\documentclass[reqno]{amsart} \usepackage{graphicx} \AtBeginDocument{{\noindent\small 2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline {\em Electronic Journal of Differential Equations}, Conference 13, 2005, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{1} \begin{document} \title[\hfilneg EJDE/Conf/13 \hfil Controllability, applications, and simulations] {Controllability, applications, and numerical simulations of cellular neural networks} \author[W. Aziz, T. Lara \hfil EJDE/Conf/13 \hfilneg] {Wadie Aziz, Teodoro Lara} % in alphabetical order \address{Wadie Aziz \hfill\break Departmento de F\'{i}sica y Matem\'{a}ticas, N\'{u}cleo Universitario ``Rafael Rangel", Universidad de los Andes, Trujillo, Venezuela} \email{wadie@ula.ve} \address{Teodoro Lara \hfill\break Departmento de F\'{i}sica y Matem\'{a}ticas, N\'{u}cleo Universitario ``Rafael Rangel", Universidad de los Andes, Trujillo, Venezuela} \email{tlara@ula.ve \quad teodorolara@cantv.net} \date{} \thanks{Published May 30, 2005.} \subjclass[2000]{37N25, 34K20, 68T05} \keywords{Cellular Neural Networks; circulant matrix; tridiagonal matrix; \hfill\break\indent Controllability} \begin{abstract} In this work we consider the model of cellular neural network (CNN) introduced by Chua and Yang in 1988. We impose the Von-Neumann boundary conditions and study the controllability of corresponding system, then these results are used in image detection by means of numerical simulations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{defn}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{example}{Example} \allowdisplaybreaks \section{Introduction} Since its introduction (\cite{chuya:cnnap,chuya:cnnth}) Cellular neural networks (CNN) have been used in numerous problems. Among them we have: Chua's circuit (\cite{aren:circhu}), Hopf bifurcation model (\cite{zoussek:hopf}), Cellular Automata and systolic arrays (\cite{automata:roskchua}), image detection (\cite{tmchhs:cnndet}), population growth model (\cite{cruchu1:cacn2}). In none of these works the Von-Neumann boundary conditions have been imposed; only in (\cite{tlp:cacn}) periodic boundary conditions were considered. The system obtained, after some changes (\cite{chuya:cnnap,chuya:cnnth}) is \begin{equation}\label{intro:eq2} \dot{v}=-v+AG(v)+Bu+f(u,v) \end{equation} where, $u,v\in \mathbb{R}^{mn\times 1}$, are column vectors; $A, B$ are matrices in $\mathbb{R}^{mn\times mn}$, $f(u,v)$ is a nonlinear perturbation, and $G(v)$ is a function which can be either linear or non-linear. In this paper we set the Von-Neumann boundary conditions, consider $G(v)=v$ and study the controllability of the resulting system which is \begin{equation}\label{intro:eq3} \dot{v} = (A-I)v+Bu+I \end{equation} where $A , B\in \mathbb{R}^{mn\times mn}$, after using the boundary conditions, are tridiagonal matrices, $I$ is the identity matrix in $\mathbb{R}^{mn\times mn}$. Also, we implement some numerical simulations of these results to show image detection; specifically Chinese characters. \begin{figure}[ht] \centering \setlength{\unitlength}{0.01in} \begin{picture}(480,236)(51,595) \thinlines \put( 80,700){\circle{42}} \put(112,800){\circle{16}} \put(140,700){\circle{40}} \put(140,620){\circle*{10}} \put(200,620){\circle*{10}} \put(260,620){\circle*{10}} \put(320,620){\circle*{10}} \put(380,620){\circle*{10}} \put(440,620){\circle*{10}} \put(200,800){\circle*{10}} \put(260,800){\circle*{10}} \put(320,800){\circle*{10}} \put(80,800){\line( 1, 0){ 25}} \put( 80,800){\line( 0,-1){ 60}} \put( 80,620){\line( 0, 1){ 60}} \put( 80,740){\line( 0,-1){ 20}} \put(140,800){\line( 0,-1){ 80}} \put(140,620){\line( 0, 1){ 60}} \put(140,800){\line( 1, 0){160}} \put(300,800){\line( 1, 0){ 40}} \put(350,800){\line( 1, 0){ 5}} \put(360,800){\makebox(0.1111,0.7778){.