Periodic Solution for Impulsive Cellar Neural Networks with Time-Varying Delays in the Leakage Terms

Bingwen Liu,Shuhua Gong

doi:10.1155/2013/701087

Abstract

This paper is concerned with impulsive cellular neural networks with time-varying delays in leakage terms. Without assuming bounded and monotone conditions on activation functions, we establish sufficient conditions on existence and exponential stability of periodic solutions by using Lyapunov functional method and differential inequality techniques. Our results are complement to some recent ones.

Highlights

It is well known that impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth [1–3]
For i ∈ {1, 2, . . . , n}, it is followed by yi󸀠 (t) = −ci (t) (xi (t − ηi (t)) − xi∗ (t − ηi (t)))
If x∗(t) = (x1∗(t), x2∗(t), . . . , xn∗(t))T is the T-periodic solution of system (4), it follows from Lemma 3 that x∗(t) is globally exponentially stable

Summary

Introduction

It is well known that impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth [1–3]. In [8–10], the authors discussed the existence and global exponential stability of periodic solution of a class of cellular neural networks (CNNs) with impulse. In view of the fact that the coefficients and delays in neural networks are usually time varying in the real world, motivated by the above discussions, Abstract and Applied Analysis in this paper, we will consider the problem on periodic solution of the following impulsive CNNs with time-varying delays in the leakage terms:. The main purpose of this paper is to give the conditions for the existence and exponential stability of the periodic solutions for system (4). By applying Lyapunov functional method and differential inequality techniques, without assuming (A1) and (A2), we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the periodic solution for system (4), which are new and complement previously known results.

Preliminary Results

Main Results

An Example