\ell_i]; \\mathbb{R}^6)$.\nThen, for any $\\varepsilon_1>0$ small enough, there exists $\\delta_1>0$ such that the rightward problem admits a unique solution $y_i \\in C^1([0, \\ell_i]\\times[0, T]; \\mathbb{R}^{12})$ with $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon_1$, provided that $\\|\\overline{\\overline{y}}_i\\|_{C_{t}^1} + \\|v_i^0\\|_{C_x^1} + \\|\\overline{q}_i\\|_{C_x^1}\\leq \\delta_1$ for all $i \\in \\{1, 2\\}$.\n\n\n\\begin{figure} \\centering\n\\includegraphics[scale=0.55]{AC}\n\\caption{Steps 1.1, 1.3, 1.4, 1.5 of the construction of the solution (top to bottom, left to right), where ``A.C.'' stands for ``artificial conditions''.}\n\\label{fig:A}\n\\end{figure}\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.4 (see Fig. \\ref{fig:A} bottom-left).}\nUsing $y_1(\\ell_1, \\cdot)$, $y_2(\\ell_2, \\cdot)$ (from Step 1.3) as data, consider the forward problem on $[0, \\ell_3]\\times[0, T]$ for the edge $i=3$, with the initial conditions of \\eqref{eq:syst_physical}, and, as boundary conditions at $x=0$ and $x=\\ell_3$, the velocities prescribed as\n\\begin{linenomath}\n\\begin{align} \\label{eq:pb3full_BC_0_l3}\nv_3 (0, t) = \\overline{R}_3(0)^\\intercal \\overline{R}_1(\\ell_1) v_1(\\ell_1, t), \\qquad \nv_3 (\\ell_3, t) = \\overline{R}_3(\\ell_3)^\\intercal \\overline{R}_2(\\ell_2) v_2(\\ell_2, t),\n\\end{align}\n\\end{linenomath}\nso that the obtained solution $y_3$ together with $y_1, y_2$ (provided by Step 1.3) fulfill the continuity conditions \\eqref{eq:IGEB_cont_velo} at the nodes $n \\in \\{2, 3\\}$.\nThen, for any $\\varepsilon_2>0$ small enough, there exists $\\delta_2>0$ such that this problem admits a unique solution $y_3 \\in C^1([0, \\ell_3]\\times[0, T]; \\mathbb{R}^{12})$ with $\\|y_3\\|_{C_{x,t}^1}\\leq \\varepsilon_2$, provided that $\\|y_3^0\\|_{c_x^1}+\\|v_i(\\ell_i, \\cdot)\\|_{C_t^1}+\\|q_n\\|_{C_t^1} \\leq \\delta_2$ for all $i \\in \\{1, 2\\}$ and $n \\in \\{2, 3\\}$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.5 (see Fig. \\ref{fig:A} bottom-right).}\nFinally, using $y_1(\\ell_1, \\cdot)$, $y_2(\\ell_2, \\cdot)$ (from Step 1.1) and $y_3(0, \\cdot)$, $y_3(\\ell_3, \\cdot)$ (from Step 1.3) as data, consider the rightward problem on $[0, \\ell_i]\\times[0, T]$ for the edges $i \\in \\{4, 5\\}$ similar to that of Step 1.3 except for the choice of the ``initial data'' at $x=0$, denoted by $\\widetilde{y}_i$, that we define by\n\\begin{linenomath}\n\\begin{align}\\label{eq:v45_x=0}\n\\widetilde{y}_4 &= \\begin{bmatrix}\n\\overline{R}_4(0)^\\intercal (\\overline{R}_1v_1)(\\ell_1, \\cdot)\\\\\n\\overline{R}_4(0)^\\intercal( (\\overline{R}_1z_1)(\\ell_1, \\cdot) - (\\overline{R}_3z_3)(0, \\cdot)-q_2)\n\\end{bmatrix}\\\\\n\\label{eq:z45_x=0}\n\\widetilde{y}_5 &= \\begin{bmatrix}\n\\overline{R}_5(0)^\\intercal (\\overline{R}_2 v_2)(\\ell_2, \\cdot)\\\\\n\\overline{R}_5(0)^\\intercal ((\\overline{R}_2 z_2)(\\ell_2, \\cdot) +(\\overline{R}_3 z_3)(\\ell_3, \\cdot) - q_3)\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\nThen, for any $\\varepsilon_3>0$ small enough, there exists $\\delta_3>0$ such that this problem admits a unique solution $y_i \\in C^1([0, \\ell_i]\\times[0 , T]; \\mathbb{R}^{12})$ with $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon_3$, provided that $\\|v_i^0\\|_{C_x^1}+ \\|\\overline{q}_i\\|_{C_x^1} + \\|q_n\\|_{C_t^1} \\leq \\delta_3$ for all $i \\in \\{4,5\\}$ and $n \\in\\{2, 3\\}$, and $\\|y_k(\\ell_k, \\cdot)\\|_{C_t^1} + \\|y_3(0, \\cdot)\\|_{C_t^1} \\leq \\delta_3$ for all $k\\in \\{1, 2, 3\\}$.\n\nNote that the $\\widetilde{y}_i$ for $i \\in \\{4, 5\\}$ have been chosen in such a way that the solutions $y_4, y_5$ together with $y_1, y_2, y_3$ (provided by Step 1.1 and Step 1.3), necessarily fulfill the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} at the nodes $n \\in \\{2, 3\\}$. \n\n\n\n\\medskip\n\n\n\\noindent It remains to prove that the solution $(y_i)_{i\\in\\mathcal{I}}$ constructed in Step 1 in fact also fulfills the initial conditions \\eqref{eq:IGEB_ini_cond} of the overall network, by showing that $y_i$ coincides with $y_i^f$ on some domain including $[0, \\ell_i]\\times \\{0\\}$ for all $i \\in \\mathcal{I}$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 2.1 (see Fig. \\ref{fig:A_ini} leftmost).}\nFirst, consider the edges $i \\in \\{1, 2\\}$. We will see that not only $y_i$ fulfills \\eqref{eq:IGEB_ini_cond}, but one also has (see \\eqref{eq:def_Lambdai_Ti})\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:coinc12_BC}\ny_i(\\ell_i, t) = y_i^f(\\ell_i, t), \\quad &t \\in [0, \\max\\{T_4, T_5\\}], \\, i \\in \\{1, 2\\}.\n\\end{align}\n\\end{linenomath}\nLet $i \\in \\{1, 2\\}$, and let $\\mathbf{t}_i \\in C^1([0, \\ell_i])$ be the function with derivative $\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{t}_i(x) = \\min_{1\\leq k \\leq 12} \\frac{1}{\\lambda_i^k(x)}$ in $[0, \\ell_i]$, which is also equal to $-\\Lambda_i(x)$ (see \\eqref{eq:sign_eigval}-\\eqref{eq:def_Lambdai_Ti}), and such that $\\mathbf{t}_i(0) = T_i + \\max \\{T_4, T_5\\}$. Then, $\\mathbf{t}_i$ describes a curve in $[0, \\ell_i]\\times[0, T]$ that passes through $(0, T_i + \\max\\{T_4, T_5\\})$ and we may also write\n\\begin{linenomath}\n\\begin{align*}\n    \\mathbf{t}_i(x) = T_i + \\max \\{T_4, T_5\\} - \\int_0^x \\Lambda_i(s) ds.\n\\end{align*}\n\\end{linenomath}\nThe definition of $T_i$ in \\eqref{eq:def_Lambdai_Ti} ensures that $[0, \\ell_i]\\times[0, \\max\\{T_4, T_5\\}]$ is a subset of the domain $\\mathcal{R}(i, \\mathbf{t}_i)$ defined by\n\\begin{linenomath}\n\\begin{align}\\label{eq:def_dom_calR}\n\\mathcal{R}(i, \\mathbf{t}_i) := \\{(x,t)\\colon 0 \\leq x \\leq \\ell_i, \\ 0 \\leq t \\leq \\mathbf{t}_i(x)\\}.\n\\end{align}\n\\end{linenomath}\nBoth $y_i$ and $y_i^f$ are by definition solutions to the one-sided sidewise (rightward) problem with ``initial data'' $\\overline{\\overline{y}}_i$ at $x=0$ and boundary data $v_i^0$ at $t=0$.\nThe definition of $\\mathbf{t}_i$ ensures that any characteristic curve\\footnote{By characteristic curves passing by $(x_\\circ,t_\\circ)$, we mean the curves specified by the functions $\\mathbf{t}_i^k$ with derivative $\\frac{\\mathrm{d}}{\\mathrm{d}s} \\mathbf{t}_i^k(s) = \\lambda_i^k(s)^{-1}$ and such that $\\mathbf{t}_i^k(x_\\circ) = t_\\circ$, for $k \\in \\{1, \\ldots, 12\\}$.\n}\nof this problem passing by $(x,t) \\in \\mathcal{R}(i, \\mathbf{t}_i)$ is necessarily entering the domain $\\mathcal{R}(i, \\mathbf{t}_i)$ at $\\{0\\} \\times [0, T_i+\\max \\{T_4, T_5\\}]$ or at $[0, \\ell_i] \\times \\{0\\}$. Thus, by \\cite[Section 1.7]{li2016book} the solution in $C^1(\\mathcal{R}(i, \\mathbf{t}_i); \\mathbb{R}^{12})$ to this sidewise problem is unique, and $y_i \\equiv y_i^f$ in $\\mathcal{R}(i, \\mathbf{t}_i)$.\n\n\n\\begin{figure}\\centering\n\\includegraphics[scale=0.7]{A_iniC}\n\\caption{Recovering the initial conditions: \\textcolor{black}{meaning of the controllability time}.}\n\\label{fig:A_ini}\n\\end{figure}\n\\medskip\n\n\n\\noindent \\textit{Step 2.2 (see Fig. \\ref{fig:A_ini} center).}\nConsider the edge $i=3$. We will show that not only $y_3$ fulfills \\eqref{eq:IGEB_ini_cond}, but also\n\\begin{linenomath}\n\\begin{align} \\label{eq:coinc3_BC}\ny_3(0, t) = y_3^f(0, t), \\quad y_3(\\ell_3, t) = y_3^f(\\ell_3, t), \\quad t \\in [0, \\max \\{T_4, T_5\\}]\n\\end{align}\n\\end{linenomath}\nholds. Indeed, $y_3$ and $y_3^f$ both solve the forward problem\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:pb3}}\n\\label{eq:pb3_gov}\n\\partial_t \\mathbf{y}_3 + A_i \\partial_x \\mathbf{y}_3 + \\overline{B}_i \\mathbf{y}_3 = \\overline{g}_i(\\cdot,\\mathbf{y}_3) &\\hspace{-0.45cm}$\\text{in } (0, \\ell_3)\\times(0, \\max \\{T_4, T_5\\})$\\\\\n\\label{eq:pb3_BC_0}\n\\mathbf{v}_3 (0, t) = \\overline{R}_3(0)^\\intercal (\\overline{R}_1 v_1)(\\ell_1, t) (t) &\\hspace{-0.45cm}$t\\in (0, \\max \\{T_4, T_5\\})$\\\\\n\\label{eq:pb3_BC_l3}\n\\mathbf{v}_3 (\\ell_3, t) = \\overline{R}_3(\\ell_3)^\\intercal (\\overline{R}_2 v_2)(\\ell_2, t) (t) &\\hspace{-0.45cm}$t\\in (0, \\max \\{T_4, T_5\\})$\\\\\n\\label{eq:pb3_ini}\n\\mathbf{y}_3(x,0) = y_3^0(x) &\\hspace{-0.45cm}$x \\in (0, \\ell_i)$,\n\\end{subnumcases}\n\\end{linenomath}\nwhich admits a unique solution in $C^1([0, \\ell_3]\\times[0, \\max\\{T_4, T_5\\}];\\mathbb{R}^{12})$.\nIn fact, $y_3$ fulfills \\eqref{eq:pb3} by definition (see Step 3); concerning $y_3^f$, it fulfills \\eqref{eq:pb3_gov} and \\eqref{eq:pb3_ini} by definition, while \\eqref{eq:coinc12_BC} and \\eqref{eq:pb3full_BC_0_l3} imply that $y_3^f$ fulfills \\eqref{eq:pb3_BC_0} and \\eqref{eq:pb3_BC_l3}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 2.3 (see Fig. \\ref{fig:A_ini} rightmost).}\nFinally, consider the edges $i\\in\\{4,5\\}$.\nLet $\\mathbf{t}_i$ be the function defined just as in Step 2.1 except that $\\mathbf{t}_i(0) = T_i$. In other words, \n\\begin{linenomath}\n\\begin{align*}\n\\mathbf{t}_i(x) = T_i - \\int_0^x \\Lambda_i(s)ds\n\\end{align*}\n\\end{linenomath}\nHere, the definition of $T_i$ in \\eqref{eq:def_Lambdai_Ti} ensures that $\\mathbf{t}_i(\\ell_i) = 0$, and therefore the corresponding domain $\\mathcal{R}(i, \\mathbf{t}_i)$ defined by \\eqref{eq:def_dom_calR}, contains $[0, \\ell_i] \\times \\{0\\}$.