}} \put(360,800){\line( 1, 0){ 5}} \put(370,800){\line( 1, 0){ 10}} \put(380,800){\line( 0,-1){ 80}} \put(380,620){\line( 0, 1){ 60}} \put(380,720){\line(-1,-1){ 20}} \put(360,700){\line( 1,-1){ 20}} \put(380,680){\line( 1, 1){ 20}} \put(400,700){\line(-1, 1){ 20}} \put(320,800){\line( 0,-1){ 80}} \put(320,620){\line( 0, 1){ 40}} \put(320,660){\line( 0, 1){ 20}} \put(320,680){\line( 1, 1){ 20}} \put(340,700){\line(-1, 1){ 20}} \put(320,720){\line(-1,-1){ 20}} \put(300,700){\line( 1,-1){ 20}} \put(260,800){\line( 0,-1){ 80}} \put(260,680){\line( 0,-1){ 60}} \put(200,615){\line( 0, 1){ 70}} \put(200,800){\line( 0,-1){ 90}} \put(185,690){\line( 1, 0){ 30}} \put(185,710){\line( 1, 0){ 30}} \put( 80,620){\line( 1, 0){420}} \put(500,620){\line( 0, 1){ 60}} \put(500,720){\line( 0, 1){ 80}} \put(500,800){\line(-1, 0){ 60}} \put(440,800){\line( 0,-1){ 80}} \put(440,620){\line( 0, 1){ 60}} \put(440,720){\line(-1,-1){ 20}} \put(420,700){\line( 1,-1){ 20}} \put(440,680){\line( 1, 1){ 20}} \put(460,700){\line(-1, 1){ 20}} \put(320,620){\line( 0,-1){ 15}} \put(305,605){\line( 1, 0){ 30}} \put(315,595){\line( 1, 0){ 10}} \put(380,690){\vector( 0, 1){ 15}} \put(440,690){\vector(0,1){15}} \put(320,690){\vector( 0, 1){ 15}} \put(140,690){\vector(0,1){15}} \put(250,720){\line( 1, 0){ 20}} \put(250,680){\line( 1, 0){ 20}} \put(270,720){\line( 0,-1){ 40}} \put(250,720){\line( 0,-1){ 40}} \put(490,720){\line( 1, 0){ 20}} \put(490,680){\line( 1, 0){ 20}} \put(510,720){\line( 0,-1){ 40}} \put(490,720){\line( 0,-1){ 40}} \put(105,725){\makebox(0,0)[lb]{\smash{ij}}} \put(240,730){\makebox(0,0)[lb]{\smash{v}}} \put(225,805){\makebox(0,0)[lb]{\smash{ij}}} \put(285,635){\makebox(0,0)[lb]{\smash{vu}}} \put(345,735){\makebox(0,0)[lb]{\smash{vy}}} \put(465,805){\makebox(0,0)[lb]{\smash{ij}}} \put(410,725){\makebox(0,0)[lb]{\smash{yv}}} \put(475,730){\makebox(0,0)[lb]{\smash{y}}} \put(75,700){\makebox(0,0)[lb]{\smash{+}}} \put(150,730){\makebox(0,0)[lb]{\smash{I}}} \put(185,730){\makebox(0,0)[lb]{\smash{C}}} \put(280,640){\makebox(0,0)[lb]{\smash{I}}} \put(340,740){\makebox(0,0)[lb]{\smash{I}}} \put(405,730){\makebox(0,0)[lb]{\smash{I}}} \put(470,735){\makebox(0,0)[lb]{\smash{R}}} \put(460,815){\makebox(0,0)[lb]{\smash{y}}} \put(85,810){\makebox(0,0)[lb]{\smash{u}}} \put(90,805){\makebox(0,0)[lb]{\smash{ij}}} \put(220,810){\makebox(0,0)[lb]{\smash{v}}} \put(95,730){\makebox(0,0)[lb]{\smash{E}}} \put(230,735){\makebox(0,0)[lb]{\smash{R}}} \put(75,690){\makebox(0,0)[lb]{\smash{--}}} \end{picture} \caption{Typical circuit of CNN $ij$-position.