A set of stability criteria for delayed cellular neural networks

This work presents a set of criteria on the global asymptotic stability of delayed cellular neural networks (DCNN) by constructing suitable Lyapunov functionals, introducing ingeniously real parameters w/sub i/>0, /spl alpha//sup *//sub ij/, /spl beta//sup *//sub ij/, /spl eta//sup *//sub ij/, /spl zeta//sup *//sub ij/, /spl alpha//sub ij/, /spl beta//sub ij/, /spl eta//sub ij/, /spl zeta//sub ij//spl isin/R with /spl alpha//sup *//sub ij/+/spl beta//sup *//sub ij/=1, /spl alpha//sub ij/+/spl beta//sub ij/=1, /spl eta//sup *//sub ij/+/spl zeta//sup *//sub ij/=1, /spl eta//sub ij/+/spl zeta//sub ij/=1(i, j=1, 2, ..., n) and combining with elementary inequality technique 2ab/spl les/a/sup 2/+b/sup 2/. These criteria are of theoretical and applicable important significance in signal processing, especially in speed detection of moving objects, processing of moving images and the design of networks since they possess infinitely adjustable real parameters. This result is also discussed from the point of view of its relationship to earlier results.

I. Introduction

In a dynamical system which is globally asymptotically stable, there exists a unique equilibrium state to which every solution converges. Cellular neural networks (CNNs) were introduced by L. O. Chua and L. Yang [1], [2] in 1988. They have found important applications in signal processing, especially in static image treatment. During the past few years, the problems of stability of cellular neural networks has been one of the most active areas of research and has attracted the attention of many researchers, we refer to [1]–[7], [9]–[20], [22]. One knows that the CNNs is formed by many units called cells, the structure of the CNNs is similar to that found in cellular automata, namely, any cell in a cellular neural network is connected only to its neighbor cells. A cell contains linear and nonlinear circuit elements, which typically are linear capacitors, linear resistors, linear and nonlinear controlled sources, and independent sources. The circuit diagram and connection pattern implementing for the CNN can be found in [1], [2]. Delayed cellular neural networks (DCNNs) were first introduced in [3] and used in various types of motion-related applications such as speed detection of moving objects, processing of moving images and in pattern classification. In order to achieve these tasks, a delay parameter was introduced into the CNNs system equation. The stability analysis in this case is much more difficult than for CNNs. There exist some results of stability for CNNs and DCNNs, we refer to [1], [2], [4], [13], [14], [20] and [5]–[7], [9]–[19], [22], respectively, and the references cited therein. In this paper, we derived a set of criteria ensuring the global asymptotic stability of DCNNs with more general output functions by constructing suitable Lyapunov functionals [7]–[11], introducing ingeniously real parameters with , and combining with elementary inequality technique . The results related in [1]–[7], [9]–[20], [22] and the references cited therein are extended and improved. Moreover these criteria are independent of delays and possess infinitely adjustable real parameters. These are of prime importance and great interest in many application fields and the design of networks, and are easy to check and apply in practice.