\nBoth $y_i$ and $y_i^f$ fulfill the following one-sided rightward problem with unknown $\\mathbf{y}_i$:\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:pb45}}\n\\label{eq:pb45_gov}\n\\partial_x \\mathbf{y}_i + A_i^{-1} \\partial_t \\mathbf{y}_i + A_i^{-1} \\overline{B}_i \\mathbf{y}_i = (A_i^{-1} \\overline{g}_i)(\\cdot, y_i)  & $\\text{in } \\mathcal{R}(i, \\mathbf{t}_i)$\\\\\n\\label{eq:pb45_BC}\n\\mathbf{v}_i(x,0) = v_i^0(x) & $x \\in (0, \\ell_i)$\\\\\n\\label{eq:pb45_ini}\n\\mathbf{y}_i (0, t) = \\widetilde{y}_i(t) & $t\\in (0, T_i)$,\n\\end{subnumcases}\n\\end{linenomath}\nwhere $\\widetilde{y}_i$ is defined by \\eqref{eq:v45_x=0}-\\eqref{eq:z45_x=0}.\nIndeed, while it is clear that $y_i$ fulfills \\eqref{eq:pb45} and $y_i^f$ fulfills \\eqref{eq:pb45_gov}-\\eqref{eq:pb45_BC} by definition, one also obtains, using \\eqref{eq:coinc12_BC}, \\eqref{eq:coinc3_BC} and the fact that $y_i^f$ satisfies the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff}, that $y_i^f$ also fulfills \\eqref{eq:pb45_ini}.\nThe definition of $\\mathbf{t}_i$ ensures that any characteristic curve of \\eqref{eq:pb45} passing through $(x,t) \\in \\mathcal{R}(i, \\mathbf{t}_i)$ is necessarily entering this domain at $\\{0\\} \\times [0,T_i]$ or at $[0, \\ell_i] \\times \\{0\\}$.\nHence, similarly to Step 2.1, one can apply \\cite[Section 1.7]{li2016book} to obtain that the solution in $C^1(\\mathcal{R}(i, \\mathbf{t}_i); \\mathbb{R}^{12})$ to \\eqref{eq:pb45} is unique.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 3.}\nFinally, we choose $q_i$ defined by $q_i(t) = z_i(\\ell_i, t)$ for all $t \\in [0, T], i \\in \\{4, 5\\}$. In view of the uniqueness of the solution to \\eqref{eq:syst_physical}, $q_4, q_5$ are controls satisfying the desired conditions of Theorem \\ref{th:controllability}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Relationship between the GEB and IGEB networks}\n\\label{sec:invert_transfo}\n\nAs in Section \\ref{sec:exist}, we now consider a general network, and seek to prove Theorem \\ref{thm:solGEB}. To do so, in Lemma \\ref{lem:invert_transfo} below, we start by inverting, on some specific spaces, the transformation $\\mathcal{T}$ defined in \\eqref{eq:transfo} that relates the states of \\eqref{eq:GEB_netw} and \\eqref{eq:syst_physical}.\nHenceforth, for any functions $(u_i)_{i \\in \\mathcal{I}}$ such that $u_i \\colon [0, \\ell_i]\\times[0, T]\\rightarrow \\mathbb{R}^{12}$, we use the notation $u_i=(u_{i,1}^\\intercal, \\ldots, u_{i, 4}^\\intercal)^\\intercal$, where $u_{i,k} \\colon [0, \\ell_i]\\times[0, T]\\rightarrow \\mathbb{R}^3$ for all $k \\in \\{1, \\ldots, 4\\}$.\nLet us define the spaces\n\\begin{linenomath}\n\\begin{align*}\nE_1 &= \\big\\{(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in {\\textstyle \\prod_{i=1}^N} C^2\\left([0, \\ell_i]\\times[0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3)\\right) \\colon \\eqref{eq:GEB_condNSv_p_R}, \\eqref{eq:GEB_IC_0ord} \\text{ hold} \\big\\}\\\\\nE_2 &= \\big\\{(y_i)_{i \\in \\mathcal{I}} \\in {\\textstyle \\prod_{i=1}^N} C^1\\left([0, \\ell_i]\\times[0, T] ; \\mathbb{R}^{12}\\right) \\colon u_i := \\mathrm{diag}(\\mathbf{I}_6, \\mathbf{C}_i) y_i \\text{ satisfies} \\\\\n&\\qquad \\text{\\eqref{eq:compat_last6eq}-\\eqref{eq:compat_ini}-\\eqref{eq:compat_nod}} \\big\\},\n\\end{align*}\n\\end{linenomath}\nwhere \\eqref{eq:compat_last6eq}-\\eqref{eq:compat_ini}-\\eqref{eq:compat_nod} are the following conditions:\n\\begingroup\n\\setlength{\\tabcolsep}{1pt}\n\\renewcommand{\\arraystretch}{0.75}\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:compat_last6eq}\n&\\begin{aligned}\n&\\text{for all }i \\in \\mathcal{I}, \\text{ in }(0, \\ell_i)\\times (0, T)\\\\\n&\\partial_t \\begin{bmatrix}\nu_{i,3} \\\\ u_{i,4}\n\\end{bmatrix} - \\partial_x \\begin{bmatrix}\nu_{i,1} \\\\ u_{i,2}\n\\end{bmatrix} - \\begin{bmatrix}\n\\widehat{\\Upsilon}_c^i & \\widehat{e}_1 \\\\\n\\mathbf{0}_3 & \\widehat{\\Upsilon}_c^i\n\\end{bmatrix}\\begin{bmatrix}\nu_{i,1} \\\\ u_{i,2}\n\\end{bmatrix}  = \n\\begin{bmatrix}\n\\widehat{u}_{i,2} & \\widehat{u}_{i,1}\\\\\n\\mathbf{0}_3 & \\widehat{u}_{i,2}\n\\end{bmatrix} \\begin{bmatrix}\nu_{i,3} \\\\ u_{i,4}\n\\end{bmatrix},\n\\end{aligned}\\\\\n\\label{eq:compat_ini}\n&\\begin{aligned}\n\\text{for all }i \\in \\mathcal{I}, \\text{ in } (0, \\ell_i), \\quad \\tfrac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{p}_i^0(\\cdot) &= \\mathbf{R}_i^0(\\cdot) (u_{i,3}(\\cdot, 0) + e_1),\\\\\n\\tfrac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{R}_i^0(\\cdot) &= \\mathbf{R}_i^0(\\cdot)(\\widehat{u}_{i,4}(\\cdot, 0) + \\widehat{\\Upsilon}_c^i(\\cdot)),\n\\end{aligned}\\\\\n\\label{eq:compat_nod}\n&\\begin{aligned}\n\\text{for all } n \\in \\mathcal{N}_S^D, \\text{ in } (0, T),\\quad \\tfrac{\\mathrm{d}}{\\mathrm{d}t} f_n^\\mathbf{p} (\\cdot) &= f_n^\\mathbf{R}(\\cdot) \\widehat{u}_{{i^n},1}(\\mathbf{x}_{i^n}^n, \\cdot),\\\\\n\\tfrac{\\mathrm{d}}{\\mathrm{d}t} f_n^\\mathbf{R} (\\cdot) &= f_n^\\mathbf{R}(\\cdot) \\widehat{u}_{{i^n},2}(\\mathbf{x}_{i^n}^n, \\cdot).\n\\end{aligned}\n\\end{align}\n\\end{linenomath}\n\\endgroup \nThe following result then holds.\n\n\n\n\\begin{lemma} \\label{lem:invert_transfo}\nAssume that $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, f_n^\\mathbf{p}, f_n^\\mathbf{R})$ are of regularity \\eqref{eq:reg_Idata_GEB} and \\eqref{eq:reg_Ndata_GEB_D}, and fulfill \\eqref{eq:compat_GEB_-1}.\nThen, the transformation $\\mathcal{T}\\colon E_1 \\rightarrow E_2$, defined in \\eqref{eq:transfo}, is bijective.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:invert_transfo}]\nOne can easily verify that $(\\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i))_{i \\in \\mathcal{I}}$ belongs to $E_2$ for any given $(\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}} \\in E_1$, and $\\mathcal{T}$ is thus well defined.\n\n\nLet $(y_i)_{i\\in\\mathcal{I}} \\in E_2$. We will now show that, there exists a unique $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$ such that $\\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}}) = (y_i)_{i\\in\\mathcal{I}}$.\nConsider $(u_i)_{i \\in \\mathcal{I}}$ defined by $u_i := \\mathrm{diag}(\\mathbf{I}_6, \\mathbf{C}_i) y_i$.\nLet $i \\in \\mathcal{I}$, and let $n$ be the index of any node such that $i \\in \\mathcal{I}^n$.\n\n\\medskip\n\n\\noindent There exists a unique solution $\\mathbf{R}_i \\in C^2([0, \\ell_i]\\times[0, T]; \\mathrm{SO}(3))$ to \n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:overdetR}}\n\\label{eq:overdetR_govt}\n\\partial_t \\mathbf{R}_i = \\mathbf{R}_i \\widehat{u}_{i,2}\n& $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetR_govx}\n\\partial_x \\mathbf{R}_i = \\mathbf{R}_i (\\widehat{u}_{i,4} + \\widehat{\\Upsilon}_c^i) & $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetR_IBC}\n\\mathbf{R}_i(\\mathbf{x}_i^n, 0) = \\mathbf{R}_i^0(\\mathbf{x}_i^n).\n\\end{subnumcases}\n\\end{linenomath}\nTo prove this, a possible way is to first rewrite \\eqref{eq:overdetR}, whose state has values in $\\mathrm{SO}(3)$, as a system with a $\\mathbb{R}^4$-valued state (using \\cite[Lem. 4.1]{RL2019}) via a parametrization of rotation matrices by quaternions \\cite{chou1992}, and then use \\eqref{eq:compat_last6eq} (last three equations) as compatibility conditions in order to deduce that the obtained system is well-posed (using \\cite[Lem. 4.3]{RL2019}); this procedure is detailed in \\cite[Section 4]{RL2019}.\n\n\\medskip\n\n\\noindent Having found $\\mathbf{R}_i$, consider the following system\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:overdetp}}\n\\label{eq:overdetp_govt}\n\\partial_t \\mathbf{p}_i = \\mathbf{R}_i u_{i,1}\n& $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetp_govx}\n\\partial_x \\mathbf{p}_i = \\mathbf{R}_i (u_{i,3} + e_1) & $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetp_IBC}\n\\mathbf{p}_i(\\mathbf{x}_i^n, 0) = \\mathbf{p}_i^0(\\mathbf{x}_i^n).\n\\end{subnumcases}\n\\end{linenomath}\nNote that \\eqref{eq:overdetp_govt} is equivalent to $\\mathbf{p}_i(x,t) = \\mathbf{p}_i(x,0) + \\int_0^t (\\mathbf{R}_i u_{i,1})(x, \\tau)d\\tau$. \nWithout loss of generality, assume that $\\mathbf{x}_i^n = 0$ (in the alternative case, the end of the proof is the same with each integral $+ \\int_{0}^x$ below replaced by $- \\int_x^{\\ell_n}$).\nBy \\eqref{eq:compat_ini} (first equation) and \\eqref{eq:overdetp_IBC}, in the above expression for $\\mathbf{p}_i(x,t)$, one may express the first term as $\\mathbf{p}_i(x,0) = \\mathbf{p}_i^0(\\mathbf{x}_i^n) + \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i^0(u_{i,3}^0+e_1))(s)ds$. Also, for any $x \\in [0, \\ell_i]$ and any $\\tau \\in [0, t]$ the integrand in the second term may be expressed as $(\\mathbf{R}_i u_{i,1})(x, \\tau) = (\\mathbf{R}_i u_{i,1})(\\mathbf{x}_i^n, \\tau) + \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i u_{i,1})(s, \\tau) ds$. Hence, \\eqref{eq:overdetp_govt} and \\eqref{eq:overdetp_IBC} are equivalent to\n\\begin{linenomath}\n\\begin{align} \\label{eq:pi_candidate}\n\\begin{aligned}\n\\mathbf{p}_i(x,t) &= \\mathbf{p}_i^0(\\mathbf{x}_i^n) + \\int_0^t (\\mathbf{R}_i u_{i,1})(\\mathbf{x}_i^n, \\tau)d\\tau\\\\\n&+ \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i^0(u_{i,3}^0+e_1))(s)ds + \\int_0^t \\int_{\\mathbf{x}_i^n}^x \\partial_x (\\mathbf{R}_i u_{i,1})(s, \\tau)d\\tau ds.