\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 89, pp. 1--6.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/89\hfil Stability of cellular neural networks] {Stability of cellular neural networks with unbounded time-varying delays} \author[ L. Wang, J. Shao \hfil EJDE-2008/89\hfilneg] {Lijuan Wang, Jianying Shao} % not in alphabetical order \address{Lijuan Wang \newline College of Mathematics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China} \email{wanglijuan1976@yahoo.com.cn} \address{Jianying Shao \newline College of Mathematics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China} \email{shaojianying2008@yahoo.cn} \thanks{Submitted May 30, 2008. Published June 21, 2008.} \thanks{Supported by grant 20070605 from the Scientific Research Fund of Zhejiang Provincial \hfill\break\indent Education, China} \subjclass[2000]{34C25, 34K13, 34K25} \keywords{Cellular neural networks; stability; equilibrium; \hfill\break\indent unbounded time-varying delays} \begin{abstract} In this article, we prove the existence of local solutions and the stability of the equilibrium points for cellular neural networks. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Let $n$ correspond to the number of units in a neural network, $x_{i}(t)$ be the state vector of the $i$th unit at the time $t$, $a_{ij}(t)$ be the strength of the $j$th unit on the $i$th unit at time $t$, $b_{ij}(t)$ be the strength of the $j$th unit on the $i$th unit at time $t-\tau_{ij}(t)$, and $\tau_{ij}(t)\geq 0$ denote the transmission delay of the $i$th unit along the axon of the $j$th unit at the time $t$. It is well known that the cellular neural networks with time-varying delays are described by the differential equations \begin{equation} x_{i}'(t)=-c_{i}(t)x_{i}(t)+\sum^n_{j=1}a_{ij}(t)f_{j}(x_{j}(t)) +\sum^n_{j=1}b_{ij}(t) g_{j}(x_{j}(t-\tau_{ij}(t)))+I_{i}(t), \label{e11} \end{equation} where $i=1,2,\dots, n$, for any activation functions of signal transmission $f_{j}$ and $ g_{j}$. Here $I_{i}(t)$ denotes the external bias on the $i$th unit at the time $t$, $c_{i}(t)$ represents the rate with which the $i$th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at the time $t$. Since the cellular neural networks (CNNs) were introduced by Chua and Yang \cite{c2} in 1990, they have been successfully applied to signal and image processing, pattern recognition and optimization. Hence, CNNs have been the object of intensive analysis by numerous authors in recent years. In particular, extensive results on the problem of the existence and stability of the equilibriums and periodic solutions for \eqref{e11} are given out in the literature. We refer the reader to \cite{c1,h1,l1,l2,l3,l4,y1} and the references cited therein. Suppose that the following condition \begin{itemize} \item[(H0)] there exists a constant $\tau $ such that \begin{equation} \tau=\max_{1\leq i,j\leq n}\big\{\sup_{t\in \mathbb{R}}\tau_{ij}(t)\big\}. \label{e12} \end{equation} \end{itemize} Most authors of bibliographies listed above obtained some sufficient conditions for the existence and stability of the equilibriums and periodic solutions for system \eqref{e11}. However, to the best of our knowledge, few authors have considered dynamic behaviors of \eqref{e11} without the assumption (H0). Thus, it is worth while to continue to investigate the stability of system \eqref{e11}. The main purpose of this paper is to give new criteria for the stability of the equilibrium of system \eqref{e11}. By applying mathematical analysis techniques, without assuming (H0), we derive some sufficient conditions ensuring that the equilibrium of \eqref{e11} is locally stable, which are new and complement of previously known results. An example is provided to illustrate our results. In this paper, for $i, j=1, 2, \dots, n$, it will be assumed that $c _{i}(t)$, $I _{i}(t)$, $a _{ij}(t)$, $ b _{ij}(t)$ are constant: \begin{equation} c _{i}\equiv c _{i}(t), \quad I _{i}\equiv I _{i}(t), \quad a _{ij}\equiv a _{ij}(t), \quad b _{ij} \equiv b _{ij}(t) . \label{e13} \end{equation} It will be assumed that \begin{equation} \max_{1\leq i,j\leq n}\{\sup_{t\in \mathbb{R}}\tau_{ij}(t)\}=+\infty, \ \tau_{ij}(t)0$ and $\xi_{i}>0$, $i=1, 2, \dots, n$, such that $$ - c _{i} \xi_{i}+ \sum_{j=1}^{n}|a _{ij} |\tilde{L}_{j}\xi_{j}+ \sum_{j=1}^{n}|b _{ij} | L_{j}\xi_{j}<-\eta<0, \quad i=1, 2, \dots, n. $$ \end{itemize} Since $c _{i}(t)$, $I _{i}(t)$, $a _{ij}(t)$, $ b _{ij}(t)$ are constant, by using a similar argument as that in the proof of \cite[Theorem 2.3]{h1}, we can easily show the following lemma. \begin{lemma} \label{lem1.1} Let {\rm (H1), (H2)} hold. Then \eqref{e11} has at least one equilibrium point. \end{lemma} The initial conditions associated with \eqref{e11} are of the form \begin{equation} x_{i}(s)=\varphi_{i}(s),s\in (-\infty, \ 0], \quad i=1,2,\dots,n, \label{e16} \end{equation} where $\varphi_{i}(\cdot)$ denotes real-valued bounded continuous function defined on $(-\infty, 0]$. For $ Z(t)=(x_{1}(t), x_{2}(t),\dots,x_{n}(t))^{T} $, we define the norm \begin{equation} \|Z(t)\|_{\xi}=\max_{i=1,2,\dots,n}|\xi^{-1}_{i}x_{i}(t)|. \label{e17} \end{equation} The remaining part of this paper is organized as follows. In Section 2, we present sufficient conditions to ensure that the equilibrium of system \eqref{e11} is locally stable. In Section 3, we give some examples and remarks to illustrate our results obtained in the previous sections. \section{Main Results} \begin{theorem} \label{thm2.1} Assume {\rm (H1), (H2)} hold. Suppose that $Z^{*} =(x^{*}_{1}, x^{*}_{2} ,\dots,x^{*}_{n} )^{T}$ is the equilibrium of \eqref{e11}. Then, $Z^{*} $ is locally stable, namely, for all $\varepsilon >0$, there exists a constant $\delta>0$ such that for every solution $ Z(t)=(x_{1}(t), x_{2}(t),\dots,x_{n}(t))^{T} $ of \eqref{e11} with any initial value $ \varphi=(\varphi_{1}(t), \varphi_{2}(t), \dots, \varphi_{n}(t))^{T}$ such that $$ \|\varphi -Z ^{*}\|=\max_{1\leq j\leq n}\{\sup_{-\infty\leq t\leq 0}|\varphi_{j}-Z_{j}^{*}| \}<\delta,$$ there holds $$ |x_{ i}(t)-x^{*}_{i } | < \varepsilon, \quad \text{for all } t\geq 0, \; i= 1, 2, \dots, n. $$ \end{theorem} \begin{proof} Let $Z(t)=(x_{1}(t), x_{2}(t),\dots,x_{n}(t))^{T} $ be a solution of system \eqref{e11} with any initial value $ \varphi=(\varphi_{1}(t), \varphi_{2}(t), \dots, \varphi_{n}(t))^{T}$, and define $$ u(t)=(u_{1}(t), u_{2}(t),\dots,u_{n}(t))^{T}=Z(t)-Z^{*} . $$ Then \begin{equation} u_{i}'(t)=-c_{i} u_{i}(t)+\sum^n_{j=1}a_{ij} [f_{j}(x_{j}(t)) -f_{j} (x^{*}_{j} )] + \sum^n_{j=1}b_{ij} [ g_{j}(x_{j}(t-\tau_{ij}(t)))-g_{j} (x^{*}_{j} )] , \label{e21} \end{equation} where $i=1, 2, \dots, n$. Let $i_{t}$ be an index such that \begin{equation} \xi^{-1}_{i_{t}}|u_{i_{t}}(t)|=\|u (t)\|_{\xi}. \label{e22} \end{equation} Calculating the upper right derivative of $ |u_{i_{s}}(s)|$ along \eqref{e21}, in view of \eqref{e21} and (H1), we have \begin{equation} \begin{aligned} D^+( |u_{i_{s}}(s)|)\Big|_{s=t} & = \mathop{\rm sign} (u_{i_{t}}(t))\Big\{-c_{i_{t}} u_{i_{t}}(t)+\sum^n_{j=1}a_{i_{t}j} [f_{j}(x_{j}(t))-f_{j}(x^{*}_{j} )] \\ &\quad +\sum^n_{j=1}b_{i_{t}j} [ g_{j}(x_{j}(t-\tau_{i_{t}j}(t))) -g_{j}(x^{*}_{j} )]\Big\}\\ & \leq - c_{i_{t}} |u_{i_{t}}(t)|\xi^{-1}_{i_{t}}\xi_{i_{t}} + \sum_{j=1}^{n}a_{i_{t}j} \tilde{L}_{j}|u_{j }(t )|\xi^{-1}_{j}\xi_{j} \\ &\quad +\sum_{j=1}^{n} b_{i_{t}j} L_{j}|u_{j}(t-\tau_{i_{t}j}(t))| \xi^{-1}_{j }\xi_{j}. \end{aligned} \label{e23} \end{equation} Let \begin{equation} M(t)=\max_{ s\leq t}\{ \|u (s)\|_{\xi}\}.\label{e24} \end{equation} It is obvious that $ \|u (t)\|_{\xi}\leq M(t)$, and $M(t)$ is non-decreasing. Now, we consider two cases. \noindent\textbf{Case (i).} \begin{equation} M(t)>\|u (t)\|_{\xi}\quad\text{for all }t\geq 0. \label{e25} \end{equation} We claim that $M(t)\equiv M(0)$ is constant for all $t\geq 0$. %\label{e26} By way of contradiction, assume that this is not the case. Consequently, there exists $t_{1}>0$ such that $M(t_{1})> M(0)$. Since $$ \|u (t)\|_{\xi}\leq M(0) \quad \text{for all } t\leq 0. $$ There must exist $\beta \in (0, \ t_{1})$ such that $$ \|u (\beta)\|_{\xi}= M(t_{1})\geq M(\beta), $$ which contradicts \eqref{e25}. This contradiction implies that $M(t)$ is constant and \begin{equation} \|u (t)\|_{\xi} < M(t)= M(0) \quad \text{for all } t\geq 0. \label{e27} \end{equation} \noindent \textbf{ Case (ii).} There is a point $t_{0}\geq 0$ such that $M(t_{0})= \|u (t_{0})\|_{\xi}$. Then, using the \eqref{e21} and \eqref{e23}, for all $\varepsilon >0$, we obtain \begin{align*} D^+( |u_{i_{s}}(s)|)\Big|_{s=t_{0}} & \leq - c_{i_{t_{0}}} |u_{i_{t_{0}}}(t_{0})|\xi^{-1}_{i_{t_{0}}}\xi_{i_{t_{0}}} +\sum_{j=1}^{n}a_{i_{t_{0}}j} \tilde{L}_{j} |u_{j }(t_{0} )|\xi^{-1}_{j } \xi_{j} \\ &\quad +\sum_{j=1}^{n} b_{i_{t_{0}}j} L_{j} |u_{j }(t_{0}-\tau_{i_{t_{0}}j}(t_{0}))| \xi^{-1}_{j } \xi_{j} \\ &\leq \big[-c_{i_{t_{0}}} \xi_{i_{t_{0}}} + \sum_{j=1}^{n}a_{i_{t_{0}}j} \tilde{L}_{j} \xi_{j} +\sum_{j=1}^{n} b_{i_{t_{0}}j} L_{j} \xi_{j} \big] M(t_{0}) \\ & < - \eta M(t_{0})+ \eta\min_{1\leq j \leq n}\{\xi^{-1}_{j }\}\varepsilon \,. \end{align*} %\eqref{e28} In addition, if $M(t_{0})\geq \min_{1\leq j \leq n}\{\xi^{-1}_{j }\}\varepsilon$, then $M(t )$ is strictly decreasing in a small neighborhood $(t_{0}, t_{0}+\delta_{0})$. This contradicts that $M(t)$ is non-decreasing. Hence, \begin{equation} \|u (t_{0})\|_{\xi}=M(t_{0})< \min_{1\leq j \leq n}\{\xi^{-1}_{j }\}\varepsilon. \label{e29} \end{equation} Furthermore, for any $t>t_{0}$, by the same approach used in the proof of \eqref{e29}, we have \begin{equation} \|u (t )\|_{\xi} < \min_{1\leq j \leq n}\{\xi^{-1}_{j }\}\varepsilon, \quad \text{if } M(t )= \|u (t )\|_{\xi}. \label{e210} \end{equation} On the other hand, if $M(t )> \|u (t )\|_{\xi}, t>t_{0}$. We can choose $t_{0}\leq t_{3} \|u (s )\|_{\xi} \quad \text{for all } s\in ( t_{3}, t]. $$ Using a similar argument as in the proof of Case (i), we can show that $M(s)\equiv M(t_{3})$ is constant for all $s\in ( t_{3}, t]$, %\label{e211} which implies that $$ \|u (t )\|_{\xi} < M(t)= M(t_{3})=\|u (t_{3})\|_{\xi} <\min_{1\leq j \leq n}\{\xi^{-1}_{j }\}\varepsilon. $$ In summary, for all $t\geq 0$, we obtain \begin{equation} \|u (t )\|_{\xi}< \max\big\{M(0), \min_{1\leq j \leq n} \{\xi^{-1}_{j }\}\varepsilon \big\}. \label{e212} \end{equation} Hence, for $\varepsilon >0$, set $$ \delta=\frac{\min_{1\leq j \leq n}\{\xi^{-1}_{j }\}\epsilon}{\max_{1\leq j \leq n}\{\xi^{-1}_{j }\}}>0. $$ Then, for every solution $Z(t)=(x_{1}(t), x_{2}(t),\dots,x_{n}(t))^{T} $ of \eqref{e11} with any initial value $ \varphi=(\varphi_{1}(t), \varphi_{2}(t), \dots, \varphi_{n}(t))^{T} $ and $$ \|\varphi -Z ^{*}\|=\max_{1\leq j\leq n} \big\{\sup_{-\infty\leq t\leq 0}|\varphi_{j}-Z_{j}^{*}| \big\}<\delta, $$ in view of \eqref{e212}, we have $|x_{ i}(t)-x^{*}_{i } | < \varepsilon$, for all $t\geq 0$, $i= 1, 2, \dots, \ n$. This completes the proof. \end{proof} \section{An Example} To illustrate the results obtained in previous sections, consider the CNNs, with unbounded time-varying delays, \begin{equation} \begin{aligned} x_{1}'(t)&=- x_{1}(t) + \dfrac{1}{4} f_{1}(x_{1}(t )) + \dfrac{1}{36} f_{2}(x_{2}(t )) + \dfrac{1}{4} g_{1}(x_{1}(t-\dfrac{1}{2}|t\sin t|))\\ &\quad + \dfrac{1}{36} g_{2}(x_{2}(t-\dfrac{1}{3}|t\sin t|))+ 1 , \\ x_{2}'(t)&= - x_{2}(t) + f_{1}(x_{1}(t )) + \dfrac{1}{4} f_{2}(x_{2}(t ))+ g_{1}(x_{1}(t-\dfrac{1}{4}|t\sin t|))\\ &\quad + \dfrac{1}{4} g_{2}(x_{2}(t-\dfrac{1}{5}|t\sin t|)) + 2 , \end{aligned} \label{e31} \end{equation} where $f_{1}(x)=f_{2}(x)=g_{1}(x)=g_{2}(x)=\arctan x $. Note that \begin{gather*} c _{1} =c _{2} = L _{1}=L _{2}=\tilde{L} _{1}=\tilde{L} _{2}=1, \quad a _{11} = b _{11} =\frac{1}{4}, \\ a _{12} = b _{12} = \frac{1}{36}, \quad a _{21} = b _{21} =1, \quad a _{22} = b _{22} =\frac{1}{4} . \end{gather*} Then $$ d_{ij}=\frac{1}{c _{i}} ( a _{ij} \tilde{L}_j + b _{ij} L_j) \quad i,j=1,2, \quad D =(d_{ij})_{2\times 2}=\begin{pmatrix} 1/2& 1/18 \\ 2& 1/2 \end{pmatrix}. $$ Hence, we have $\rho(D )=\frac{5}{6}<1$. Therefore, it follows from theory of $M$-matrix in \cite{b1} that there exist constants $\eta>0 $ and $\xi_{i}>0, i=1, 2,$ such that for all $t>0$, there holds $$ - c _{i} \xi_{i}+ \sum_{j=1}^{2}|a _{ij} |\tilde{L}_{j}\xi_{j}+ \sum_{j=1}^{2}|b _{ij} | L_{j}\xi_{j}<-\eta<0, \quad i=1, 2, $$ which implies that \eqref{e31} satisfied (H1) and (H2). Hence, from Lemma \ref{lem1.1} and Theorem \ref{thm2.1}, system \eqref{e31} has at least one equilibrium $Z^{*} $, and $Z^{*} $ is locally stable. 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