Elsevier

Neurocomputing

Volume 68, October 2005, Pages 196-207
Neurocomputing

Letters
Existence and exponential stability of almost periodic solutions for Hopfield neural networks with delays

https://doi.org/10.1016/j.neucom.2005.05.002Get rights and content

Abstract

In this paper Hopfield neutral networks with delays are considered. Sufficient conditions for the existence and exponential stability of the almost periodic solutions are established by using the fixed point theorem and differential inequality techniques. The results of this paper are new and they complement previously known results.

Introduction

Consider the following model for the delayed Hopfield neutral networks (HNNs):xi(t)=-cixi(t)+j=1nbij(t)gj(xj(t-τij(t)))+Ii(t),i=1,2,,n,in which n is the number of units in a neural network, xi(t) is the state vector of the ith unit at the time t, ci represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, gj(xj) denotes the conversion of the membrane potential of the jth unit into its firing rate, bij(t) denotes the strength of the jth unit on the ith unit at time t-τij(t), τij(t)0 corresponds to the transmission delay of the ith unit along the axon of the jth unit at time t, and Ii(t) denotes the external bias on the ith unit at time t.

It is well known that the HNNs have been successfully applied to signal and image processing, pattern recognition and optimization. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic solutions of system (1.1) in the literature. We refer the reader to [10], [11], [13], [4], [6], [16], [7] and the references cited therein. However, there exist few results on the existence and exponential stability of the almost periodic solutions of system (1.1).

The main purpose of this paper is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (1.1). By applying fixed point theorem and differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness and exponential stability of the almost periodic solution, which are new and complement previously known results.

Throughout this paper, it will be assumed that ci>0,τ=max1i,jn{maxt[0,ω]τij(t)}, and Ii,bij,τij:RR are almost periodic functions, where ci and τ0 are constants, and i,j=1,2,,n. Then, we can choose constants bij¯ and Ii¯ such thatsuptR|bij(t)|=bij¯,suptR|Ii(t)|=Ii¯,i,j=1,2,,n.

We also assume that the following condition (H0) holds.

(H0) for each j{1,2,,n}, gj:RR is Lipschitz with Lipschitz constant Lj, i.e.,|gj(uj)-gj(vj)|Lj|uj-vj|,foralluj,vjR.For convenience, we introduce some notations. We will use x=(x1,x2,,xn)TRn to denote a column vector, in which the symbol (T) denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x|=(|x1|,|x2|,,|xn|)T, and define x=max1in|xi|. For matrix A=(aij)n×n,AT denotes the transpose of A, A-1 denotes the inverse of A, |A| denotes the absolute-value matrix given by |A|=(|aij|)n×n, and ρ(A) denotes the spectral radius of A. A matrix or vector A0 means that all entries of A are greater than or equal to zero. A>0 is defined similarly. For matrices or vectors A and B, AB (resp. A>B) means that A-B0 (resp. A-B>0). LetD=diag(c1,c2,,cn),E¯=(bij¯)n×n,L=diag(L1,L2,,Ln).For V(t)C((a,+),R), letD-V(t)=limsuph0-V(t+h)-V(t)h,D-V(t)=liminfh0-V(t+h)-V(t)h,t(a,+).As usual, we introduce the phase space C([-τ,0];Rn) as a Banach space of continuous mappings from [-τ,0] to Rn equipped with the supremum norm defined byϕ=max1insup-τt0|ϕi(t)|,for all ϕ=(ϕ1(t),ϕ2(t),,ϕn(t))TC([-τ,0];Rn).

The initial conditions associated with system (1.1) are of the formxi(s)=ϕi(s),s[-τ,0],i=1,2,,n,where ϕ=(ϕ1(t),ϕ2(t),,ϕn(t))TC([-τ,0];Rn).

Definition 1 see Fink [5], He [9]

Let u(t):RRn be continuous in t. u(t) is said to be almost periodic on R if, for any ε>0, the set T(u,ε)={δ:|u(t+δ)-u(t)|<ε,tR} is relatively dense, i.e., for ε>0, it is possible to find a real number l=l(ε)>0, for any interval with length l(ε), there exists a number δ=δ(ε) in this interval such that |u(t+δ)-u(t)|<ε, for tR.

Definition 2

Let Z*(t)=(x1*(t),x2*(t),,xn*(t))T be an almost periodic solution of system (1.1) with initial value ϕ*=(ϕ1*(t),ϕ2*(t),,ϕn*(t))TC([-τ,0];Rn). If there exist constants α>0 and M>1 such that for every solution Z(t)=(x1(t),x2(t),,xn(t))T of system (1.1) with any initial value ϕC([-τ,0];Rn),|xi(t)-xi*(t)|Mϕ-ϕ*e-αt,t>0,i=1,2,,n.Then Z*(t) is said to be global exponential stable.

