Global robust stability of delayed neural networks

This work presents a sufficient condition for the existence, uniqueness, and global robust stability of the equilibrium point for Hopfield-type delayed neural networks. The result imposes constraint conditions on the boundary values of the network parameters independently of the delay parameter. This result is compared with the previous results derived in the literature.

I. Introduction

In electronic implementation of neural networks, a time delay will occur in the interaction between the neurons, which will affect the stability of the neural system. In recent years, considerable efforts have been devoted to the analysis of the stability of delayed neural networks [1]–[25], where various sufficient conditions were presented for the global stability of this class of neural networks with constant network parameters. On the other hand, besides time-delayed features of such neural networks, there might also be some uncertainties such as perturbations and component variations, which might lead to very complex dynamical behavior. In the design of neural networks, it is important to ensure that systems be stable with respect to these uncertainties, (this type of stability is called robust stability). This brief aims to derive new results for the existence, uniqueness and global asymptotic stability of the equilibrium point for delayed neural networks with intervalised network parameters. The delayed neural-network model we consider is defined by the following state equations:

$$\displaylines{{{du_{i}(t)}\over{dt}}=-a_{i}u_{i}(t)+\sum_{j=1}^{n}w_{ij}f_{j}(u_{j}(t))+ \sum_{j=1}^{n}w_{ij}^{\tau}f_{j}(u_{j}(t-\tau))+I_{i},\hfill\cr \hfill i=1,2,.., n \quad {\hbox{(1)}}}$$ where the are positive constants, is the transmission delay, and represent the weight coefficients of the neurons, the are the constant external inputs, and the are the activation functions.