\n\\end{aligned}\n\\end{align}\n\\end{linenomath}\nOn the other hand, we know that \\eqref{eq:pi_candidate} fulfills  $\\partial_t \\mathbf{p}_i(\\mathbf{x}_i^n, \\cdot) = (\\mathbf{R}_i u_{i,1})(\\mathbf{x}_i^n, \\cdot)$, while by \\eqref{eq:compat_last6eq} (first three equations), one has $\\partial_x (\\mathbf{R}_i u_{i,1}) = \\partial_t(\\mathbf{R}_i(u_{i,3}+e_1))$. The latter two facts, together with \\eqref{eq:compat_ini} (second equation), permit us to deduce that \\eqref{eq:pi_candidate} also writes as $\\mathbf{p}_i(x, t) = \\mathbf{p}(\\mathbf{x}_i^n, t) + \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i(u_{i,3} + e_1))(t,s)ds$.\nThus, \\eqref{eq:pi_candidate} is the unique solution to \\eqref{eq:overdetp}.\n\n\n\nFinally, note that, because of \\eqref{eq:compat_ini}, requiring \\eqref{eq:overdetR_IBC} and \\eqref{eq:overdetp_IBC} is equivalent to imposing the initial conditions \\eqref{eq:GEB_IC_0ord}. Moreover, in the case of $n \\in \\mathcal{N}_S^D$, due to \\eqref{eq:compat_GEB_-1} and \\eqref{eq:compat_nod}, requiring \\eqref{eq:overdetR_IBC} and \\eqref{eq:overdetp_IBC} is equivalent to imposing the nodal conditions \\eqref{eq:GEB_condNSv_p_R}. This concludes the proof of Lemma \\ref{lem:invert_transfo}.\n\\end{proof}\n\n\n\n\n\n\nWe now have the tools to prove Theorem \\ref{thm:solGEB}.\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:solGEB}]\nWe divide the proof in seven steps. Let $(y_i)_{i \\in \\mathcal{I}}$ be as in Theorem \\ref{thm:solGEB}, and let $(u_i)_{i \\in\\mathcal{I}}$ be defined by $u_i = \\mathrm{diag}(\\mathbf{I}_6, \\mathbf{C}_i) y_i$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1: inverting the transformation.}\nSince the last six equations in \\eqref{eq:IGEB_gov} hold for $(y_i)_{i\\in\\mathcal{I}}$, we know that \\eqref{eq:compat_last6eq} is fulfilled. On the other hand, the last six equations of the initial conditions \\eqref{eq:IGEB_ini_cond} with initial data \\eqref{eq:rel_inidata} yield \\eqref{eq:compat_ini}. Finally, the definition of the boundary data \\eqref{eq:def_qn_D}, together with the nodal conditions \\eqref{eq:IGEB_condNSv} on velocities, yield \\eqref{eq:compat_nod}. Hence, $(y_i)_{i\\in\\mathcal{I}} \\in E_2$, and by Lemma \\ref{lem:invert_transfo} there exists a unique $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in E_1$ such that\n\\begin{linenomath} \n\\begin{align} \\label{eq:transfo_inverted}\ny_i = \\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i), \\quad \\text{for all }i \\in \\mathcal{I}.\n\\end{align}\n\\end{linenomath}\n\nNow, we want to check that this ``candidate'' $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$, satisfies the rest of system \\eqref{eq:GEB_netw}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 2: governing equations.}\nUsing \\eqref{eq:transfo_inverted} and the first six governing equations in \\eqref{eq:IGEB_gov}, one can deduce that $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$ satisfies the governing system \\eqref{eq:GEB_gov} after some computations.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 3: conditions at simple nodes.}\nFor $n \\in \\mathcal{N}_S^N$, from \\eqref{eq:transfo_inverted} together with the nodal conditions \\eqref{eq:IGEB_condNSz} on forces and moments and the definition of $f_n$ (see \\eqref{eq:def_fn}), one can directly deduce that the nodal conditions \\eqref{eq:GEB_condNSz} hold.\n\n\nFor $n \\in \\mathcal{N}_S^D$, from \\eqref{eq:transfo_inverted} together with the nodal conditions \\eqref{eq:IGEB_condNSv} on velocities and initial conditions \\eqref{eq:GEB_IC_0ord}, we deduce that $(\\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot), \\mathbf{R}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot))$ satisfies\n\\begin{linenomath}\n\\begin{align} \\label{eq:nodPDE}\n\\left\\{ \\begin{aligned}\n&\\frac{\\mathrm{d}\\beta}{\\mathrm{d}t}(t) = \\beta(t)  \\widehat{q_n^W}(t), \\ \\ \\frac{\\mathrm{d}\\alpha}{\\mathrm{d}t}(t) = \\beta(t) q_n^V(t) && \\ \\text{ in }(0, T)\\\\\n&(\\alpha, \\beta)(0) = (\\mathbf{p}_{i^n}^0, \\mathbf{R}_{i^n}^0)(\\mathbf{x}_{i^n}^n),\n\\end{aligned} \\right.\n\\end{align}\n\\end{linenomath}\nof unknown state $(\\alpha, \\beta)$, where we denote $q_n = ((q_n^V)^\\intercal, (q_n^W)^\\intercal)^\\intercal$ with $q_n^V, q_n^W \\in C^1([0, T]; \\mathbb{R}^3)$.\nDue to \\eqref{eq:def_qn_D} and \\eqref{eq:compat_GEB_-1}, $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$ also satisfies \\eqref{eq:nodPDE}.\nOne may see that \\eqref{eq:nodPDE} admits a unique solution in $C^2([0, T]; \\mathbb{R}^{3}\\times \\mathrm{SO}(3))$. Indeed, as in the proof of Lemma \\ref{lem:invert_transfo}, one may replace \\eqref{eq:nodPDE} (first equation) by an equivalent equation whose unknown state is the quaternion \\cite{chou1992} parametrizing the rotation matrix $\\beta = \\beta(t)$ (see \\cite[Section 4]{RL2019} for more detail). Having then only vector valued unknowns, one can use the classical ODE theory. Thus, $(\\mathbf{p}_{i^n},\\mathbf{R}_{i^n})(\\mathbf{x}_{i^n}^n, \\cdot) \\equiv (f_n^\\mathbf{p}, f_n^\\mathbf{R})$, and the nodal conditions \\eqref{eq:GEB_condNSv_p_R} hold.\n \n\n\\medskip\n\n\n\\noindent \\textit{Step 4: remaining initial conditions.}\nOne recovers the initial conditions \\eqref{eq:GEB_IC_1ord} directly from the first six equations in \\eqref{eq:IGEB_ini_cond} and the definition of $y_i^0$ \\eqref{eq:rel_inidata}, together with \\eqref{eq:transfo_inverted}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 5: rigid joint condition.}\nIn order to show that $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$ fulfills the transmission conditions of \\eqref{eq:GEB_netw}, we start with the rigid joint condition. Let $n \\in \\mathcal{N}_M$.  For all $i \\in \\mathcal{I}^n$, let us define $\\Lambda_i \\in  C^1([0, T]; \\mathbb{R}^{3 \\times 3})$ by $\\Lambda_i(t) = (R_i \\mathbf{R}_i^\\intercal)(\\mathbf{x}_i^n, t)$. By the continuity condition \\eqref{eq:IGEB_cont_velo} (last three equations),\n\\begin{linenomath}\n\\begin{align} \\label{cont_derivative_angle}\n\\textstyle\n\\left(\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_i \\right) \\Lambda_i^\\intercal  = \\left(\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_{i^n} \\right)\\Lambda_{i^n}^\\intercal, \\quad \\text{in }(0, T), \\text{ for all }i\\in\\mathcal{I}^n.\n\\end{align}\n\\end{linenomath}\nLet $F_n := \\left(\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_{i^n} \\right)\\Lambda_{i^n}^\\intercal$ and $a_n := (R_{i^n}{\\mathbf{R}_{i^n}^0}^\\intercal)(\\mathbf{x}_{i^n}^n)$. By \\eqref{cont_derivative_angle}, \\eqref{eq:compat_GEB_transmi} (second equation) and the fact that \\eqref{eq:GEB_IC_0ord} holds (by Step 1), for all $i \\in \\mathcal{I}^n$, $\\Lambda_i$ fulfills\n\\begin{linenomath}\n\\begin{align*}\n\\begin{dcases}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_i(t) = F_n(t) \\Lambda_i(t) & \\text{for all }t \\in (0, T)\\\\\n\\Lambda_i(0) = a_n,\n\\end{dcases}\n\\end{align*}\n\\end{linenomath}\nwhich admits a unique $C^1([0, T]; \\mathbb{R}^{3 \\times 3})$ solution (see \\cite[Sec. 2.1 and Th. 4.1.1 or Coro. 2.4.4]{vrabie2004}, for instance). Hence, $\\Lambda_i \\equiv \\Lambda_j$ for all $i,j\\in\\mathcal{I}^n$, and the rigid joint condition \\eqref{eq:GEB_rigid_angles} holds. \n\n\nAs \\eqref{eq:GEB_rigid_angles} holds, we can now deduce the transmission conditions of \\eqref{eq:GEB_netw} that remain.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 6: continuity of the displacement.}\nLet $n \\in \\mathcal{N}_M$.\nBy \\eqref{eq:transfo_inverted} together with the rigid joint condition \\eqref{eq:GEB_rigid_angles} and the continuity condition \\eqref{eq:IGEB_cont_velo} (first three equations), one deduces that \n\\begin{linenomath}\n\\begin{align*}\n\\partial_t \\mathbf{p}_i(\\mathbf{x}_i^n, t)  = \\partial_t \\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, t), \\quad \\text{in }(0, T), \\text{ for all } i\\in\\mathcal{I}^n.\n\\end{align*}\n\\end{linenomath}\nUsing additionally \\eqref{eq:GEB_IC_0ord} with \\eqref{eq:compat_GEB_transmi} (first equation), we deduce that for all $i \\in \\mathcal{I}^n$, the function $\\mathbf{p}_i(\\mathbf{x}_i^n, \\cdot)$ fulfills the problem\n\\begin{linenomath}\n\\begin{align} \\label{eq:contdipl_ODE}\n\\begin{dcases}\n\\partial_t \\mathbf{p}_i(\\mathbf{x}_i^n, t) = h_n(t) & \\text{for all }t \\in (0, T)\\\\\n\\mathbf{p}_i(\\mathbf{x}_i^n, 0) = \\alpha_n,\n\\end{dcases}\n\\end{align}\n\\end{linenomath}\nwhere we denote $h_n := \\partial_t \\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot)$ and $\\alpha_n := \\mathbf{p}_{i^n}^0(\\mathbf{x}_{i^n}^n)$. Since the $C^1([0, T];\\mathbb{R}^3)$ solution to \\eqref{eq:contdipl_ODE} is unique, we conclude that \\eqref{eq:GEB_continuity_pi} holds.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 7: Kirchhoff condition.}\nOne recovers the Kirchhoff condition \\eqref{eq:GEB_Kirchhoff} from the rigid joint assumption \\eqref{eq:GEB_rigid_angles} together with \\eqref{eq:IGEB_Kirchhoff} and \\eqref{eq:transfo_inverted}.