Exponential Stability of Discrete-Time Hopfield Neural Networks

In this paper, some sufficient conditions for the local and global exponential stability of the discrete-time Hopfield neural networks with general activation functions are derived, which generalize those existing results. By means of M-matrix theory and some inequality analysis techniques, the exponential convergence rate of the neural networks to the equilibrium is estimated, and for the local exponential stability, the basin of attraction of the stable equilibrium is also characterized. @ 2004 Elsevier Ltd. All rights reserved. K e y w o r d s D i s c r e t e t i m e Hopfield neural networks, Equilibrium, Global exponential stability, Exponential convergence rate, Local exponential stability. 1. I N T R O D U C T I O N The well-known Hopfield neural network and its variations (delayed Hopfield neural networks, discrete-time Hopfield neural networks) have attracted many researchers' attention [1-12] since Hopfield's pioneering work [1], and have been successfully applied to many fields, such as intelligent control, optimization solvers, and associative memories (or pattern recognition), etc. In those applications, the stability of the neural networks is crucial and needs to be prescribed before designing a powerful network model with a fast convergence ability. Especially in associative memories, each particular pattern is stored in the networks as an equilibrium, the stability of the associated equilibrium shows the networks have the ability to retrieve the related pattern. Usually, in associative memory neural networks, one expects the networks can store as many patterns as possible. In that sense, information about the basin of attraction of each stable equilibrium *Author to whom all correspondence should be addressed. Research supported by National Natural Science Foundation of P.R. China (10371034), by the Foundation for University Excellent Teacher by the Ministry of Education, and by the Key Project of Chinese Ministry of Education (No. [2002178), and by the Science Foundation of Hunan University. 0898-1221/04/$ see front matter @ 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.camwa.2004.04.010 Typeset by A.A, CS-TEX