Stochastic Synchronization of Reaction‐Diffusion Neural Networks under General Impulsive Controller with Mixed Delays

Xinsong Yang,Chuangxia Huang,Zhichun Yang

doi:10.1155/2012/603535

Abstract

This paper investigates drive‐response synchronization of a class of reaction‐diffusion neural networks with time‐varying discrete and distributed delays via general impulsive control method. Stochastic perturbations in the response system are also considered. The impulsive controller is assumed to be nonlinear and has multiple time‐varying discrete and distributed delays. Compared with existing nondelayed impulsive controller, this general impulsive controller is more practical and essentially important since time delays are unavoidable in practical operation. Based on a novel impulsive differential inequality, the properties of random variables and Lyapunov functional method, sufficient conditions guaranteeing the global exponential synchronization in mean square are derived through strict mathematical proof. In our synchronization criteria, the distributed delays in both continuous equation and impulsive controller play important role. Finally, numerical simulations are given to show the effectiveness of the theoretical results.

Highlights

Since the pioneering work of Pecora and Carroll 1, the issue of synchronization and chaos control has been extensively studied 2 due to its potential engineering applications such as secure communication, biological systems, and information processing see 3–10
Being motivated by the above discussions, this paper aims to study the global exponential derive-response synchronization of reaction-diffusion neural networks with multiple time-varying discrete delays and unbounded distributed delays via general impulsive control
Under the impulsive controller 2.7, the controlled system 2.8 is globally exponentially synchronized with system 2.2 in mean square if the following inequalities hold 0 < bk < 1, k ∈ N, 3.19 ξk − λmin R C

Summary

Introduction

Since the pioneering work of Pecora and Carroll 1 , the issue of synchronization and chaos control has been extensively studied 2 due to its potential engineering applications such as secure communication, biological systems, and information processing see 3–10. It is known that many pattern formation and wave propagation phenomena that appear in nature can be described by systems of coupled nonlinear differential equations, Abstract and Applied Analysis generally known as reaction-diffusion equations. These wave propagation phenomena are exhibited by systems belonging to very different scientific disciplines. In 22 , the authors investigated synchronization of reaction-diffusion neural networks with discrete and unbounded distributed delays. In 24 , the authors investigated the boundedness and exponential stability for nonautonomous fuzzy cellular neural networks with unbounded distributed delays and reaction-diffusion terms. The authors of 25 studied exponential stability of reaction-diffusion Cohen-Grossberg neural networks with time-varying discrete delays and stochastic perturbations

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Conclusion