Elsevier

Neurocomputing

Volume 74, Issue 17, October 2011, Pages 3404-3411
Neurocomputing

Stability and synchronization for Markovian jump neural networks with partly unknown transition probabilities

https://doi.org/10.1016/j.neucom.2011.05.018Get rights and content

Abstract

This paper addresses the problems of stability and synchronization for a class of Markovian jump neural networks with partly unknown transition probabilities. We first study the stability analysis problem for a single neural network and present a sufficient condition guaranteeing the mean square asymptotic stability. Then based on the Lyapunov functional method and the Kronecker product technique, the chaos synchronization problem of an array of coupled networks is considered. Both the stability and the synchronization conditions are delay-dependent, which are expressed in terms of linear matrix inequalities. The effectiveness of the developed methods is shown by simulation examples.

Introduction

In the past decades, there exists increasing interest in the study of dynamical properties of neural networks due to their extensive applications in associative memories, pattern recognition, signal processing and the other fields [4]. It is well known that the stability of neural networks is a prerequisite for these applications. Therefore, the stability analysis of neural networks with or without time delay is of much importance and has received a great deal of attention; see, for instance, [6], [14], [19], [21], [22], [25], [27], [30], [40] and the references therein.

Recently, it has been found that the neural networks can show complicated and unpredictable dynamics and even chaotic behaviors if the time delay and the parameters of the networks are chosen appropriately [12]. The problem of synchronization in various coupled systems and networks has been investigated and significant results have been obtained; see, for example, [1], [2], [3], [5], [8], [9], [13], [17], [28], [20], [29], [37] and the references therein. Synchronization in coupled delayed neural networks has been shown to be an important step toward both fundamental science and technological practice, such as brain science, secure communication, and harmonic oscillation generation [2], [13].

In reality, neural networks may exhibit the network mode jumping (switching) characteristic; that is, a neural network may have finite modes, which may switch from one to another at different time. It is shown that such jumping can be determined by a Markovian chain [18]. A great number of significant results on analysis and synthesis of Markovian jump systems have been reported in the literature [15], [24], [26], [31], [36], [38]; dynamical analysis results on the Markovian jump neural networks can be found in [10], [11], [32], [40], [41]. It is worth pointing out that all of the above-mentioned references assume that the information on transition probabilities in the jumping process is completely known. Some extended results concern the uncertain transition probabilities [7], [23]. However, such uncertainties have to require the knowledge of bound or structure of uncertainties. It is known that in most cases the transition probabilities of Markovian jump systems and networks are not exactly known, thus, it is of great significance to consider partly unknown transition probabilities. The stability and stabilization problems of continuous-time and discrete-time Markovian jump systems with partly unknown transition probabilities were investigated in [33], where the partly unknown transition probabilities did not need any knowledge of the unknown elements. With the similar concept and method, H control, filtering design and fuzzy control for Markovian jump systems with partly unknown transition probabilities were studied in [34], [35], [16], respectively. Recently, a sufficient condition on stabilization problem was derived in [39], which was shown to be less conservative than that in [33].

In this paper, we aim to deal with the problem of stability analysis and synchronization for a class of delayed Markovian jump neural networks with partly unknown transition probabilities. Based on the Lyapunov–Krasovskii functional method and stochastic analysis technique, a delay-dependent condition is presented to guarantee the mean square asymptotic stability of the considered networks. The chaos synchronization problem is then addressed for an array of coupled neural networks. Both the stability criterion and synchronization criterion are expressed in terms of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed method is demonstrated via numerical examples.

Notation: Throughout this paper, for real symmetric matrices X and Y, the notation XY (respectively, X>Y) means that the matrix XY is positive semi-definite (respectively, positive definite). In is an n-dimensional identity matrix. Rn denotes the n-dimensional Euclidean space, and the notation |·| refers to the Euclidean vector norm. The notation MT represents the transpose of the matrix M. The symmetric terms in a symmetric matrix are denoted by . The Kronecker product of matrices X and Y is denoted as XY. (Ω,F,{Ft}t0,P) denotes a complete probability space with a filtration {Ft}t0, where Ω is a sample space, F is the σ-algebra of subset of the sample space and P is the probability measure on F. Denote by LF0p([η,0],Rn) the family of all F0-measurable C([-η,0],Rn)-valued random variables ξ={ξ(θ):ηθ0} such that supηθ0E|ξ(θ)|2<, where E{·} is the expectation operator with respect to some probability measure P. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

Section snippets

Problem formulation

We consider the following delayed neural network with Markovian jumping parameters:x˙(t)=C(rt)x(t)+A(rt)f(x(t))+B(rt)f(x(tτ(t)))+D(rt)tσ(t)tf(x(s))ds,x(s)=ϕ(s),s[η,0],η=max{τ¯,σ¯},where x(t)=[x1(t),x2(t),,xn(t)]T is the state vector; f(x(t))=[f1(x1(t)),f2(x2(t)),,fn(xn(t))]T is the neuron activation function. C(rt)=diag{c1(rt),c2(rt),,cn(rt)}>0 describes the rate with which each neuron will reset its potential to the resting state in isolation when disconnected from the networks and

Stochastic stability

In this section, we will investigate the problem of asymptotic stability in the mean square of Markovian jump neural networks with partial information on transition probability. The main result is given in the following theorem.