} \label{fig:neurona} \end{figure} \section{Cellular Neural Networks} A CNN consists, basically, in a collection of non linear circuit displayed in a 2-dimensional array. The basic circuit of CNN is called cell. A cell is made of elements of linear and non-linear circuit which usually are linear capacitors, linear resistors, linear and non linear controlled sources, and independents sources. Each cell receives external signals through its input. The state voltage of a give cell is influenced no only by its own input through a feedback, its output; but also by the input and output of the neighboring cells. These interactions are implemented by voltage-controlled current sources. In the initial papers (\cite{chuya:cnnap,chuya:cnnth}) any cell in CNN is connected only to its neighbor cells; this is accomplished by using the so called 1-neighborhood or simply neighborhood and consequently $3 \times 3$-cloning templates. The adjacent cells can interact directly with each other in the sense that are made of a massive aggregate of regularity spaced cells which communicate with each other directly only through its nearest neighbors. In the figure \ref{fig:neurona} the basic circuit of a CNN of a cell (located at, say, position $ij$ of the array) is depicted. Here $v_{ij}$ is the voltage across the cell (state of the cell) with its initial condition satisfying $| v_{ij}(0)| \leq 1$. $E_{ij}$ is an independent voltage source, and $u_{ij}=E_{ij}$ is called the input or control, also assumed to satisfy $| u_{ij}| \leq 1$. $I$ is an independent current source, $C$ is a linear capacitors, $R_{v}$ and $R_{y}$ are linear resistors. $I_{vu}$, $I_{vy}$ are linear voltage-controlled currents sources such that at each neighbor cell, say $kl$, $I_{vy}=(I_{vy})_{kl}=a_{kl}g(v_{kl})$ are current source; $I_{vu}=(I_{vy})_{kl}=b_{kl}u_{kl}$; is nonlinear voltage-controlled source give by $I_{vy}=\frac{1}{R_y}g(v_{ij})$ where, $a_{kl}, \, b_{kl} \in \mathbb{R}$ and $g$ is an output sigmoid function. \section{Dynamics of CNN} \begin{defn}[$r$-neighborhood] \rm %3.1 The $r$-neighborhood of a cell $c_{ij}$, in a cellular neural network is defined by \begin{equation}\label{arantxa1:eq:vecindad} N^{ij}=\{c_{i_1 j_1}: \max\{|i-i_{1}|;|j-j_{1}|\} \leq r ; \; 1\leq i_{1} \leq m, \; 1 \leq j_{1} \leq n\} \end{equation} where $r$ is a positive integer. \end{defn} We consider the case $r=1$ which produces a couple of $3 \times 3$-matrices (cloning templates); the feedback and control operator, given as \begin{equation}\label{arantxa1:eq:marrano} \widetilde{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}, \quad \widetilde{B} = \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{pmatrix}, \end{equation} The output feedback depends on the interactive parameters $a_{ij}$ and the input control depends an parameters $b_{ij}$, $v\in \mathbb{R}^{mn}$ is the voltage and represents the state vector, and $u=(u_{11},u_{12}, \dots,u_{mn})^{T} \in \mathbb{R}^{mn}$ is the control (input), and the output $y=G(v)$ \begin{equation}\label{arantxa1:eq:defineg} G:\mathbb{R}^{mn}\to \mathbb{R}^{mn};\quad \quad G(v)= (g(v_{11}),g(v_{12}), \dots, g(v_{mn}))^{T} \end{equation} $g$ is differentiable, bounded and $\| g \|\leq 1$ (in the most general case $\| g \|\leq K$) and non decreasing ($g'\geq 0$); that is a sigmoid function. We also assume $\| u\|\leq 1$, $\| u(0) \|\leq 1$. \begin{defn} \rm %3.2 Let $K$ and $L$ be two square matrices of the same size and elements $k_{ij}$, $l_{ij}$ respectively; we define $\odot$ product \begin{equation}\label{arantxa1:eq:productoo} K \odot L =\sum_{i,j}k_{ij}l_{ij}. \end{equation} \end{defn} By imposing the Von-Neumann boundary conditions \begin{equation} \begin{array}{cc} \left. \begin{array}{c} v_{ik} = v_{ik+1} \\ v_{ik-1} = v_{ik+2} \end{array}\right\} & i=-1, \dots,n+2, \quad k = 0,m \\ &\\ \left. \begin{array}{c} v_{kj} = v_{k+1j} \\ v_{k-1j} = v_{k+2j} \end{array}\right\} & j=-1,\dots,m+2, \quad k = 0,n; \end{array} \end{equation} and applying the Kirchhoff Law of Voltage and Current, we obtain the equation at cell $c_{ij}$, \begin{equation} \dot{v}_{ij}=-v_{ij} + \widetilde{A} \odot \widehat{G}(v_{ij}) + \widetilde{B} \odot \widehat{u}_{ij} + I , \end{equation} and in its vector form, by taking the row order in this vector, that is, the first $n$-elements are formed by the first row of matrix and so on, the resulting system is \begin{equation}\label{problem} \dot{v}=-v + AG(v) + B u + I , \end{equation} where \begin{gather*} AG(v) = (\widetilde{A} \odot \widehat{G}(v_{11}), \dots , \widetilde{A} \odot \widehat{G}(v_{mn}))^{T}, \\ Bu = (\widetilde{B} \odot \widehat{u}_{11}, \dots , \widetilde{B} \odot \widehat{u}_{mn})^{T}, \quad I = (I,\dots, I)^{T} \end{gather*} matrices $A$, $B$ are block tridiagonal, $AG(v)=(\widehat{A}+\overset{\circ}{A})v$ and $Bu=(\widehat{B}+\overset{\circ}{B})u$ with $$\widehat{A} = \begin{pmatrix} A_2 & A_3 & 0 & \dots & 0 & 0 \\ A_1 & A_2 & A_3 & 0 & \dots & 0 \\ \vdots & \ddots & \ddots & \ddots & 0 & 0 \\ 0 & \dots & A_1 & A_2 & A_3 & 0 \\ 0 & 0 & \dots & A_1 & A_2 & A_3 \\ 0 & 0 & 0 & \dots & A_1 & A_2 \end{pmatrix}, \quad \widehat{B} =\begin{pmatrix} B_2 & B_3 & 0 & \dots & 0 & 0 \\ B_1 & B_2 & B_3 & 0 & \dots & 0 \\ \vdots & \ddots & \ddots & \ddots & 0 & 0 \\ 0 & \dots & B_1 & B_2 & B_3 & 0 \\ 0 & 0 & \dots & B_1 & B_2 & B_3 \\ 0 & 0 & 0 & \dots & B_1 & B_2 \end{pmatrix}, $$ $$ A_i= \begin{pmatrix} a_{i2} & a_{i3} & 0 & \dots & 0 \\ a_{i1} & \ddots & \ddots & &\vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots &\ddots & a_{i3} \\ 0 & \dots & \dots & a_{i1} & a_{i2} \end{pmatrix}, \quad i=1,2,3.$$ The matrix $\widehat{B}$ has the same blocks. The perturbation matrices look like, $$\overset{\circ}{A} = \begin{pmatrix} L_1 + \Gamma_2 & \Gamma_3 & 0 & \dots & 0 & 0 \\ \Gamma_1 & \Gamma_2 & \Gamma_6 & 0 & \dots & 0 \\ 0 & \ddots & \ddots & \ddots & & 0 \\ \vdots & \ddots & \Gamma_1 & \Gamma_2 & \Gamma_3 & 0 \\ \vdots &\dots & \ddots & \Gamma_1 & \Gamma_2 & \Gamma_3 \\ 0 &\dots & \dots & L_2 & \Gamma_1 & \Gamma_2 \end{pmatrix},\quad \begin{cases} L_1 = A_1 + \Gamma_1 \\ L_2 = A_3 + \Gamma_3 \end{cases}. $$ $$ \Gamma_i=\begin{pmatrix} a_{i1} & 0 & &\dots & 0 \\ 0 & 0 & & & \\ 0 &\ddots &\ddots & & \\ & \ddots & \ddots & \ddots & \\ 0 & \dots & a_{i3} & 0 & 0 \end{pmatrix}\quad i=1,2,3.$$ The matrix $\overset{o}{B}$ is defined similarly. \begin{rem} \rm %3.3 Other types of order were tested but they produce the same type of matrix, block tridiagonal. \end{rem} \begin{lem} %3.4 If $A$, $B$ are two arbitrary square matrices of size $l \times l$ and real entries, then $(A \otimes B)^{n}= A^{n} \otimes B^{n},$ for all $n \in \mathbb{N}.