Global stability of Hopfield neural networks under dynamical thresholds with distributed delays

Fei-Yu Zhang,Hai-Feng Huo

doi:10.1155/ddns/2006/27941

Abstract

We study the dynamical behavior of a class of Hopfield neural networks with distributed delays under dynamical thresholds. Some new criteria ensuring the existence, uniqueness, and global asymptotic stability of equilibrium point are derived. In the results, we do not require the activation functions to satisfy the Lipschitz condition, and also not to be bounded, differentiable, or monotone nondecreasing. Moreover, the symmetry of the connection matrix is not also necessary. Thus, our results improve some previous works in the literature. These conditions have great importance in designs and applications of the global asymptotic stability for Hopfield neural networks involving distributed delays under dynamical thresholds.

Highlights

During the last 30 years, Hopfield neural networks (Hopfield [9]) have been extensively studied and developed and have found many applications in different areas such as pattern recognition, model identification, and optimization
For Hopfield neural networks, one of the most investigated problems in dynamical behaviors is that of the existence, uniqueness, and global asymptotic stability of the equilibrium point
The property of global asymptotic stability, which means that the domain of attraction of the equilibrium point is the whole space and many pseudostable points will be eliminated, is of importance from the theoretical point of view as well as in practical applications in several fields

Summary

Introduction

During the last 30 years, Hopfield neural networks (Hopfield [9]) have been extensively studied and developed and have found many applications in different areas such as pattern recognition, model identification, and optimization. For Hopfield neural networks, one of the most investigated problems in dynamical behaviors is that of the existence, uniqueness, and global asymptotic stability of the equilibrium point. Gopalsmy and Leung [6] first considered the Hopfield neural networks with distributed delays under dynamical thresholds as follows:. By using Lyapunov function, they established a sufficient condition ensuring global asymptotic stability of the unique equilibrium point x∗ = 0 of system (1.1) with the case c = 0. Motivated by the above discussion, our aim in this paper is to study further the existence, uniqueness, and global asymptotic stability for the equilibrium point of the following Hopfield neural network (1.7) with distributed delays under dynamical thresholds: xi (t) = −dixi(t) + aij fj xj(t) − bij kij (s)xj(t − s)ds − cj , t ≥ 0, j=1. A is called a nonnegative matrix if A ≥ 0

Existence and uniqueness of the equilibrium point

Global asymptotic stability of the equilibrium point