Fixed Points and Exponential Stability for Impulsive Time-Delays BAM Neural Networks via LMI Approach and Contraction Mapping Principle

Ruofeng Rao,Zhilin Pu,Xinggui Li,Shouming Zhong

doi:10.1155/2016/3154683

Abstract

The fixed point technique has been employed in the stability analysis of time-delays bidirectional associative memory (BAM) neural networks with impulse. By formulating a contraction mapping in a product space, a new LMI-based exponential stability criterion was derived. Lately, fixed point methods have educed various good results inspiring this work, but those criteria cannot be programmed by a computer. In this paper, LMI conditions of the obtained result can be applicable to computer Matlab LMI toolbox which meets the need of the large-scale calculation in real engineering. Moreover, a numerical example and a comparable table are presented to illustrate the effectiveness of the proposed methods.

Highlights

Bidirectional associative memory (BAM) neural networks model was originally introduced by Kosko [1, 2]: p ẋi = −aixi (t) + ∑wjigj (yj (t)) + Ii, i = 1, 2, . . . , n, j=1 (1)
A stable equilibrium of BAM neural networks is the important precondition of the successful applications
In 2015, Zhou utilized Brouwer’s fixed point theorem to prove the existence and uniqueness of equilibrium of the hybrid BAM neural networks with proportional delays and constructed appropriate delay differential inequalities to derive the stability of equilibrium [28]

Summary

Introduction

In [29], Banach fixed point theorem was applied to show the existence of the unique equilibrium of BAM neural networks with time-varying delays in the leakage terms, and the Lyapunov functional method was for demonstrating the global exponential stability. Different from [28, 29], we shall use Banach fixed point theorem deriving straightway the stability criterion of impulsive time-delay BAM neural networks, in which LMI conditions facilitate computer programming. (ii) L = (lij)n×n ⩾ 0(⩽ 0): a semipositive (seminegative) definite matrix; that is, yTLy ⩾ 0(⩽ 0) for any y ∈ Rn. Different from the methods of [28, 29], it will be the first time to utilize contraction mapping principle to infer directly the LMI-based stability criterion of BAM neural networks, convenient for computer programming. We shall propose the LMI-based criterion, novel against the existing results, published from 2013 to 2016 (see Remark 10 and Table 1)

Preliminaries

Main Result

Numerical Example

Conclusion