Theorem 1

For given scalars τ¯>0, σ¯>0 and μ, system (1) is mean square asymptotically stable if there exist matrices Pi>0, Q1>0, Q2>0, Q3>0, Q4>0, R>0, diagonal matrices D>0, H>0, and matrices Zi=ZiT, such that the following LMIs hold for all iS:Ξ=Ξ11R0Ξ14PiBiτ¯2CiTRBiPiDiτ¯2C

Synchronization

In this section, we will consider the synchronization problem of an array of coupled identical Markovian neural networks with partly unknown transition probabilities.

We consider the following array of delayed neural networks for k=1,2,,m:x˙k(t)=C(rt)xk(t)+A(rt)f(xk(t))+B(rt)f(xk(tτ(t)))+D(rt)tσ(t)tf(xk(s))ds+l=1mαklΓ1xl(t)+l=1mβklΓ2xl(tτ(t)),where xk(t)=[xk1(t),xk2(t),,xkn(t)]TRn is the state vector of the kth neural network, n corresponds to the number of neurons. Γ1,Γ2Rn×n are

Numerical examples

In this section, we will give two examples to demonstrate the effectiveness of the proposed approach.

Example 1

Consider the Markovian neural network (1) with three modes and the following parameters: C1=3.0002.5,C2=2.5002.4,C3=2.0002.2,A1=0.20.50.40.3,A2=0.30.10.20.4,A3=0.10.90.30.2,B1=0.30.80.50.4,B2=0.10.90.81.2,B3=0.50.40.10.8,D1=0.50.50.20.7,D2=0.30.20.50.4,D3=0.20.70.50.3.The activation functions are taken as f1(x)=f2(x)=12(|x+1|+|x1|). Thus, we have Λ1=I, Λ2=I. We assume τ¯=1, μ=0.15, σ¯=0.1

Conclusions

In this paper, we have investigated the problems of stability and synchronization for a class of Markovian jump neural networks with partly unknown transition probabilities. By the Lyapunov–Krasovskii functional method, stochastic analysis technique and some properties of Kronecker product, delay-dependent stability condition and synchronization condition have been obtained, respectively. These conditions are given in terms of LMIs, which can be checked easily by Matlab LMI Toolbox. Numerical

Acknowledgments

This work was supported by the National Natural Science Foundation of PR China under Grant 61074043, the Natural Science Foundation of Jiangsu Province under Grant BK2008047, and Qing Lan Project.

Qian Ma received the B.Sc. degree in Computational Mathematics from Jiangsu University of Science and Technology, Zhenjiang, China, in 2005, and the M.Sc. degree in Fluid Dynamics from Nanjing University of Science and Technology, Nanjing, China, in 2007. She is now a Ph.D. candidate at the School of Automation, Nanjing University of Science and Technology, Nanjing, China. Her current research interests include neural networks, stochastic systems and genetic regulatory networks.

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Qian Ma received the B.Sc. degree in Computational Mathematics from Jiangsu University of Science and Technology, Zhenjiang, China, in 2005, and the M.Sc. degree in Fluid Dynamics from Nanjing University of Science and Technology, Nanjing, China, in 2007. She is now a Ph.D. candidate at the School of Automation, Nanjing University of Science and Technology, Nanjing, China. Her current research interests include neural networks, stochastic systems and genetic regulatory networks.

Shengyuan Xu received his B.Sc. degree from the Hangzhou Normal University, China in 1990, M.Sc. degree from the Qufu Normal University, China in 1996, and Ph.D. degree from the Nanjing University of Science and Technology, China 1999. From 1999 to 2000 he was a Research Associate in the Department of Mechanical Engineering at the University of Hong Kong, Hong Kong. From December 2000 to November 2001, and December 2001 to September 2002, he was a Postdoctoral Researcher in CESAME at the Université catholique de Louvain, Belgium, and the Department of Electrical and Computer Engineering at the University of Alberta, Canada, respectively. From September 2002 to September 2003, and September 2003 to September 2004, he was a William Mong Young Researcher and an Honorary Associate Professor, respectively, both in the Department of Mechanical Engineering at the University of Hong Kong, Hong Kong. Since November 2002, he has joined the School of Automation at the Nanjing University of Science and Technology as a Professor. He is now an Honorary Professor at the University of Hong Kong.

Dr. Xu was a recipient of the National Excellent Doctoral Dissertation Award in the year 2002 from the Ministry of Education of China. He obtained a grant from the National Science Foundation for Distinguished Young Scholars of PR China in the year 2006. He was awarded a Cheung Kong Professorship in the year 2008 from the Ministry of Education of China.

Dr. Xu is a member of the Editorial Boards of the Multidimensional Systems and Signal Processing, and the Circuits Systems and Signal Processing. His current research interests include robust filtering and control, singular systems, time-delay systems, neural networks, and multidimensional systems and nonlinear systems.

Yun Zou was born in Xian, PR China, in 1962. He received the B.Sc. degree in Numerical Mathematic form Northwestern University, Xian, China, in 1983. In 1987 and 1990, he received, respectively, the M.Tech degree and Ph.D degree in Automation from Nanjing University of Science and Technology (NUST), Nanjing 210094, China. Since July, 1990, he has been with NUST and now a professor in the Department of Automation, NUST. His recent research interests include singular systems, MD systems, and transient stability of power systems.

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