Elsevier

Neurocomputing

Volume 73, Issues 13–15, August 2010, Pages 2671-2680
Neurocomputing

Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays

https://doi.org/10.1016/j.neucom.2010.05.002Get rights and content

Abstract

In this paper, the global asymptotic stability is investigated for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. The jumping parameters are modeled as a continuous-time, finite-state Markov chain. By employing the Lyapunov–Krasovskii functional and stochastic analysis theory as well as linear matrix inequality technique, some novel sufficient conditions are derived to guarantee the global asymptotic stability of the equilibrium point in the mean square. The proposed model of neutral type is quite general since many factors such as noise perturbations, Markovian jump parameters and mixed time delays are considered in this paper. The activation functions in this paper may be neither monotonically increasing nor continuously differentiable, and they are more general than those usual Lipschitz conditions. The results obtained in this paper comprise and generalize those given in the previous literature. Two numerical examples are provided to show the effectiveness of the theoretical results.

Introduction

Recently, neutral type systems have been intensively studied due to the reason that many practical processes can be modeled as general neutral type descriptor systems, such as circuit analysis, computer aided design, real time simulation of mechanical systems, power systems, chemical process simulation, population dynamics and automatic control. For example, Arino and Nosov [1] studied the stability and asymptotic properties of a class of neutral type functional differential equations based on the pattern equation method; by employing a new way on functional analysis of semi-group of operators, Hadd [6] investigated a class of singular functional differential equations of neutral type in Banach spaces; Han et al. [7] discussed robust absolute stability criteria for uncertain Lur’e systems of neutral type; Rabah et al. [15] investigated the asymptotic stability properties of neutral type systems in Hilbert space, and [16] studied the problem of strong stabilizability of linear systems of neutral type; Sahiner [19] discussed the oscillatory behavior of the second order neutral type delay differential equations; Wang and Liu [22] studied the existence, uniqueness and global attractivity of periodic solution for a kind of neutral functional differential systems with delays; Xu et al. [26] considered the problem of guaranteed cost control for uncertain neutral stochastic systems. Moreover, it is inspiring that the neutral type phenomenon has been already taken into account in delayed neural networks; see, for instance Lien et al. [4], Lou and Cui [10], Park et al. [13,14], Rakkiyappan and Balasubramaniam [17,18], and so on.

On the other hand, systems with Markov jump parameters, which driven by continuous-time Markov chains, have been widely used to model many practical systems where they may experience abrupt changes in their structure and parameters. For example, Boukas and Yang [3] studied the exponential stability problem for a class of stochastic systems with Markovian jump parameters; Liu et al. [9] investigated the delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching; Mao [11] considered the stability of nonlinear stochastic differential equations with Markovian jump parameters; Mariton and Bertrand [12] investigated the feedback control problem for a class of linear systems with Markovian jump parameters; Sun and Cao [20] discussed the stabilization of stochastic delayed neural networks with Markovian switching; Wang et al. [21] studied the exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters; Wang et al. [23] investigated the exponential stability of delayed recurrent neural networks with Markovian jumping parameters; Yuan and Lygeros [27] discussed the problem of exponential stabilization by the state feedback for a class of stochastic differential equations with Markovian jump parameters; Yuan and Mao [28] investigated the almost surely asymptotic stability for nonlinear stochastic differential delay equations with Markovian jump parameters; Zhu and Cao [29] discussed the global asymptotic stability of Markovian jump neural networks with continuously distributed delays. However, there have been few results of an investigation for the stability of neural networks of neutral type with Markovian jump parameters, which motivates the present study.

Based on the above discussion, we consider a class of neutral-type stochastic neural networks with both Markovian jump parameters and mixed time delays. The main purpose of this paper is to study the global asymptotic stability for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. By employing the Lyapunov–Krasovskii functional and stochastic analysis theory as well as linear matrix inequality technique, some novel sufficient conditions are derived to guarantee the global asymptotic stability of the equilibrium point in the mean square. The proposed model of neutral type is quite general since many factors such as noise perturbations, Markovian jump parameters and mixed time delays are considered in this paper. The results obtained in this paper comprise and generalize those given in the previous literature. Two numerical examples are provided to show the effectiveness of the theoretical results.

The remainder of this paper is organized as follows. In Section 2, the model of stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays is introduced, and some assumptions needed in this paper are presented. By means of the Lyapunov–Krasovskii functional approach, our main results are established in Section 3. In Section 4, two numerical examples are given to show the effectiveness of the obtained results. Finally, the paper is concluded with some general conclusions in Section 5.

