Stability of Stochastic $$\theta $$ -Methods for Stochastic Delay Hopfield Neural Networks Under Regime Switching
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Stability of Stochastic θ-Methods for Stochastic Delay Hopfield Neural Networks Under Regime Switching Feng Jiang · Yi Shen
© Springer Science+Business Media New York 2013
Abstract This paper is concerned with the general mean-square (GMS) stability and meansquare (MS) stability of stochastic θ -methods for stochastic delay Hopfield neural networks under regime switching. The sufficient conditions to guarantee GMS-stability and MS-stability of stochastic θ -methods are given. Finally, an example is used to illustrate the effectiveness of our result. Keywords Stochastic delay Hopfield neural networks · Regime switching · MS-stability · GMS-stability · Stochastic θ -methods
1 Introduction Since the 1980s, stability analysis of various neural networks has been studied and applied successfully in many fields such as combinatorial optimization, parallel computing, image processing, signal processing and pattern recognition [1–6]. However, in implementation or application of neural networks, neural networks are often subject to environmental noise (white noise) or regime switching (color noise). The regime switching can be modeled by a finite-state Markov chain. The presence of such noise often affects neural networks. Many researchers have made a lot of progress in Markovian switching control theory [7,8]. Meanwhile, the problem of stochastic stability for stochastic delay neural networks with Markovian switching has been investigated extensively
F. Jiang (B) School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China e-mail: [email protected] Y. Shen Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China e-mail: [email protected]
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via the linear matrix inequality (LMI) technique [9,11]. For example, Zhu et al. [12] discussed exponential stability of uncertain stochastic neural networks with Markovian switching. At present, the stability study of stochastic neural networks is mainly based on Lyapunov method. However, it is often difficult to find an appropriate Lyapunov function or Lyapunov functional. When we fail to find an appropriate Lyapunov function or Lyapunov functional, we can employ the numerical approximation. Recently, there are many papers concerned with the numerical methods of stochastic differential equations [13–16]. Jiang and Shen [17] discussed the stability of the Split-Step Backward Euler method for the linear hybrid stochastic delay integro-differential equations. Zhou and Wu [18] discussed the convergence of the Euler-Maruyama method for stochastic delay differential equations with Markovian switching. Therefore it is very useful to deal with the stability by establishing the appropriate numerical methods of stochastic Hopfield neural networks under regime switching. Li et al. [19] discussed exponential stability of the Euler-Maruyama method for stochastic delay Hopfield neural networks. Further, Jiang and Shen [20] discussed mean-square stability of the Split-Step Backwa
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