$ \end{lem} \begin{cor} %3.5 If $A$ is a matrix of order $n \times n$ and $\Pi = \mathop{\rm circ}(0,1,0, \dots,0)$ is circulant matrix, then $(A \otimes \Pi)^k =A^{k} \otimes \Pi^{k}$; for $k=1, \dots, m$. \end{cor} \section{CNN and Controllability} In this section we study the controllability of the general system (\ref{problem}) by means of the properties of block tridiagonal matrices Instead of (\ref{problem}) we study the linear case \begin{equation}\label{ctte1} \dot{v} = (A - I)v + Bu + I\,. \end{equation} The study of the controllability of (\ref{ctte1}) is equivalent to study the controllability of \begin{equation}\label{ctte} \dot{v} = (A - I)v + Bu. \end{equation} Note that $A-I$ is tridiagonal matrix same type as $A$. \begin{lem} %4.1 Any block tridiagonal matrix $$ A =\begin{pmatrix} A_2 & A_3 & 0 & \dots & 0 & 0 \\ A_1 & A_2 & A_3 & 0 & \dots & 0 \\ \vdots & \ddots & \ddots & \ddots & 0 & 0 \\ 0 & \dots & A_1 & A_2 & A_3 & 0 \\ 0 & 0 & \dots & A_1 & A_2 & A_3 \\ 0 & 0 & 0 & \dots & A_1 & A_2 \end{pmatrix} $$ can be written as $A=A_{3}\otimes \Pi + A_{1} \otimes \Pi ^{n-1} + A_{2} \otimes \Pi^{n}$. \end{lem} \begin{lem} %4.2 For every block tridiagonal matrix $A$, the following takes place \[ A^{k} = \sum_{i=0}^{k} \sum_{j=0}^{k-i}\begin{pmatrix} k \\ i \end{pmatrix} \begin{pmatrix} k-i \\ j \end{pmatrix} (A_{3}^{k-i-j}A_{1}^{nj-j}A_2^{i} \otimes \Pi)^{k-i-2j}; \quad k \in \mathbb{N}. \] \end{lem} \begin{proof} For $l \in \mathbb{N}$ fixed \begin{align*} A^{l} & = [A_{3}\otimes \Pi + A_{1} \otimes \Pi ^{n-1} + A_{2} \otimes \Pi^{n}]^{l} \\ & = \sum_{i=0}^{l} \sum_{j=0}^{l-i} \begin{pmatrix} l \\ i \end{pmatrix} \begin{pmatrix} l-i \\ i \end{pmatrix} (A_{3} \otimes \Pi)^{l-i-j}(A_{1} \otimes \Pi^{n-1})^{j}(A_2 \otimes \Pi^{n})^{i} \\ & = \sum_{i=0}^{l} \sum_{j=0}^{l-i} \begin{pmatrix} l \\ i \end{pmatrix}\begin{pmatrix} l-i \\ i \end{pmatrix} A_{3}^{l-i-j}A_{1}^{j}A_2^{i} \otimes \Pi^{l-i-2j}. \end{align*} \end{proof} \begin{thm} %4.3 Let $A$ and $B$ be two $n \times n$ block tridiagonal matrices. Then \begin{align*} A^k B =& \sum_{i=0}^{k} \sum_{j=0}^{k-i} \begin{pmatrix} k \\ i \end{pmatrix} \begin{pmatrix} k-i \\ j \end{pmatrix} (A_3^{k-i-j}A_1^{j}A_2^{i})\\ &\times [B_3 \otimes \Pi + B_1 \otimes \Pi^{n-1} + B_2 \otimes \Pi^{n}] \Pi^{k-(i+2j)} \end{align*} for $k \in \mathbb{N}$. \end{thm} \begin{proof} By induction: for $k=1$, $$AB=\sum_{i=0}^{1} \sum_{j=0}^{1-i} \begin{pmatrix} 1 \\ i \end{pmatrix} \begin{pmatrix} 1-i \\ j \end{pmatrix} (A_3^{1-i-j}A_1^{j}A_2^{i}) [B_3 \otimes \Pi + B_1 \otimes \Pi^{n-1} + B_2 \otimes \Pi^{n}] \Pi^{1-(i+2j)}. $$ Assume the statement of the theorem is true for $k=m$. Then for for $k=m+1$, we have \begin{align*} A^{m+1} B & = AA^m B \\ & = \sum_{i=0}^{m+1} \sum_{j=0}^{m+1-i} \begin{pmatrix} m+1 \\ i \end{pmatrix}\begin{pmatrix} m+1-i \\ j \end{pmatrix} (A_3^{m+1-i-j}A_1^{j}A_2^{i}) \\ & \times [B_3 \otimes \Pi + B_1 \otimes \Pi^{n-1} + B_2 \otimes \Pi^{n}] \Pi^{m+1-(i+2j)} \end{align*} \end{proof} According to \cite[Theorem 3]{sontag:control}, the controllability of (\ref{ctte}) depends on the rank of $(A,B)$. However, \begin{align*} \mathbf{Rg}[\mathbb{R}(A,B)] & = \mathbf{Rg}([ B, AB, \dots, A^{n-1}B ]) \\ & = \mathbf{Rg}\left[\left( \begin{array}{ll} \begin{bmatrix} C_1 &C_2 &\dots &C_{n-1} \\ C_1 &C_2 &\dots &C_{n-1} \\ \vdots &\vdots & & \vdots \\ C_1 &C_2 &\dots &C_{n-1} \end{bmatrix} \otimes \mathbf{B} \end{array} \right) \mathbf{D}\right], \end{align*} where \[ \mathbf{C}= \begin{bmatrix} C_1 &C_2 &\dots &C_{n-1} \\ C_1 &C_2 &\dots &C_{n-1} \\ \vdots &\vdots &\vdots & \vdots \\ C_1 &C_2 &\dots &C_{n-1} \end{bmatrix}, \] $\mathbf{B}=B_3 \otimes \Pi + B_1 \otimes \Pi^{n-1} + B_2 \otimes \Pi^{n}$, \begin{gather*} C_1 = \sum_{i=0}^{0} \sum_{j=0}^{0-i} \begin{pmatrix} 0 \\ i \end{pmatrix}\begin{pmatrix} 