\n\n\nTo finish, the uniqueness of the solution to \\eqref{eq:GEB_netw} is a consequence of the uniqueness of the solution to \\eqref{eq:syst_physical} (Theorem \\ref{th:existence}) and of the bijectivity of the transformation $\\mathcal{T}$ (Lemma \\ref{lem:invert_transfo}). This concludes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Concluding remarks and outlook}\n\n\\label{sec:conclusion}\n\nIn this article, we have studied networks, possibly with cycles, of geometrically exact beams. Notably, we considered the representations of such beams in terms of either displacements and rotations expressed in a fixed coordinate system (GEB model), or velocities and internal forces\/moments expressed in a moving coordinate system attached to the beam (IGEB model), reflecting on the advantages and drawbacks of these two points of view, and the relationship between them. \nFor these beam networks, we addressed the problem of local exact controllability of nodal profiles in the special case of a network containing one cycle: the A-shaped network depicted in Fig. \\ref{subfig:AshapedNetwork}.\n\nThe fact that one has the possibility of expressing the beam model as a first-order semilinear hyperbolic system -- the IGEB model -- while keeping track of the link with the GEB model, permits us to give a proof of nodal profile controllability in line with works done on other one-dimensional hyperbolic systems -- e.g., wave equation, Saint-Venant equations, Euler equations \\cite{gu2011, gu2013, gugat10, li2010nodal, li2016book, kw2011, kw2014, YWang2019partialNP, Zhuang2018}. Namely, we used the existence and uniqueness theory of semi-global classical solutions to the network system, combined with a constructive method as in \\cite{Zhuang2018} to obtain adequate controls.\n\n\\medskip\n\n\\noindent \\textbf{Local nature of the results.}\n{\\color{black} \nLet us give some comments about the local nature of the nodal profile controllability result, Theorem \\ref{th:controllability}. This theorem notably implies that even though there might be large displacements and rotations of the beam -- due to the use of a geometrically exact (thus nonlinear) beam model --, we apply controls that subsequently keep these motions small. As noted in Remark \\ref{rem:controllability_thm} \\ref{subrem:global}, Theorem \\ref{th:controllability} focuses on the small data scenario and could possibly be a preliminary step in view of obtaining a global result.\n\nBesides, in the proof of Theorem \\ref{th:controllability}, some ``degree of freedom'' has not been used, as we rely on an existence and uniqueness result which has been established for general one-dimensional first-order quasilinear hyperbolic systems. Since we are considering a very specific model -- the IGEB model -- it would be interesting to establish an appropriate well-posedness result and keep track of the bounds on the initial and boundary data to obtain more quantitative information.\n\nOn another hand, as explained in the introduction, the GEB and IGEB models are valid as long as the strains $s_i(x,t)$ are small enough. The latter being proportional to the the internal forces and moments $z_i(x,t)$ (more precisely, they are given by $s_i = \\mathbf{C}_i z_i$ where we recall that $\\mathbf{C}_i(x)$ denotes the flexibility matrix), comparing this assumption to the smallness of the internal forces and moments required in Theorem \\ref{th:controllability} would also be of interest.\n}\n\n\\medskip\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale = 0.8]{otherControllableNet_colorC.pdf}\n    \\caption{Other networks for which local exact controllability of nodal profiles is achievable by following Algorithm \\ref{algo:control} (the numbers refer to the variable \\textsf{step}).}\n    \\label{fig:otherControllableNet}\n\\end{figure}\n\n\n\\noindent \\textbf{More general networks.} The A-shaped network is an illustrative example where the controllability of nodal profiles is achievable for a network with a cycle, but let us stress that similar arguments to those used in Section \\ref{sec:controllability} apply for various other networks, and with controls at different locations.\n\n\n\\begin{algorithm}\n\\DontPrintSemicolon\n\n\n\\Input{%\n$\\mathcal{I}, \\, \\mathcal{N}, \\, \\mathcal{N}_S, \\, k_n$\\tcp*{edges, nodes, simple nodes, degrees}\n\\hspace{1.2cm}$\\mathcal{I}^n$ for all $n\\in \\mathcal{N}$\\tcp*{edges incident to the node $n$}\n\\hspace{1.2cm}$\\mathcal{N}^i$ for all $i \\in \\mathcal{I}$\\tcp*{nodes at the tips of the edge $i$}\n\\hspace{1.2cm}$\\mathcal{P}$, $\\mathcal{C}$\\tcp*{charged nodes, controlled nodes}\n\\hspace{1.2cm}$\\mathcal{S}$\\tcp*{edges on control paths}\n}\n\n\n\\setcounter{AlgoLine}{0}\n\\ShowLn\n$J$ $\\leftarrow$ $[\\, 0, \\quad $for $n = 1 \\ldots \\#\\mathcal{N}\\, ]$; \\ \\lFor{all $n \\in \\mathcal{P}$}{($J(n)$ $\\leftarrow$ $k_n - 1$);} \n\\tcp*{amount $J(n)$ of data available at $n$ to solve sidewise}\n\n\\ShowLn\n$\\mathcal{F}$ $\\leftarrow$ $\\emptyset$;\\tcp*{solved edges}\n \n\\ShowLn\n\\textsf{step} $\\leftarrow$ $1$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{false};\\tcp*{to count the steps}\n\n\\ShowLn\n$\\mathcal{M}$  $\\leftarrow$ $\\mathcal{P}\\cup \\{n \\in \\mathcal{N} \\colon\\text{$n$ not incident with any edge in $\\mathcal{S}$}\\};$\\;\n \n\\ShowLn\n\\While(\\hfill \\tcp*[h]{while entire network not solved}){$\\mathcal{F} \\neq \\mathcal{I}$}{\n \n\\ShowLn\n\\For(\\hfill \\tcp*[h]{Principle 1}){$m =\\#\\mathcal{M}, \\ldots, 3, 2, 1$}{\n\n\\ShowLn\n\\For{all ${\\mathcal{N}}^\\dagger \\subseteq \\mathcal{M}$ such that $\\#\\mathcal{N}^\\dagger = m$}{\n\n\\ShowLn\n\\If{there exists a connected subgraph with nodes ${\\mathcal{N}}^\\dagger$ and edges ${\\mathcal{I}}^\\dagger$, such that ${\\mathcal{I}}^\\dagger \\cap ( \\mathcal{S}\\cup \\mathcal{F}) = \\emptyset$}{\n\n\\ShowLn\n  solve forward problem for the network $({\\mathcal{I}^\\dagger}, {\\mathcal{N}}^\\dagger)$;\\;\n  \n\\ShowLn\n\\lFor{all $n \\in {\\mathcal{N}}^\\dagger$}{\n  ($J(n)$ $\\leftarrow$ $J(n) + 1$);}\n  \n\\ShowLn\n$\\mathcal{M}$ $\\leftarrow$ $\\mathcal{M} \\cup {\\mathcal{N}}^\\dagger$; \\, $\\mathcal{F}$ $\\leftarrow$ $\\mathcal{F} \\cup {\\mathcal{I}}^\\dagger$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{true};\\;\n}}}\n\n\\ShowLn\n\\lIf{\\upshape \\textsf{moved} $=$ \\textsf{true} }{(\\textsf{step} $\\leftarrow$ \\textsf{step} $+ 1$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{false});}  \n   \n\\ShowLn\n\\For(\\hfill \\tcp*[h]{Principle 2}){all $n \\in \\mathcal{M}$}{\n\n\\ShowLn\n\\If(\\hfill \\tcp*[h]{if enough data at $n$}){$J(n) = k_n - 1$}{\n\n\\ShowLn\n\\For{all $i \\in \\mathcal{I}^n \\cap (\\mathcal{S} \\setminus \\mathcal{F}$)}{\n\n\\ShowLn\nsolve sidewise problem for the edge $i$ with ``initial \\;\n  conditions'' at the node $n$;\\;\n\n\\ShowLn\n$\\mathcal{M}$ $\\leftarrow$ $\\mathcal{M} \\cup \\mathcal{N}^i$; \\, $\\mathcal{F}$ $\\leftarrow$ $\\mathcal{F}\\cup \\{i\\}$;\\;\n\n\\ShowLn\n\\lFor{all $m \\in \\mathcal{N}^i$}{\n  ($J(m)$ $\\leftarrow$ $J(m) + 1$);}\n  } \n  \n\\ShowLn \n\\textsf{moved} $\\leftarrow$ \\textsf{true};\\;\n}}\n\n\\ShowLn\n\\lIf{\\upshape \\textsf{moved} $=$ \\textsf{true}}{(\\textsf{step} $\\leftarrow$ \\textsf{step} $+ 1$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{false});} \n}\n\n\\ShowLn  \nCompute controls $q_n$ by evaluating the trace at nodes $n\\in\\mathcal{C}$;\\;\n \n\\caption{Steps of controllability proof for other networks}\n\\label{algo:control}\n\\end{algorithm}\n\n\n\n\n\nLet us introduce some more notation. For any given network, we denote by $\\mathcal{P}$ and $\\mathcal{C}$ the set of indexes of the \\emph{charged nodes} and \\emph{controlled nodes} (see Section \\ref{sec:intro}), respectively.\nGiven a charged node $n\\in\\mathcal{P}$ and a controlled node $m\\in\\mathcal{C}$, a \\emph{control path} between $n$ and $m$ \\cite{YWang2019partialNP}, is any connected subgraph (of the current graph representing the beam network) forming a path graph\\footnote{A path graph is an oriented graph without cycle such that two of its nodes are of degree $1$, and all other nodes have a degree equal to $2$.} whose nodes of degree $1$ are $n$ and $m$. In Fig. \\ref{fig:otherControllableNet}, examples of control paths are highlighted by blue arrows.\n\n\n\\medskip\n\n\n\n\n\n\\noindent On a \\emph{tree-shaped} network -- hence without loop --, some conditions were proved to be \\emph{sufficient} for the exact controllability of nodal profiles to be achieved \\cite{gu2011, gu2013, li2016book, kw2011, kw2014, YWang2019partialNP}.\nIn \\cite{YWang2019partialNP} the authors are concerned with the wave equation and provide a controllability result for any given tree-shaped network with possibly several charged nodes. Moreover, in \\cite{YWang2019partialNP}, at a charged node $n \\in \\mathcal{P}$, profiles may be prescribed for only some (rather than all) of the edges incident with $n$, and profiles may be prescribed for only part of the state (which would translate in the case of \\eqref{eq:syst_physical} to prescribing the velocities only, or the internal forces and moments only, for example). This type of problem, which is then called \\emph{partial nodal profile controllability}, is not considered here.