摘要： 研究一類具有時(shí)滯的二階Hopfield神經(jīng)網(wǎng)絡(luò)的動(dòng)力學(xué)行為.利用Brouwer不動(dòng)點(diǎn)定理和反證法證明了唯一平衡點(diǎn)的存在性，并通過(guò)構(gòu)造合適的Lyapunov函數(shù)，結(jié)合不等式放縮，得到了平衡點(diǎn)全局漸近穩(wěn)定的幾個(gè)充分條件.

關(guān)鍵詞： Hopfield神經(jīng)網(wǎng)絡(luò);Lyapunov函數(shù);存在性與唯一性;全局漸近穩(wěn)定性

Stability of second order Hopfield neural networks with time delays

Wang Shuna， Liu Jiang

（School of Mathematics & Statistics，Jiangsu Normal University，Xuzhou 221116，Jiangsu，China）

Abstract： Dynamical behaviors of a class of second order Hopfield neural networks with time delays is investigated. The existence of a unique equilibrium point is proved by using Brouwer’s fixed point theorem and the counter proof method， and some sufficient conditions for the global asymptotic stability of the equilibrium point are obtained through the combination of a suitable Lyapunov function and an algebraic inequality technique

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Key words： Hopfield neural network; Lyapunov function; existence and uniqueness; global asymptotic stability

CLC number： O193，TP183

Document code：A

doi： 10.3969/j.issn.2095-4298..0.009

0 Introduction

As is known to all， neural networks have attracted much attention because of their wide application in many engineering and technical fields， such as signal processing， pattern recognition， automatic control engineering， image shadowing， associative memory， date science and engineering computing， combinatorial optimization， e.g.^［1-3］. The Hopfield neural network presented in 1982^［4］ is one of the most used neural network architectures， it has been used to solve ill-posed problems with great success， which has been the evergreen hot topics in recent several decades because of their important applications in many fields. Therefore， the study of stability of Hopfield neural networks has caught many researchers’ attention^［5-8］. Moreover， it is well known that the signal transmission between neurons needs to take some time， either in biological or artificial neural networks. Hopfield neural networks with time delays have been extensively investigated over the years， and various sufficient conditions for the stability of the equilibrium point of such neural networks have been presented via different approaches^［9-15］.

For example，F(xiàn)aydasicok^［10］ conducted an investigation into the stability issue for a more general class of neutral-type Hopfield neural networks that involves multiple time delays in the states of neurons and multiple neutral delays in the time derivatives of the states of neurons. By constructing a new proper Lyapunov functional， the global asymptotic stability was derived. Li and Chen^［12］ discussed the uniform asymptotic stability， global asymptotic stability and global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation by using the Lyapunov functional method and the linear matrix inequality approach.

There are numerous interesting problems warrant further study. For instance， the time delay discussed in this paper is a constant time delay. We can also consider time-varying delays τ_j（t）. On the other hand， it is worth to study the exponential stability of second order Hopfield neural networks with time delays.

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