0-i \\ j \end{pmatrix} (A_3^{0-i-j}A_1^{j}A_2^{i}) \\ C_2 = \sum_{i=0}^{1} \sum_{j=0}^{1-i} \begin{pmatrix} 1 \\ i \end{pmatrix}\begin{pmatrix} 1-i \\ j \end{pmatrix} (A_3^{1-i-j}A_1^{j}A_2^{i}) \\ \vdots \\ C_{n-1} = \sum_{i=0}^{n-1} \sum_{j=0}^{n-1-i} \begin{pmatrix} n-1 \\ i \end{pmatrix}\begin{pmatrix} n-1-i \\ j \end{pmatrix}(A_3^{n-1-i-j}A_1^{j}A_2^{i}), \end{gather*} and \[ \mathbf{D}=\begin{bmatrix} \Pi^{n} &0 &0 &0 \\ 0 &\Pi^{1-(i+2j)} &0 &0 \\ 0 &0 &\ddots &0 \\ 0 &0 &0 &\Pi^{(n-1)-(i+2j)} \end{bmatrix}. \] \begin{prop} %4.4 Let \begin{equation*} \mathbf{D}=\begin{bmatrix} \Pi^{n} &0 &0 &0 \\ 0 &\Pi^{1-(i+2j)} &0 &0 \\ 0 &0 &\ddots &0 \\ 0 &0 &0 &\Pi^{(n-1)-(i+2j)} \end{bmatrix}\,. \end{equation*} Then $| \det(\mathbf{D})|= 1$. \end{prop} The proof of the above proposition can be found in \cite{waa:tesis} We are now ready to give the main result of this section, which is quite technical, but applicable to several situations discussed later. \begin{thm} %4.5 The system (\ref{ctte}) is controllable if and only if $\mathbf{Rg}(C \otimes B)=n.$ \end{thm} \begin{proof} By \cite[Theorem 3]{sontag:control}, the system (\ref{ctte}) is controllable if and only if \begin{equation*} \mathbf{Rg}[\mathbb{R}(A,B)] = \mathbf{Rg} [B, AB, \dots, A^{n-1}B]\,. \end{equation*} By the above proposition this is true if and only if $\mathbf{Rg}(C \otimes B)=n$. \end{proof} \subsection*{Example} Let $m=3$ and $n=3$; let matrices $\widetilde{A}$ and $\widetilde{B}$ be as in (\ref{arantxa1:eq:marrano}); let the output $y=G_2(v)$, with $G_2:\mathbb{R}^{3 \times 3}\to \mathbb{R}^{3 \times 27}$ given as \[ G_2(v) = \big(G(v_{11}),G(v_{12}),G(v_{13}),G(v_{21}),G(v_{22}),G(v_{23}), G(v_{31}),G(v_{32}),G(v_{33})\big)^{T}\,. \] We impose Von-Neumann the boundary conditions and get \newcommand{\matr}[9]{\begin{pmatrix} v_{#1}&v_{#2}&v_{#3}\\ v_{#4}&v_{#5}&v_{#6}\\ v_{#7}&v_{#8}&v_{#9} \end{pmatrix}} \begin{gather*} G(v_{11}) = \matr{11}{11}{12}{11}{11}{12}{21}{21}{22},\quad G(v_{12}) = \matr{11}{12}{13}{11}{12}{13}{21}{22}{23}, \\ G(v_{13}) = \matr{12}{13}{11}{12}{13}{11}{22}{23}{21},\quad G(v_{21}) = \matr{11}{11}{12}{21}{21}{22}{31}{31}{32}, \\ G(v_{22}) = \matr{11}{12}{13}{21}{22}{23}{31}{32}{33},\quad G(v_{23}) = \matr{12}{13}{11}{22}{23}{21}{32}{33}{31}, \\ G(v_{31}) = \matr{21}{21}{22}{31}{31}{32}{11}{11}{12},\quad G(v_{32}) = \matr{21}{22}{23}{31}{32}{33}{11}{12}{13}, \\ G(v_{33}) = \matr{22}{23}{21}{32}{33}{31}{12}{13}{11}\,. \end{gather*} Now $AG_2(v)$ has the form {\scriptsize \newcommand{\sa}[2]{a_{#1}+a_{#2}} \[ \begin{pmatrix} (\sa{11}{12}+ & & & & & & & & \\ \sa{21}{22}) & \sa{13}{23} & 0 & \sa{31}{32} & a_{33} & 0 & 0 & 0 & 0 \\ \sa{11}{21} & \sa{12}{22} & \sa{13}{23} & a_{31} & a_{32} & a_{33} & 0 & 0 & 0 \\ \sa{13}{23} & \sa{11}{21} & \sa{12}{22} & a_{33} & a_{31} & a_{32} & 0 & 0 & 0 \\ \sa{11}{12} & a_{13} & 0 & \sa{21}{22} & a_{23} & 0 & \sa{31}{32} & a_{33} & 0 \\ a_{11} & a_{12} & a_{13} & a_{21} & a_{22} & a_{23} & a_{31} & a_{32} & a_{33}\\ a_{13} & a_{11} & a_{12} & a_{23} & a_{21} & a_{22} & a_{33} & a_{31} & a_{32}\\ \sa{31}{32} & a_{33} & 0 & \sa{11}{12} & a_{13} & 0 & \sa{21}{22} & a_{23} & 0 \\ a_{31} & a_{32} & a_{33} & a_{11} & a_{12} & a_{13} & a_{21} & a_{22} & a_{23}\\ a_{33} & a_{31} & a_{32} & a_{13} & a_{11} & a_{12} & a_{23} & a_{21} & a_{22}\\ \end{pmatrix} \begin{pmatrix} v_{11} \\ v_{12} \\ v_{13} \\ v_{21} \\ v_{22} \\ v_{23} \\ v_{31} \\ v_{32} \\ v_{33} \\ \end{pmatrix}. \] }%\input{tacha} We write $AG_2(v)$ as \begin{equation*} AG_2(v)=(\widehat{A}+\overset{\circ}{A})G_2(v)=\widehat{A}G_2(v) + \overset{\circ}{A}G_2(v). \end{equation*} Then we do the same for matrix $Bu$. Now (\ref{ctte1}) becomes \begin{equation}\label{arantxa1:eq:general} \dot{v}= -v + \widehat{A}G_2(v) + \widehat{B}u + f(u,v) \end{equation} Note that $A$ and $B$ are tridiagonal matrices and $f(u,v)=I+ \overset{\circ}{A}G_2(v) + \overset\circ{B}u$ is a perturbation of ~(\ref{ctte}); if $f(u,v)=0$, (\ref{arantxa1:eq:general}) is controllable, for \cite{lee:markus} (Theorem 11), then ~(\ref{arantxa1:eq:general}) also is controllable, where $\overset\circ{A}G_2(v)$ and $\overset\circ{B}u$ have the form, respectively, \[ \overset{\circ}{A}G_2(v) = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ a_{13}+a_{23} & 0 & 0 & a_{33} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ a_{13} & 0 & 0 & a_{23} & 0 & 0 & a_{33} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ a_{33} & 0 & 0 & a_{13} & 0 & 0 & a_{23} & 0 & 0\\ \end{pmatrix} \begin{pmatrix} v_{11} \\ v_{12} \\ v_{13} \\ v_{21} \\ v_{22} \\ v_{23} \\ v_{31} \\ v_{32} \\ v_{33} \\ \end{pmatrix}\,, \] %\input{tacha1} \input{tacha2} \[ \overset{\circ}{B}u = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ b_{13}+b_{23} & 0 & 0 & b_{33} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ b_{13} & 0 & 0 & b_{23} & 0 & 0 & b_{33} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ b_{33} & 0 & 0 & b_{13} & 0 & 0 & b_{23} & 0 & 0\\ \end{pmatrix} \begin{pmatrix} u_{11} \\ u_{12} \\ u_{13} \\ u_{21} \\ u_{22} \\ u_{23} \\ u_{31} \\ u_{32} \\ u_{33} \\ \end{pmatrix}. \] \begin{rem} %4.6 \rm So far we have studied the case where $G(v)= \alpha v$, $\alpha > 0$; that is, the linear case. The non-linear case \begin{equation}\label{hugo} \dot{v} = -v + AG(v) + Bu \end{equation} can be attacked just writing down \begin{equation}\label{hugo2} \dot{v} = (A-I)v + Bu + (AG(v) - Av) = (A-I)v + Bu + A(G(v) - v) \end{equation} and imposing the condition of $A(G(v)-v)$ being globally Lipschitz. In this case we guarantee controllability of (\ref{hugo}) if (\ref{hugo2}) is controllable. \end{rem} \begin{figure}[htb] \begin{center} \includegraphics[width=0.6 \textwidth]{fig2} %diamant} \end{center} \caption{Input and some iterations by a $30\times 30$ matrix.} \label{diamante} \end{figure} \section{Numerical Simulations} In this section we use our model of CNN in image detection; most of our examples are Chinese characters. The idea is input an image and iterate equation (\ref{intro:eq3}) by using Runge-Kutta 4-order method. We shall use the corner detecting CNN since in \cite{crounchu:detection}, but taking $b_{22}=5$; in other words \[ \widetilde{A}=\begin{pmatrix} 0 &0 &0 \\ 0 &2 &0 \\ 0 &0 &0 \end{pmatrix}, \quad \widetilde{B}=\begin{pmatrix} -7/20 & -1/4 &-7/20 \\ -1/4 & 5 & -1/4 \\ -7/20 & -1/4 & -7/20 \end{pmatrix}, \quad I=3 \times 10^{-4} {\rm Amp.} \] First, we consider figure \ref{diamante} a diamond as input and some iterations, we detect the main character of the stroke in the first three steps of this process. In a $30 \time 30$ array; and after a few iterations we reach the maximum detections. In figure \ref{diamiter}, we find the same behavior as in the figure \ref{diamante}; by taking now $k$ (number of iterations) a little bigger. \begin{figure}[htb] \includegraphics[width=0.7\textwidth]{fig3} % diamiter \caption{More iterations in case of the diamond.