Section snippets

Model description and main assumptions

Notation: Throughout this paper, the following notations will be used. Rn and Rn×n denote the n-dimensional Euclidean space and the set of all n×n real matrices, respectively. The superscript “T” denotes the transpose of a matrix or vector, and the symbol “ within a matrix represents the symmetric term of the matrix. Trace (·) denotes the trace of the corresponding matrix and I denotes the identity matrix with appropriate dimensions. For square matrices M1 and M2, the notation M1>(,<,)M2

Main results and proofs

In this section, the stochastically asymptotic stability in the mean square of the equilibrium point for the system (2) is investigated under Assumption 1, Assumption 2, Assumption 3, Assumption 4.

Theorem 1

Under Assumptions Assumption 1, Assumption 2, Assumption 3, Assumption 4 the equilibrium point of Eq. (2) (or Eq. (1) equivalently) is stochastically asymptotic stability in the mean square, if there exist positive scalars λi(iS), positive diagonal matrices H,M1,M2 and positive definite matrices F1,F

Illustrative examples

In this section, two numerical examples are given to illustrate the effectiveness of the obtained results.

Example 1

Consider a two-dimensional stochastic neural network of neutral type with both Markovian switching and mixed time delays:dx(t)=D(r(t))x(t)+A(r(t))f˜(x(t),r(t))+B(r(t))g˜(x(tτ1(t)),r(t))+C(r(t))tτ2(t)th˜(x(s),r(s))ds+Jdt+E1(r(t))d(x(tτ1(t)))+E2(r(t))d(x(tτ2(t)))+σ(x(t),x(tτ1(t)),x(tτ2(t)),t,r(t))dw(t),where x(t)=(x1(t),x2(t))T, J=(0,0)T, w(t) is a two dimensional Brownian motion,

Concluding remarks

In this paper, we have introduced and studied the stability of a new class of stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. By employing the Lyapunov–Krasovskii functional and stochastic analysis theory as well as linear matrix inequality (LMI) technique, we have investigated the stochastically asymptotic stability analysis problem for this new model of neutral type. The obtained results generalize and comprise those results with/without

Acknowledgements

The authors would like to thank the chief editor, the associate editor and two anonymous reviewers for their good comments and valuable suggestions, which have helped us to improve the paper greatly.

Quanxin Zhu received the B.S. and M.S. degrees from Hunan Normal University, Changsha, China, in 1999 and 2002, respectively. In 2005, he received the Ph.D. degree from Sun Yatsen (Zhongshan) University, Guangzhou, China, in probability and statistic. From July 2005 to May 2009, he was with the South China Normal University. From February 2009 to June 2009, he was a visiting scholar in the Department of Mathematics, Southeast University, Nanjing, China. Dr. Zhu is a reviewer of Mathematical

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Quanxin Zhu received the B.S. and M.S. degrees from Hunan Normal University, Changsha, China, in 1999 and 2002, respectively. In 2005, he received the Ph.D. degree from Sun Yatsen (Zhongshan) University, Guangzhou, China, in probability and statistic. From July 2005 to May 2009, he was with the South China Normal University. From February 2009 to June 2009, he was a visiting scholar in the Department of Mathematics, Southeast University, Nanjing, China. Dr. Zhu is a reviewer of Mathematical Reviews and Zentralblatt-Math, and he is a reviewer of more than 18 other journals. He is currently a professor of Ningbo University. He is the author or coauthor of more than 25 journal papers. His research interests include random processes, stochastic control, stochastic differential equation, stability theory, nonlinear systems, Markovian jump systems and stochastic neural networks.

Jinde Cao received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, in 1986, 1989, and 1998, respectively, all in mathematics/applied mathematics. From March 1989 to May 2000, he was with the Yunnan University, where he was a professor during 1996–2000. In May 2000, he was in the Department of Mathematics, Southeast University, Nanjing, China. From July 2001 to June 2002, he was a postdoctoral research fellow in the Department of Automation and Computer-aided Engineering, Chinese University of Hong Kong, Hong Kong. During 2006–2008, he was a visiting research fellow and a visiting professor in the School of Information Systems, Computing and Mathematics, Brunel University, London, UK. He is currently a TePin professor and a doctoral advisor at the Southeast University. He is the author or coauthor of more than 160 journal papers and five edited books and a reviewer of the Mathematical Reviews and the Zentralblatt-Math. He is an associate editor of the Journal of The Franklin Institute, the Mathematics and Computers in Simulation, the Neurocomputing, the International Journal of Differential Equations, the Differential Equations and Dynamical Systems, and the Open Electrical and Electronic Engineering Journal. His research interests include nonlinear systems, neural networks, complex systems and complex networks, stability theory, and applied mathematics. Prof. Cao is an Associate Editor of the IEEE Transactions on Neural Networks.

This work was jointly supported by the National Natural Science Foundation of China (10801056, 60874088), the Natural Science Foundation of Ningbo (2010A610094), K.C. Wong Magna Fund in Ningbo University and the Specialized Research Fund for the Doctoral Program of Higher Education (20070286003).

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