Globally Exponential Stability of Impulsive Neural Networks with Given Convergence Rate

This paper deals with the stability problem for a class of impulsive neural networks. Some sufficient conditions which can guarantee the globally exponential stability of the addressed models with given convergence rate are derived by using Lyapunov function and impulsive analysis techniques. Finally, an example is given to show the effectiveness of the obtained results. 1. Introduction Recently, special interest has been devoted to the dynamics analysis of neural networks due to their potential applications in different areas of science. Particularly, there has been a significant development in the theory of neural networks with impulsive effects [1–9], since such neural networks with impulsive effect can be used as an appropriate description of the phenomena of abrupt qualitative dynamical changes of essential continuous time systems. Based on the theory of impulsive differential equations [10–17], some sufficient conditions guaranteeing the exponential stability are derived [18–24]. For example, in [8], the author has obtained a criterion of exponential stability for a Hopfield neural network with periodic coefficients; in [18], by constructing the extended impulsive delayed Halanay inequality and Lyapunov functional methods, authors have got some sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive Hopfield neural networks with time delays. They all have obtained exponential stability for some kinds of neural networks through different methods. However, most of the existing results about the exponential stability of impulsive neural networks have a common feature that the exponential convergence rate cannot be derived, or derived but not the given one [8, 18, 23, 24]. The purpose of this paper is to establish some criteria which can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate by using Lyapunov function and impulsive analysis techniques. This work is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, the main results are presented. In Section 4, an example is discussed to illustrate the results. 2. Preliminaries Let denote the set of real numbers, denote the set of nonnegative real numbers, denote the set of positive integers and denote the -dimensional real space equipped with the Euclidean norm . Consider the following impulsive neural networks: where . corresponds to the number of units in a neural network; the impulse times satisfy , ; corresponds to the state of the neurons, denotes the