Definition 3

A real n×n matrix K=(kij) is said to be an M-matrix if kij0,i,j=1,2,,n,ij, and K-10.

The remaining part of this paper is organized as follows. In Section 2, we shall derive new sufficient conditions for checking the existence of almost periodic solutions. In Section 3, we present some new sufficient conditions for the uniqueness and exponential stability of the almost periodic solution of (1.1). In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections.

Section snippets

Existence and uniqueness of almost periodic solutions

The following lemma will be very essential to prove our main results of this paper.

Lemma 2.1

see Bermon and Plemmons [1], Lasalle [14]

Let A0 be an n×n matrix and ρ(A)<1, then (En-A)-10, where En denotes the identity matrix of size n.

Theorem 2.1

Let Condition (H0) hold and ρ(D-1E¯L)<1. Then, there exists exactly one almost periodic solution of system (1.1).

Proof

Let X={φ|φ=(φ1(t),φ2(t),,φn(t))T},where φi:RR is a continuous almost periodic function, i=1,2,,n. Then, X is a Banach space with the norm defined by φX=suptRmax1in|φi(t)|.

To proceed further, we

Exponential stability of almost periodic solutions

In this section, we establish some results for the uniqueness and exponential stability of the almost periodic solution of (1.1).

Theorem 3.1

Suppose that all the conditions of Theorem 2.1 hold. Then system (1.1) has exactly one almost periodic solution Z*(t). Moreover, Z*(t) is globally exponentially stable.

Proof

Since ρ(D-1E¯L)<1, it follows from Theorem 3.1 that system (1.1) has a unique almost periodic solution Z*(t)=(x1*(t),x2*(t),,xn*(t))T. Let Z(t)=(x1(t),x2(t),,xn(t))T be an arbitrary solution of

Two examples

In this section, we give an example to demonstrate the results obtained in previous sections.

Example 4.1

Consider the following HNN with delays:x1(t)=-x1(t)+12(sint)g1(x1(t-sin2t))+118(cost)g2(x2(t-2sin2t))+I1(t),x2(t)=-x2(t)+2(sin2t)g1(x1(t-3cos2t))+12(cos4t)g2(x2(t-4sin2t))+I2(t),where g1(x)=g2(x)==12(|x+1|-|x-1|), and I1(t) and I2(t) are almost periodic functions. Observe that c1=c2=L1=L2=1, and D-1E¯L=12118212.So, by easy computation, we can see that ρ(D-1E¯L)=56<1. Thus, from Theorem 3.1, Eq. (4.1)

Conclusions

In this letter, Hopfield neural networks with delays have been studied. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions have been established. These obtained results are new and they complement previously known results. Moreover, two examples are given to illustrate the effectiveness of the new results

Acknowledgements

The authors would like to express their sincere appreciation to the reviewer for his/her helpful comments in improving the presentation and quality of the Letter.

Bingwen Liu was born in Hunan Province, China, in 1971. He was received the B.S. degree in Mathematics, from the Hunan Normal University, Changsha China, in 1994, and the M.S. and Ph.D. degrees in Applied Mathematics from the Hunan University, Changsha China, in 2000, and 2005, respectively.

He is currently an Associate Professor in the Department of mathematics, Hunan University of Arts and Science, Changde, China. He is also the author or co-author of more than 30 journal papers. His research

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Bingwen Liu was born in Hunan Province, China, in 1971. He was received the B.S. degree in Mathematics, from the Hunan Normal University, Changsha China, in 1994, and the M.S. and Ph.D. degrees in Applied Mathematics from the Hunan University, Changsha China, in 2000, and 2005, respectively.

He is currently an Associate Professor in the Department of mathematics, Hunan University of Arts and Science, Changde, China. He is also the author or co-author of more than 30 journal papers. His research interests include nonlinear dynamic systems, neural networks.

Lihong Huang was born in Hunan, China, in 1963. He received the B.S. degree in Mathematics in 1984 from the Hunan Normal University, Changsha, China, and the M.S. degree in Applied Mathematics and the Ph.D. degree in Applied Mathematics from the Hunan University, Changsha, China, in 1988 and 1994, respectively.

From July 1988 to June 2000, he was with the Department of Applied Mathematics at Hunan University, where he was an Associate Professor of Applied Mathematics from July 1994 to May 1997, in June 1997 he became a Professor and Doctoral Advisor of Applied Mathematics and Chair of the Department of Applied Mathematics. Since July 2000 he has been Dean of the College of Mathematics and Econometrics at Hunan Unuversity, Changsha, China.

He is the author or co-author of more than 100 journal papers, five edited books. His research interests are in the areas of dynamics of neural networks, and qualitative theory of differential equations and difference equations.

This work was supported by the NNSF (10371034) of China, the Doctor Program Foundation of the Ministry of Education of China (20010532002) and Key Project of Chinese Ministry of Education ([2002]78).

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