\n\n\nThe nodal profile controllability has also been established for the Saint-Venant system \\cite{Zhuang2021, Zhuang2018} for numerous networks \\emph{with cycles}, of various shapes and with several charged nodes.\n\n\\medskip\n\n\\noindent It arises, from these works, a series of conditions on the number and location of the charged nodes, which are sufficient to achieve the respective controllability goals. We refer notably to \\cite[Theorem 5.1]{YWang2019partialNP}, and to \\cite[Sections 7 and 8]{Zhuang2021}.\n\n\nIn the case of the beam networks considered in this article, these conditions become (recall that $k_n$ is defined as the degree of the node $n$)\n\\begin{enumerate}\n\\item The total number of controlled nodes $\\#\\mathcal{C}$ is equal to $\\sum_{n\\in \\mathcal{P}} k_n$.\n\\item For any $n \\in \\mathcal{P}$, there are $k_n$ controlled nodes connecting with it through control paths. These control paths have the charged node $n$ for sole common node. \n\\item The control paths corresponding to different charged nodes do not have any common node.\n\\end{enumerate}\nLet us stress again that we are restricting ourselves to the type of systems presented in Subsection \\ref{subsec:network_systems}. Namely, if a multiple node is controlled, then the control is applied at the Kirchhoff condition, while if a simple node is controlled, then the control is applied at either the first six (velocities) or last six (internal forces and moments) components of the state $y_i$, and at any charged node $n \\in\\mathcal{P}$ profiles are prescribed for all incident beams $i \\in \\mathcal{I}^n$ and for the entire state $y_i$.\n\n\n\\medskip\n\n\\noindent Then, one may use the \\emph{constructive} method as in Section \\ref{sec:controllability}, by following the steps instructed by Algorithm \\ref{algo:control}, for different networks; see Fig. \\ref{fig:otherControllableNet}.\nWe can assert that this algorithm yields a proof of controllability for the networks defined in Fig. \\ref{fig:otherControllableNet}, but not that it constitutes a proof for any given network.\n\nIn Algorithm \\ref{algo:control}, edges belonging to control paths are solved according to the \\emph{Principle 2} -- solving a sidewise problem as in the Steps 1.3 and 1.5 of the proof of Theorem \\ref{th:controllability} -- while the other edges are solved according to the \\emph{Principle 1} -- solving a forward problem similar to the Step 1.4 of the proof of Theorem \\ref{th:controllability}.\n\nAs noted here and in the above cited works, the conditions given to obtain controllability of nodal profiles are only \\emph{sufficient} to ensure the controllability result and the search for necessary and sufficient conditions is open.\n\n\n\n\n\n\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{Introduction}\n\nWe present a derivation of the spin multiplicities that occur in $n$-fold\ntensor products of spin-$j$ representations, $j^{\\otimes n}$. \\ We make use of\ngroup characters, properties of special functions, and asymptotic analysis of\nintegrals. \\ While previous derivations for some of our results are scattered\nthroughout the literature, especially for specific values of $j$, we provide\nhere a treatment that is self-contained, and valid for any $j$ and for any\n$n$. \\ We emphasize two types of novel features: \\ patterns that arise when\ncomparing different values of $j$, and asymptotic behavior for large $n$. \\ \n\nOur methods and results should be useful for various calculations. \\ In\nparticular, the asymptotic behavior that we obtain should be helpful in the\nanalysis of statistical problems such as the determination of partition\nfunctions. \\ In the last section some other applications are briefly\ndiscussed, including a problem of interest for quantum computing, namely, an\nestimation of the number of entangled states.\n\n\\section*{Basic Theory of Group Characters}\n\nThe \\href{https:\/\/en.wikipedia.org\/wiki\/Character_theory}{character\n$\\chi\\left(  R\\right)  $ of a group representation $R$} succinctly encodes\nconsiderable information about $R$, as is well-known \\cite{Texts}. \\ For\nirreducible representations the characters are orthogonal\n\\begin{equation}\n\\sum\\hspace{-0.2in}\\int~\\mu~\\chi^{\\ast}\\left(  R_{1}\\right)  \\chi\\left(\nR_{2}\\right)  =\\delta_{R_{1},R_{2}}\\ , \\label{OrthoChar\n\\end{equation}\nwhere the sum or integral is over the group parameter space with an\nappropriate measure $\\mu$.\n\nFor a Kronecker product of $n$ representations, the character is given by the\nproduct of the individual characters\n\\begin{equation}\n\\chi\\left(  R_{1}\\otimes R_{2}\\otimes\\cdots\\otimes R_{n}\\right)  =\\chi\\left(\nR_{1}\\right)  \\chi\\left(  R_{2}\\right)  \\cdots\\chi\\left(  R_{n}\\right)  \\ ,\n\\end{equation}\nfrom which follows an explicit expression for the number of times that a given\nrepresentation $R$ appears in the product (e.g., see \\cite{V} Chapter I\n\\S 4.7). \\ This multiplicity is\n\\begin{equation}\nM\\left(  R;R_{1},\\cdots,R_{n}\\right)  =\\sum\\hspace{-0.2in}\\int~\\mu~\\chi^{\\ast\n}\\left(  R\\right)  \\chi\\left(  R_{1}\\right)  \\chi\\left(  R_{2}\\right)\n\\cdots\\chi\\left(  R_{n}\\right)  \\ .\n\\end{equation}\nFor real characters, this is totally symmetric in $\\left\\{  R,R_{1\n,\\cdots,R_{n}\\right\\}  $, and it immediately shows that the number of times\n$R$ appears in the product $R_{1}\\otimes\\cdots\\otimes R_{n}$ is equal to the\nnumber of times the trivial or \\textquotedblleft singlet\\textquotedblrigh\n\\ representation appears in the product $R\\otimes R_{1}\\otimes\\cdots\\otimes\nR_{n}$.\n\n\\section*{The $SU\\left(  2\\right)  $ Case}\n\nConsider now the Lie group $SU\\left(  2\\right)  $. \\ In this case the\nirreducible representations are labeled by angular momentum or spin, $j$ or\n$s$, the classes of the group are specified by the angle of rotation about an\naxis, $\\theta$, and the characters are\n\\href{https:\/\/en.wikipedia.org\/wiki\/Chebyshev_polynomials#Trigonometric_definition}{Chebyshev\npolynomials of the second kind}, $\\chi_{j}\\left(  \\theta\\right)\n=U_{2j}\\left(  \\cos\\left(  \\theta\/2\\right)  \\right)  $. \\ Explicitly, for\neither integer or semi-integer $j$\n\\begin{equation}\n\\chi_{j}\\left(  \\theta\\right)  =\\frac{\\sin\\left(  \\left(  2j+1\\right)\n\\theta\/2\\right)  }{\\sin\\left(  \\theta\/2\\right)  }\\ .\n\\end{equation}\nThese characters are all real. \\ Therefore the number of times that spin $s$\nappears in the product $j_{1}\\otimes\\cdots\\otimes j_{n}$ i\n\\begin{equation}\nM\\left(  s,j_{1},\\cdots,j_{n}\\right)  =\\frac{1}{\\pi}\\int_{0}^{2\\pi}\\chi\n_{s}\\left(  2\\vartheta\\right)  \\chi_{j_{1}}\\left(  2\\vartheta\\right)\n\\cdots\\chi_{j_{n}}\\left(  2\\vartheta\\right)  \\sin^{2}\\vartheta~d\\vartheta\\ ,\n\\label{IntegralForm\n\\end{equation}\nwhere we have taken $\\theta=2\\vartheta$ to avoid having half-angles appear in\nthe invariant measure and the Chebyshev polynomials (e.g., see \\cite{V}\nChapter III \\S 8.1) thereby mapping the $SU\\left(  2\\right)  $ group manifold\n$0\\leq\\theta\\leq4\\pi$ to $0\\leq\\vartheta\\leq2\\pi$. \\ To re-emphasize earlier\nremarks, we note that (\\ref{IntegralForm}) is totally symmetric in $\\left\\{\ns,j_{1},\\cdots,j_{n}\\right\\}  $ and valid if $s$ or any of the $j$s are\ninteger or semi-integer, and we also note that $M\\left(  s,j_{1},\\cdots\n,j_{n}\\right)  =M\\left(  0,s,j_{1},\\cdots,j_{n}\\right)  $. \\ In general\n$M\\left(  s,j_{1},\\cdots,j_{n}\\right)  $ will obviously reduce to a finite sum\nof integers through use of the Chebyshev\n\\href{https:\/\/en.wikipedia.org\/wiki\/Chebyshev_polynomials#Products_of_Chebyshev_polynomials}{product\nidentity}, $U_{m}U_{n}=\\sum_{k=0}^{n}U_{m-n+2k}$ for $m\\geq n$.\n\\ Alternatively, the integral form (\\ref{IntegralForm}) for the multiplicity\nalways reduces to a finite sum of hypergeometric functions (e.g. see\n(\\ref{Hyper1}) and (\\ref{Hyper2})\\ to follow).\n\nIn particular, for $j_{1}=\\cdots=j_{n}=j$, the $n$-fold product $j^{\\otimes\nn}$ can yield spin $s$ a number of times, as given b\n\\begin{equation}\nM\\left(  s;n;j\\right)  =\\frac{1}{\\pi}\\int_{0}^{2\\pi}\\sin\\left(  \\left(\n2s+1\\right)  \\vartheta\\right)  \\left(  \\frac{\\sin\\left(  \\left(  2j+1\\right)\n\\vartheta\\right)  }{\\sin\\left(  \\vartheta\\right)  }\\right)  ^{n}\\sin\n\\vartheta~d\\vartheta\\ , \\label{ProdMultInt\n\\end{equation}\nfor $s,j\\in\\left\\{  0,\\frac{1}{2},1,\\frac{3}{2},2,\\cdots\\right\\}  $. \\ Yet\nagain, we note that $M\\left(  j;n;j\\right)  =M\\left(  0;n+1;j\\right)  $.\n\\ Moreover, the symmetry of the integrand in (\\ref{ProdMultInt}) permits us to\nwrit\n\\begin{equation}\nM\\left(  s;n;j\\right)  =\\int_{0}^{2\\pi}\\frac{\\exp\\left(  2is\\vartheta\\right)\n}{2\\pi}\\left(  \\frac{\\sin\\left(  \\left(  2j+1\\right)  \\vartheta\\right)  \n{\\sin\\left(  \\vartheta\\right)  }\\right)  ^{n}~d\\vartheta-\\int_{0}^{2\\pi\n\\frac{\\exp\\left(  2i\\left(  s+1\\right)  \\vartheta\\right)  }{2\\pi}\\left(\n\\frac{\\sin\\left(  \\left(  2j+1\\right)  \\vartheta\\right)  }{\\sin\\left(\n\\vartheta\\right)  }\\right)  ^{n}~d\\vartheta\\ .