} \label{diamiter} \end{figure} Figure \ref{chi2entra} is a Chinese character with an $35 \times 35$ array. After some iterations for $k=3$ and $k=10$ maximum detection is achieved. \begin{figure}[htb] \includegraphics[width=0.7\textwidth]{fig4} %chi2entra} \caption{Input and some iterations for a $35\times 35$ matrix with an ideogram.} \label{chi2entra} \end{figure} Figure \ref{chi2ite} is made by an iteration of the input in figure \ref{chi2entra} with $k$ bigger, the output is the same as in the previous figure. \begin{figure}[htb] \includegraphics[width=0.7 \textwidth]{fig5} %chi2ite \caption{$35 \times 35$-array; some more iterations.} \label{chi2ite} \end{figure} As a concluding remark, we want to mention that the two input figures chosen here are the same as two of the chosen in (\cite{chuya:cnnap}, \cite{tlp:cacn}), but now we are imposing Von-Neumann boundary conditions. In our case maximum detection is attained in fewer steps that the ones in the mentioned papers. \newpage \begin{thebibliography}{10} \bibitem{waa:tesis} Wadie Aziz, \emph{Redes neuronales celulares}, Master's thesis, Universidad de Los Andes, N\'{u}cleo Universitario Rafael Rangel, Trujillo - Venezuela, January 2003. \bibitem{chuya:cnnap} Leon~O. Chua and L.~Yang, \emph{Cellular neural networks: Applications}, IEEE. Transc. Circuits Syst. \textbf{35} (1988), 1273--1290. \bibitem{chuya:cnnth} Leon~O. Chua and L.~Yang, \emph{Cellular neural networks: Theory}, IEEE. Transc. Circuits Syst. \textbf{35} (1988), 1257--1271. \bibitem{tmchhs:cnndet} K.~R. Crounse and L.~O. Chua, \emph{Methods for image processing and pattern formation in cnn: A tutorial}. \textbf{42}, no.~10, (1995), 583--601 \bibitem{cruchu1:cacn2} J.~Cruz and L.~O. Chua, \emph{Application of cellular neural networks to model population dynamics}, IEEE. Transc. Circuits Syst. \textbf{42}, no.~10, (1995), 715--720. \bibitem{aren:circhu} S.~Baglio, L.~Fortuna, P.~Arenas and G.~Manganaro, \emph{Chua's circuit can be generated by cnn cells}. IEEE Transc. on Circuit Sys.I: Fundamental Theory and Applic. \textbf{42}, no.~2, (1995), 123--126. \bibitem{tlp:cacn} T.~Lara, \emph{Controllability and applications of cnn}, Ph.D. thesis, Georgia Institute of Technology, USA, December 1997. \bibitem{lee:markus} E.~B. Lee and L.~Markus, \emph{Foundations of optimal control theory}. John Wiley and Sons, New York, 1967. \bibitem{automata:roskchua} T.~Roska and L.~Chua, \emph{Cellular neural network with non-linear and delay-type templates elements}. IEEE Transc. on Circuit Sys.I: Fundamental Theory and Applic. \textbf{37}, (1990), 12--25. \bibitem{sontag:control} E.~D. Sontag, \emph{Mathematical control theory}. Springer--Verlag, New York 1990. \bibitem{crounchu:detection} T.~Boros, A.~Radva'nyi, T.~Roska, Leon~Chua and P.~Thiran, \emph{Detecting moving and standing objects using cellular neural networks}. Int. Journal on Circ. Theory and Applictions, \textbf{20}, (1992), 613--628. \bibitem{zoussek:hopf} Fan Zou and Josef Nossek, \emph{Bifurcation and chaos in cellular neural networks}, IEEE Transc. on Circuit Sys.I: Fundamental Theory and Applic. \textbf{40} (1993), no.~3, 157--164. \end{thebibliography} \end{document}