\n\\end{equation}\nEach integral in the last expression reduces to a simple residue\n\\begin{equation}\n\\int_{0}^{2\\pi}\\frac{\\exp\\left(  2is\\vartheta\\right)  }{2\\pi}\\left(\n\\frac{\\sin\\left(  \\left(  2j+1\\right)  \\vartheta\\right)  }{\\sin\\left(\n\\vartheta\\right)  }\\right)  ^{n}~d\\vartheta=\\frac{1}{2\\pi i}\\oint\nz^{2s}\\left(  \\frac{z^{2j+1}-z^{-2j-1}}{z-z^{-1}}\\right)  ^{n}\\frac{dz\n{z}=c_{0}\\left(  s,n,j\\right)  \\ ,\n\\end{equation}\nwhere $c_{k}$ are the coefficients in the Laurent expansion of the integrand\n\\begin{equation}\nz^{2s}\\left(  \\frac{z^{2j+1}-z^{-2j-1}}{z-z^{-1}}\\right)  ^{n}=z^{2s}\\left(\n\\sum_{m=0}^{2j}z^{2\\left(  m-j\\right)  }\\right)  ^{n}=\\sum_{k=-2\\left(\njn-s\\right)  }^{2\\left(  jn+s\\right)  }z^{k}~c_{k}\\left(  s,n,j\\right)  \\ .\n\\end{equation}\nThat is to say, $c_{0}$ is the coefficient of $z^{-2s}$ (or of $z^{+2s}$) in\nthe Laurent expansion of $(z^{-2j}+z^{-2j+2}+\\cdots\\allowbreak+z^{2j-2\n+z^{2jn})$ \\cite{Katriel}, a coefficient that is easily obtained, e.g. using\neither Maple$^{\\textregistered}$ or Mathematica$^{\\textregistered}$. \\ \n\n\\subsection*{Explicit $SU\\left(  2\\right)  $ Results as Binomial Coefficients}\n\nSo then, the multiplicity is always given by a difference,\n\\begin{equation}\nM\\left(  s;n;j\\right)  =c_{0}\\left(  s,n,j\\right)  -c_{0}\\left(\ns+1,n,j\\right)  \\ , \\label{MIsADifference\n\\end{equation}\nwhere $2s$ is any integer such that $0\\leq2s\\leq2nj$, and where $s=0$ is\nalways allowed when $j$ is an integer but is only allowed for even $n$ when\n$j$ is a semi-integer. \\ To be more explicit, the expansion of $\\left(\nz^{-2j}+z^{-2j+2}+\\cdots+z^{2j-2}+z^{2j}\\right)  ^{n}$ involves so-called\n\\textquotedblleft generalized binomial coefficients\\textquotedblright\\ (see\nEqn(3) in \\cite{Bollinger}) which can be written as sums of products of the\nusual binomial coefficients. \\ Eventually (see Lemma 6 in \\cite{Kirillov} and\nthe Appendix in \\cite{Mendonca}) this leads t\n\\begin{equation}\nc_{0}\\left(  s,n,j\\right)  =\\sum_{k=0}^{\\left\\lfloor \\frac{nj+s\n{2j+1}\\right\\rfloor }\\left(  -1\\right)  ^{k}\\binom{n}{k}\\binom{nj+s-\\left(\n2j+1\\right)  k+n-1}{nj+s-\\left(  2j+1\\right)  k}\\ .\n\\end{equation}\nFor example, if $j=1\/2$ the $c_{0}$s reduce to a single binomial coefficient\n\\cite{Bethe}\n\\begin{equation}\nc_{0}\\left(  s,n,1\/2\\right)  =\\binom{n}{n\/2-s}\\ ,\\ \\ \\ M\\left(\ns;n;1\/2\\right)  =\\binom{n}{n\/2-s}-\\binom{n}{n\/2-s-1}\\ , \\label{SpinHalf\n\\end{equation}\nwhere $0\\leq2s\\leq n$, with $s=0$ allowed only for even $n$.\n\n\\subsection*{A Lattice of Multiplicities}\n\nOne may visualize $M\\left(  s;n;j\\right)  $ as a 3-dimensional semi-infinite\nlattice of points $\\left(  s;n;j\\right)  $ with integer multiplicities\nappropriately assigned to each lattice point. \\ There are many straight lines\non this lattice such that the multiplicities are polynomial in the line\nparameterization. \\ For example, along some of the lattice diagonals\n\\begin{equation}\nM\\left(  n;n;1\\right)  =1\\ ,\\ \\ \\ M\\left(  n-1;n;1\\right)\n=n-1\\ ,\\ \\ \\ M\\left(  n-2;n;1\\right)  =\\tfrac{1}{2}~n\\left(  n-1\\right)  \\ .\n\\end{equation}\nThese are, respectively, the number of ways the highest possible spin (i.e.\n$s=n$), the 2nd highest spin ($s=n-1$), and the 3rd highest spin ($s=n-2$)\noccur in the Kronecker product of $n$ vector (i.e. $s=1$) representations.\n\\ The form for the number of spins farther below the maximum $s=n$, that occur\nin products of $n$ vectors, i\n\\begin{subequations}\n\\begin{align}\nM\\left(  n-\\left(  2k+2\\right)  ;n;1\\right)   &  =\\tfrac{1}{\\left(\n2k+2\\right)  !}~n\\left(  n-1\\right)  \\left(  n-2\\right)  \\cdots\\left(\nn-k\\right)  \\times p_{k+1}\\left(  n\\right)  \\ ,\\\\\nM\\left(  n-\\left(  2k+3\\right)  ;n;1\\right)   &  =\\tfrac{1}{\\left(\n2k+3\\right)  !}~n\\left(  n-1\\right)  \\left(  n-2\\right)  \\cdots\\left(\nn-k\\right)  \\times q_{k+2}\\left(  n\\right)  \\ ,\n\\end{align}\nfor $k=0,1,2,3,\\cdots$, where $p_{k+1}$ and $q_{k+2}$ are polynomials in $n$\nof order $k+1$ and $k+2$, as follows.\n\\end{subequations}\n\\begin{subequations}\n\\begin{align}\np_{k+1}\\left(  n\\right)   &  =n^{k+1}+\\tfrac{1}{2}\\left(  k+1\\right)  \\left(\n5k-2\\right)  n^{k}+\\tfrac{1}{24}\\left(  k\\right)  \\left(  k+1\\right)  \\left(\n75k^{2}-205k-134\\right)  n^{k-1}+\\cdots\\ ,\\\\\nq_{k+2}\\left(  n\\right)   &  =n^{k+2}+\\tfrac{1}{2}\\left(  k\\right)  \\left(\n5k+7\\right)  n^{k+1}+\\tfrac{1}{24}\\left(  k+1\\right)  \\left(  75k^{3\n-85k^{2}-410k-168\\right)  n^{k}+\\cdots\\ .\n\\end{align}\nAs an exercise, the reader may verify the complete polynomials for orders $2$,\n$3$, $4$, and $5$\n\\end{subequations}\n\\begin{gather}\np_{2}\\left(  n\\right)  =n^{2}+3n-22\\ ,\\ \\ \\ q_{2}\\left(  n\\right)\n=n^{2}-7\\ ,\\\\\np_{3}\\left(  n\\right)  =n^{3}+12n^{2}-61n-192\\ ,\\ \\ \\ q_{3}\\left(  n\\right)\n=n^{3}+6n^{2}-49n+6\\ ,\\nonumber\\\\\np_{4}\\left(  n\\right)  =n^{4}+26n^{3}-37n^{2}-1622n+120\\ ,\\ \\ \\ q_{4}\\left(\nn\\right)  =n^{4}+17n^{3}-91n^{2}-587n+1200\\ ,\\nonumber\\\\\np_{5}\\left(  n\\right)  =n^{5}+45n^{4}+205n^{3}-5565n^{2\n-17486n+48720\\ ,\\ \\ \\ q_{5}\\left(  n\\right)  =n^{5}+33n^{4}-23n^{3\n-3393n^{2}+2542n+21000\\ .\\nonumber\n\\end{gather}\nAt the time of writing, the authors have not managed to identify the $p_{k}$\nand $q_{k}$ polynomial sequences with any that were previously studied.\n\n\\subsection*{Tabulating Some Examples}\n\nFor more explicit examples, we tabulate the number of singlets that appear in\nproducts $j^{\\otimes n}$ for $j=1,\\cdots,9$ and for $n=1,\\cdots,10$. \\ The\nTable entries below were obtained just by evaluation of the integrals in\n(\\ref{ProdMultInt}) for $s=0$\n\\\n\\begin{array}\n[c]{cccccccccc\n\\mathsf{M}\\left(  \\mathsf{0;n;j}\\right)  &\n\\text{\\text{\\href{https:\/\/oeis.org\/A005043}{\\text{j = 1}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A007043}{\\text{j = 2}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A264608}{\\text{j = 3}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A272393}{\\text{j = 4}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A272395}{\\text{j = 5}}}} & \\mathsf{j=6} &\n\\mathsf{j=7} & \\mathsf{j=8} & \\mathsf{j=9}\\\\\n\\mathsf{n=1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\\mathsf{n=2} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n\\mathsf{n=3} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n\\mathsf{n=4} & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19\\\\\n\\text{\\textsf{\\text{\\href{https:\/\/oeis.org\/A005891}{\\text{n = 5}}}}} & 6 &\n16 & 31 & 51 & 76 & 106 & 141 & 181 & 226\\\\\n\\text{\\textsf{\\text{\\href{https:\/\/oeis.org\/A005917}{\\text{n = 6}}}}} & 15 &\n65 & 175 & 369 & 671 & 1105 & 1695 & 2465 & 3439\\\\\n\\mathsf{n=7} & 36 & 260 & 981 & 2661 & 5916 & 11\\,516 & 20\\,385 & 33\\,601 &\n52\\,396\\\\\n\\mathsf{n=8} & 91 & 1085 & 5719 & 19\\,929 & 54\\,131 & 124\\,501 & 254\\,255 &\n474\\,929 & 827\\,659\\\\\n\\mathsf{n=9} & 232 & 4600 & 33\\,922 & 151\\,936 & 504\\,316 & 1370\\,692 &\n3229\\,675 & 6836\\,887 & 13\\,315\\,996\\\\\n\\mathsf{n=10} & 603 & 19\\,845 & 204\\,687 & 1178\\,289 & 4779\\,291 &\n15\\,349\\,893 & 41\\,729\\,535 & 100\\,110\\,977 & 217\\,915\\,579\n\\end{array}\n\\]\n\n\n\\subsubsection*{Nonpolynomial Columns}\n\nThe columns of the Table are \\emph{not} expressible as polynomials in $n$, for\nany fixed $j$, but they may be written as sums of hypergeometric or rational\nfunctions of $n$. \\ For example, the first two columns may be written a\n\\begin{align}\nM\\left(  0;n;1\\right)   &  =3^{n}\\sum_{k=0}^{n}\\binom{n}{k}\\binom{2k+1\n{k+1}\\left(  -\\frac{1}{3}\\right)  ^{k}=\\frac{4^{n}}{\\sqrt{\\pi}}\\frac\n{\\Gamma\\left(  \\frac{1}{2}+n\\right)  }{\\Gamma\\left(  2+n\\right)  }\\left.\n_{2}F_{1}\\right.  \\left(  -n,-1-n;\\frac{1}{2}-n;\\frac{1}{4}\\right)\n\\ ,\\label{Hyper1}\\\\\nM\\left(  0;n;2\\right)   &  =\\frac{1}{2}\\sum_{k=0}^{n}\\frac{\\left(  -6\\right)\n^{k}~\\Gamma\\left(  \\frac{1}{2}+k\\right)  }{\\Gamma\\left(  1+\\frac{k}{2}\\right)\n\\Gamma\\left(  \\frac{3}{2}+\\frac{k}{2}\\right)  }\\binom{n}{k}\\left.  _{3\nF_{2}\\right.  \\left(  \\frac{1}{4}+\\frac{k}{2},\\frac{3}{4}+\\frac{k\n{2},k-n;1+\\frac{k}{2},\\frac{3}{2}+\\frac{k}{2};-16\\right)  \\ . \\label{Hyper2\n\\end{align}\nTo obtain these and other multiplicities as hypergeometric functions, for\ninteger $s$ and $j$, it is useful to change variables, to $t=\\cos^{2\n\\vartheta$, so that (\\ref{ProdMultInt}) becomes\n\\begin{equation}\nM\\left(  s;n;j\\right)  =\\frac{2}{\\pi}~4^{s+nj}\\int_{0}^{1}\\left(\n\\prod\\limits_{k=1}^{s}\\left(  t-r_{k}\\left(  s\\right)  \\right)  \\right)\n\\left(  \\prod\\limits_{l=1}^{j}\\left(  t-r_{l}\\left(  j\\right)  \\right)\n\\right)  ^{n}\\sqrt{\\frac{1-t}{t}}~dt\\ . \\label{HyperForm\n\\end{equation}\nThe products here involve the known roots $r_{l}\\left(  j\\right)  $ of the\nChebyshev polynomials. \\ For integer $j$,\n\\begin{equation}\nU_{2j}\\left(  \\cos\\left(  \\vartheta\\right)  \\right)  =4^{j}\\prod\n\\limits_{l=1}^{j}\\left(  t-r_{l}\\left(  j\\right)  \\right)  \\ ,\\ \\ \\ t=\\cos\n^{2}\\vartheta\\ ,\\ \\ \\ r_{l}\\left(  j\\right)  =\\cos^{2}\\left(  \\frac{l\\pi\n}{2j+1}\\right)  \\ , \\label{ChebPolys&Roots\n\\end{equation}\nwhile for semi-integer $j$, for comparison to the integer case,\n\\begin{equation}\nU_{2j}\\left(  \\cos\\left(  \\vartheta\\right)  \\right)  =4^{j}\\sqrt{t\n~\\prod\\limits_{l=1}^{j-\\frac{1}{2}}\\left(  t-\\ r_{l}\\left(  j\\right)  \\right)\n\\ ,\n\\end{equation}\nwith the usual convention that the empty product is $1$.\n\nThe columns of the Table should be compared to the multiplicities of integer\nspins that appear in the product of $2m$ spin $1\/2$ representations. \\ These\nare well-known to be given by the\n\\href{https:\/\/en.wikipedia.org\/wiki\/Catalan's_triangle}{Catalan triangle}\n\\cite{SU(2)q}\n\\begin{equation}\nM\\left(  s;2m;1\/2\\right)  =\\frac{\\left(  1+2s\\right)  \\left(  2m\\right)\n!}{\\left(  m-s\\right)  !\\left(  m+s+1\\right)  !}\\ , \\label{Catalan\n\\end{equation}\nas follows from (\\ref{SpinHalf}). \\ As an aside, it is perhaps not so\nwell-known that multiplicities of all $SU\\left(  N\\right)  $ representations\noccurring in the product of $n$ fundamental $N$-dimensional representations\nare given by \\href{http:\/\/oeis.org\/A005789}{$N$-dimensional Catalan\nstructures} \\cite{MultiCat,SU(N)}. \\ \n\nBe that as it may, this aside suggests an alternate route to obtain and to\nre-express some of the above results, especially for $j=1$, a route that\n\\emph{retraces} [pun intended] many of the logical steps. \\ This other route\nuses the explicit formula \\cite{SU(N)} for products of fundamental triplets of\nthe group $SU\\left(  3\\right)  $\\ and the \\textquotedblleft tensor\nembedding\\textquotedblright\\ $SU\\left(  3\\right)  \\supset SU\\left(  2\\right)\n$ (where the triplet of $SU\\left(  3\\right)  $ is identified with the $s=1$\nvector representation) to deduce the number of $s=0$ singlets appearing in the\nproduct of $n$ vector representations of $SU\\left(  2\\right)  $, namely,\n\\begin{equation}\nM\\left(  0;n;1\\right)  =\\left(  -1\\right)  ^{n}\\left.  _{2}F_{1}\\right.\n\\left(  -n,\\tfrac{1}{2};2;4\\right)  \\ .\n\\end{equation}\nThis is in exact agreement with the seemingly different result (\\ref{Hyper1}).\n\\ Combining this with the elementary recursion relation that follows from\n$\\overrightarrow{s}\\otimes\\overrightarrow{1}=\\overrightarrow{s+1\n\\oplus\\overrightarrow{s}\\oplus\\overrightarrow{s-1}$, namely\n\\begin{equation}\nM\\left(  s;n;1\\right)  =M\\left(  s+1;n-1;1\\right)  +M\\left(  s;n-1;1\\right)\n+M\\left(  s-1;n-1;1\\right)  \\ ,\n\\end{equation}\none then obtains $M\\left(  s;n;1\\right)  $ as a sum of Gauss hypergeometric\nfunctions. \\ Relations between contiguous functions then simplify the result\nto a single hypergeometric function\n\\begin{equation}\nM\\left(  s;n;1\\right)  =\\left(  -1\\right)  ^{n+s}\\binom{n}{s}\\left.  _{2\nF_{1}\\right.  \\left(  s-n,s+\\frac{1}{2};2+2s;4\\right)  \\ .\n\\end{equation}\nFinally, the standard integral representation for $\\left.  _{2}F_{1}\\right.  $\neventually leads to the same integral form for $M\\left(  s;n;1\\right)  $ as\ngiven by (\\ref{ProdMultInt}) for $j=1$.\n\n\\subsubsection*{Polynomial Rows}\n\nIn contrast to the columns, the rows of the Table \\emph{are} expressible as\npolynomials in $j$ for any fixed $n$. \\ Starting with $n=3$, the entries in\nthe $n$th row of the Table are polynomials in $j$ of order $n-3$. \\ The fourth\nrow is obviously just the dimension of the spin $j$ representation, and the\nfifth row is less obviously $1+\\frac{5}{2}c_{j}$, where $c_{j}$ is the\nquadratic $su\\left(  2\\right)  $ Casimir for spin $j$. \\ In fact, based on the\nnumbers displayed above and some modest extensions of the Table, the row\nentries are seen to be of the form $poly_{\\left(  n-3\\right)  \/2}\\left(\nc_{j}\\right)  $ for odd $n\\geq3$ and $poly_{\\left(  n-4\\right)  \/2}\\left(\nc_{j}\\right)  \\times d_{j}$ for even $n\\geq4$, where $poly_{k}\\left(\nc\\right)  $ is a polynomial in $c$ of order $k$. \\ For the\\ last eight rows of\nthe Table these polynomials are given by\n\\begin{gather\n\\begin{array}\n[c]{ccccc\n\\mathsf{n=3} & 1\\medskip &  & \\mathsf{n=4} & d_{j}\\medskip\\\\\n\\mathsf{n=5} & 1+\\frac{5}{2}c_{j}\\medskip &  & \\mathsf{n=6} & \\left(\n1+2c_{j}\\right)  d_{j}\\medskip\\\\\n\\mathsf{n=7} & 1+\\frac{14}{3}c_{j}+\\frac{77}{12}c_{j}^{2}\\medskip &  &\n\\mathsf{n=8} & \\left(  1+4c_{j}+\\frac{16}{3}c_{j}^{2}\\right)  d_{j}\\medskip\\\\\n\\mathsf{n=9} & 1+\\frac{27}{4}c_{j}+\\frac{73}{4}c_{j}^{2}+\\frac{289}{16\nc_{j}^{3}\\medskip &  & \\mathsf{n=10} & \\left(  1+6c_{j}+\\frac{143}{9}c_{j\n^{2}+\\frac{140}{9}c_{j}^{3}\\right)  d_{j}\\medskip\n\\end{array}\n\\\\\n\\text{where\\ }d_{j}=1+2j\\ ,\\ \\ \\ \\text{and\\ \\ \\ }c_{j}=j\\left(  1+j\\right)\n\\ .\n\\end{gather}\nThus the\\ ten rows of the Table may be effortlessly extended to arbitrarily\nlarge $j$. \\ Moreover, to obtain the polynomial that gives any row for $n>10$,\nfor arbitrary values of $j$, it is only necessary to evaluate $M\\left(\n0;n;j\\right)  $ for $1\\leq j\\leq\\left\\lfloor \\frac{n-1}{2}\\right\\rfloor $.\n\\ Once again, at the time of writing, the authors have not managed to identify\nthis polynomial sequence with any that were previously studied.\n\n\\subsection*{Asymptotic Behavior}\n\nFinally, consider the extension of the columns of the Table to arbitrarily\nlarge $n$, or more generally, consider the asymptotic behavior of $M\\left(\ns;n;j\\right)  $ as $n\\rightarrow\\infty$ for fixed $s$ and $j$. \\ This behavior\ncan be determined in a straightforward way, for any $s$ and $j$, by a careful\nasymptotic analysis of the integral in (\\ref{ProdMultInt}). \\ Such\n$n\\rightarrow\\infty$ behavior may be of interest in various statistical problems.\n\nThe simplest illustration is $M\\left(  0;n;1\/2\\right)  $ for even $n$. \\ For\nthis particular case, (\\ref{Catalan}) and Stirling's approximation,\n$n!\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{2\\pi n}\\left(  \\frac{n}{e}\\right)\n^{n}$, give directly the main term in the asymptotic behavior\n\\begin{equation}\nM\\left(  0;2m;1\/2\\right)  \\underset{m\\rightarrow\\infty}{\\sim}\\frac{4^{m\n}{m^{3\/2}\\sqrt{\\pi}}\\left(  1+O\\left(  \\frac{1}{m}\\right)  \\right)  \\ .\n\\label{SpinHalfSingAsymp\n\\end{equation}\nOn the other hand, upon setting $t=\\cos^{2}\\vartheta$ the integral\n(\\ref{ProdMultInt}) has a form like that in (\\ref{HyperForm}), namely\n\\begin{equation}\nM\\left(  0;2m;1\/2\\right)  =\\frac{2}{\\pi}~4^{m}\\int_{0}^{1}t^{m}\\sqrt\n{\\frac{1-t}{t}}~dt=\\frac{2}{\\pi}~4^{m}B\\left(  m+\\frac{1}{2},\\frac{3\n{2}\\right)  \\ . \\label{SpinHalfSingInt\n\\end{equation}\nThe $t$ integral is just a beta function, $B\\left(  m+\\frac{1}{2},\\frac{3\n{2}\\right)  =\\Gamma\\left(  m+\\frac{1}{2}\\right)  \\Gamma\\left(  \\frac{3\n{2}\\right)  \/\\Gamma\\left(  m+2\\right)  $, which leads back to exactly\n(\\ref{Catalan}) for $s=0$. \\ But rather than using Stirling's approximation,\nit is more instructive to determine the asymptotic behavior directly from the\nintegral (\\ref{SpinHalfSingInt}) using\n\\href{https:\/\/en.wikipedia.org\/wiki\/Watson's_lemma}{Watson's lemma}. \\ Thu\n\\begin{equation}\nM\\left(  0;2m;1\/2\\right)  \\underset{m\\rightarrow\\infty}{\\sim}2\\sqrt{2\n~\\frac{2^{2m}}{\\left(  2m\\right)  ^{3\/2}\\sqrt{\\pi}}\\left(  1-\\frac{9\n{8m}+O\\left(  \\frac{1}{m^{2}}\\right)  \\right)  \\ .\n\\end{equation}\nNaively it might be expected that\\emph{ }the leading asymptotic behavior\n(\\ref{SpinHalfSingAsymp}) follows from a heuristic, saddle-point-Gaussian\nevaluation of the integration in (\\ref{SpinHalfSingInt}). \\ Unfortunately,\nthat expectation is not fulfilled.\\ \\ The correct $m$ dependence is obtained\nfor $M$, but with an incorrect overall coefficient. \\ To obtain the correct\ncoefficient, a more careful analysis of the asymptotic behavior is needed, as\nprovided by Watson's lemma.\n\nSimilarly, for large $n$ the number of singlets occurring in the product of\n$n$ spin $1$ representations behaves a\n\\begin{equation}\nM\\left(  0;n;1\\right)  \\underset{n\\rightarrow\\infty}{\\sim}\\frac{3\\sqrt{3}\n{8}~\\frac{3^{n}}{n^{3\/2}\\sqrt{\\pi}}\\left(  1-\\frac{21}{16n}+O\\left(  \\frac\n{1}{n^{2}}\\right)  \\right)  \\ , \\label{SpinOneSingAsymp\n\\end{equation}\nand the number of singlets in the product of $n$ spin $2$ representations\nbehaves a\n\\begin{equation}\nM\\left(  0;n;2\\right)  \\underset{n\\rightarrow\\infty}{\\sim}\\frac{1}{8\n~\\frac{5^{n}}{n^{3\/2}\\sqrt{\\pi}}\\left(  1-\\frac{15}{16n}+O\\left(  \\frac\n{1}{n^{2}}\\right)  \\right)  \\ . \\label{SpinTwoSingAsymp\n\\end{equation}\n\n\nIn general, the number of spin $s$ representations occurring in the product of\n$n$ spin $j$ representations for large $n$ has asymptotic behavior \\cite{ADF}\n\\begin{equation}\nM\\left(  s;n;j\\right)  \\underset{n\\rightarrow\\infty}{\\sim}\\left(  1+2s\\right)\n\\left(  \\frac{3}{2j\\left(  j+1\\right)  }\\right)  ^{3\/2}~\\frac{\\left(\n1+2j\\right)  ^{n}}{n^{3\/2}\\sqrt{\\pi}}\\left(  1-\\frac{3}{4n}-\\frac{9}{8n\n\\frac{1}{j\\left(  j+1\\right)  }-\\frac{3}{2n}\\frac{s\\left(  s+1\\right)\n}{j\\left(  j+1\\right)  }+O\\left(  \\frac{1}{n^{2}}\\right)  \\right)  \\ .\n\\label{SpinJSpinSAsymp\n\\end{equation}\nThis is correct for either integer or semi-integer $s$ or $j$, although of\ncourse $n$ must be (odd) even to obtain (semi-)integer $s$ from products of\nsemi-integer $j$, and only integer $s$ are produced by integer $j$.\n\\ Asymptotically then, for integer $j$,\n\\begin{equation}\nM\\left(  j;n;j\\right)  \/M\\left(  0;n;j\\right)  =M\\left(  0;n+1;j\\right)\n\/M\\left(  0;n;j\\right)  \\underset{n\\rightarrow\\infty}{\\sim}1+2j+O\\left(\n\\frac{1}{n}\\right)  \\ .\n\\end{equation}\nRemarkably, this behavior is approximately seen in the Table, with errors\n$\\lessapprox10\\%$. \\ On the other hand, for semi-integer $j$ and even $n$\n\\begin{equation}\nM\\left(  j;n+1;j\\right)  \/M\\left(  0;n;j\\right)  =M\\left(  0;n+2;j\\right)\n\/M\\left(  0;n;j\\right)  \\underset{n\\rightarrow\\infty}{\\sim}\\left(\n1+2j\\right)  ^{2}+O\\left(  \\frac{1}{n}\\right)  \\ .\n\\end{equation}\n\n\nFor integer $j$, the result in (\\ref{SpinJSpinSAsymp}) follows directly,\nalbeit tediously, from an application of Watson's lemma to (\\ref{HyperForm})\nafter switching to exponential variables. \\ In that case the overall\ncoefficient in (\\ref{SpinJSpinSAsymp}) arises as a simple algebraic function\nof the Chebyshev roots in (\\ref{ChebPolys&Roots}), namely, $1\/\\left(\n\\sum_{l=1}^{j}\\frac{1}{1-r_{l}\\left(  j\\right)  }\\right)  ^{3\/2}$. \\ This then\nreduces to the Casimir-dependent expression in (\\ref{SpinJSpinSAsymp}) by\nvirtue of the integer $j$ identity\n\\begin{equation}\n\\sum_{l=1}^{j}\\frac{1}{1-r_{l}\\left(  j\\right)  }=\\frac{2}{3}~j\\left(\nj+1\\right)  \\ . \\label{C(j)\n\\end{equation}\nSimilar statements apply when $j$ is semi-integer leading again to\n(\\ref{SpinJSpinSAsymp}). \\ For semi-integer $j$ the relevant identity i\n\\begin{equation}\n\\frac{1}{2}+\\sum_{l=1}^{j-1\/2}\\frac{1}{1-r_{l}\\left(  j\\right)  }=\\frac{2\n{3}~j\\left(  j+1\\right)  \\ ,\n\\end{equation}\nwith the usual convention that the empty sum is $0$.\n\n\\subsection*{All-Order Extensions of the Asymptotics}\n\nThe asymptotic behavior given by (\\ref{SpinJSpinSAsymp}) is useful for fixed\n$s$ and $j$ in the limit as $n\\rightarrow\\infty$. \\ If the resulting spin $s$\nproduced by the $n$-fold product is also allowed to become large in the limit,\ne.g. $s=O\\left(  \\sqrt{n}\\right)  $, then (\\ref{SpinJSpinSAsymp})\\ is\n\\emph{not} useful. \\ However, in that particular case it is possible to use\nrenomalization group methods \\cite{RG} to sum the series of terms involving\npowers of $\\frac{1}{n}\\frac{s\\left(  s+1\\right)  }{j\\left(  j+1\\right)  }$ to\nobtain an exponential, and hence an improved approximation. \\ The result i\n\\begin{equation}\nM\\left(  s;n;j\\right)  \\underset{n\\rightarrow\\infty}{\\sim}\\left(  1+2s\\right)\n\\left(  \\frac{3}{2j\\left(  j+1\\right)  }\\right)  ^{3\/2}~\\frac{\\left(\n1+2j\\right)  ^{n}}{n^{3\/2}\\sqrt{\\pi}}~e^{-\\frac{3}{2n}\\frac{s\\left(\ns+1\\right)  }{j\\left(  j+1\\right)  }}~\\left(  1-\\frac{3}{4n}-\\frac{9}{8n\n\\frac{1}{j\\left(  j+1\\right)  }+O\\left(  \\frac{1}{n^{2}}\\right)  \\right)  \\ .\n\\label{TSvKAsymptotics\n\\end{equation}\nFor large $n$ this last expression gives\n\\href{https:\/\/cgc.physics.miami.edu\/SpinAsymptotics.html}{an excellent\napproximation} out to values\\ of $s$ of order $\\sqrt{n}$ and beyond.\n\\ Moreover, the peak in the distribution of spins $s$ produced by the product\nof $n$ spin $j$s is given for large $n$ by\n\\begin{equation}\ns_{\\text{mult}}\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{nj\\left(  j+1\\right)\n\/3}\\ . \\label{peak\n\\end{equation}\nThis follows from the exact result (\\ref{Catalan}) for spin $1\/2$, or from\n(\\ref{TSvKAsymptotics}) for any $j$. \\ Alternatively, for specific $j$ the\ndirect numerical evaluation of either (\\ref{ProdMultInt}) or\n(\\ref{MIsADifference}) verifies (\\ref{peak}) upon taking $n$ large, say,\n$n\\approx10^{4}$.\n\nPerhaps some further insight is provided by the asymptotic behavior of the\ncontinuous function that gives the \\emph{normalized number of states with a\ngiven total spin}, $s$, as obtained from (\\ref{TSvKAsymptotics}). \\ This is \\\n\\begin{equation}\n\\frac{\\left(  1+2s\\right)  M\\left(  s;n;j\\right)  }{\\left(  1+2j\\right)  ^{n\n}~dj\\underset{n\\rightarrow\\infty}{\\sim}\\left(  1-\\frac{3}{4n}\\left(\n1+\\frac{1}{s\\left(  s+1\\right)  }\\right)  +O\\left(  \\frac{1}{n^{2}}\\right)\n\\right)  ~P\\left(  x\\right)  ~dx\\ , \\label{ChiSquared\n\\end{equation}\nwhere, with a suitable choice of the variable $x$, $P\\left(  x\\right)  $ is\nthe normalized\n\\href{https:\/\/en.wikipedia.org\/wiki\/Chi-squared_distribution}{chi-squared\nprobability distribution function} for \\emph{three} degrees of freedom\n\\begin{equation}\nx\\equiv\\frac{3\\left(  1+2j\\right)  ^{2}}{8ns\\left(  s+1\\right)  \n\\ ,\\ \\ \\ P\\left(  x\\right)  =\\frac{2}{\\sqrt{\\pi}}~\\sqrt{x}~e^{-x\n\\ ,\\ \\ \\ \\int_{0}^{\\infty}P\\left(  x\\right)  dx=1\\ .\n\\end{equation}\nIn retrospect, this may not be a total surprise since the underlying rotation\ngroup may be parameterized by \\emph{three} Euler angles. \\ Note that this last\nasymptotic form is correctly normalized to give the total number of states as\n$n\\rightarrow\\infty$, i.e.\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\\frac{1}{\\sqrt{\\pi}}\\int_{0}^{nj}\\left(  1+2s\\right)\n^{2}~\\left(  \\frac{3}{2nj\\left(  j+1\\right)  }\\right)  ^{3\/2}e^{-\\frac{3\n{2n}\\frac{s\\left(  s+1\\right)  }{j\\left(  j+1\\right)  }}~ds=1\\ .\n\\end{equation}\nAlso note that the expression for the number of states, (\\ref{ChiSquared}),\nhas a maximum at spi\n\\begin{equation}\ns_{\\text{state}}\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{2}~s_{\\text{mult\n}\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{2nj\\left(  j+1\\right)  \/3}\\text{ .\n\\end{equation}\n\\ \n\n\\section*{Some Applications}\n\nIn closing, we stress that spin multiplicities play useful roles in a wide\nrange of fields, too numerous to present in detail here. \\ But we briefly\nsketch a few applications of the results described above.\n\nSome of the $SU\\left(  2\\right)  $ results for $s=0$ have been used for\ndecades in elasticity theory \\cite{Ogden} and in quantum chemistry \\cite{AT},\nas well as in nuclear physics, as is evident from the literature we have cited\nupon recognizing that\n\\href{http:\/\/mathworld.wolfram.com\/IsotropicTensor.html}{the number of\nisotropic rank-$n$ tensors} in three dimensions is just $M\\left(\n0;n;1\\right)  $.\n\nThe theory of group characters has been widely used in lattice gauge theory\ncalculations for a long time \\cite{BGZ,Creutz} and continues to play an\nimportant role in various strong coupling calculations \\cite{Unger}.\n\\ Characters are also indispensible to determine the spin content of various\nstring theories \\cite{CGGT}$.$\n\nMore generally, generic representation composition results continually find\nnew uses. \\ Recent examples include frustration and entanglement entropy for\nspin chains, with possible applications to black hole physics \\cite{Shor}.\n\nMultiplicities such as those in the Table have also attracted some recent\nattention in the field of quantum computing, ultimately with implications for\ncryptography. \\ In particular, there are so-called \\textquotedblleft\nentanglement witness\\textquotedblright\\ (EW) operators that allow the\ndetection of entangled states \\cite{LKCH,Toth,CHI}. \\ By knowing the\ndegeneracy of the EW eigenstates for an $n$-particle state, one can determine\nthe fraction of all states for which entanglement is \\textquotedblleft\ndecidable\\textquotedblright\\ --- a fraction that is especially of interest in\nthe limit of large $n$. For systems of $n$ spin $j$ particles, with the EW\noperator taken to be the Casimir of the total spin, this fraction of decidable\nstates \\cite{CHI} is denoted $f_{j}\\left(  n\\right)  $. \\ In this case, from\nthe asymptotic expression given above in (\\ref{ChiSquared}), one readily\nobtains\n\\href{https:\/\/www.researchgate.net\/publication\/304660051_Decidable_States_in_the_Large_N_Limit}{the\nexact result\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}~f_{j}\\left(  n\\right)  =f_{j}\\left(  \\infty\\right)\n=\\operatorname{erf}\\left(  \\sqrt{\\frac{3\/2}{s+1}}\\right)  -\\sqrt{\\frac{6\/\\pi\n}{s+1}}~\\exp\\left(  -\\frac{3\/2}{s+1}\\right)  \\ ,\n\\end{equation}\nwhere $\\operatorname{erf}\\left(  x\\right)  =2\\int_{0}^{x}\\exp\\left(\n-s^{2}\\right)  ds\/\\sqrt{\\pi}$ is the conventional\n\\href{https:\/\/en.wikipedia.org\/wiki\/Error_function}{error function}. \\ \n\nMany other statistical applications of spin multiplicities for large $n$ have\nbeen proposed in a recent, independent investigation of this subject\n\\cite{Poly}.\\bigskip\n\n\\textbf{Acknowledgements:} \\ We thank J Katriel and J Mendon\\c{c}a for\npointing out elegant ways to re-express the multiplicity in the general case.\n\\ We also thank A Polychronakos and K Sfetsos for an advance copy of their\npaper. \\ Finally, we thank an anonymous reviewer for bringing \\cite{Kirillov\n\\ to our attention. \\ This work was supported in part by a University of Miami\nCooper Fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}