Elsevier

Neural Networks

Volume 20, Issue 6, August 2007, Pages 714-722
Neural Networks

R+n-global stability of a Cohen–Grossberg neural network system with nonnegative equilibria

https://doi.org/10.1016/j.neunet.2007.05.004Get rights and content

Abstract

In this paper, without assuming strict positivity of amplifier functions, boundedness of activation functions, or symmetry of the connection matrix, dynamical behaviors of delayed Cohen–Grossberg neural networks with nonnegative equilibria are studied. Based on the theory of the nonlinear complementary problem (NCP), a sufficient condition is derived guaranteeing the existence and uniqueness of the nonnegative equilibrium in the NCP sense. Moreover, this condition also guarantees the R+n-global asymptotic stability of the nonnegative equilibrium in the first orthant. The result is compared with some previous results and numerical examples are presented to indicate the viability of our theoretical analysis.

Section snippets

Introduction and model description

Cohen–Grossberg neural networks, proposed by Cohen and Grossberg (1983), have been extensively studied both in theory and applications. They have been successfully applied to signal processing, pattern recognition and associative memory, for example.

The success of these applications relies on understanding the underlying dynamical behavior of the models. A thorough analysis of the dynamics is a necessary step towards a practical design of neural networks. Such a model can be formalized as

Preliminaries

In this paper we use the following notations. A denotes the matrix transpose of A. As denotes the symmetric part of matrix A:12(A+A). A>0 denotes that A is symmetric and positive definite. Similarly we denote A0, A<0, and A0. R+n={x=(x1,x2,,xn):xi0,i=1,,n} denotes the first orthant. We denote the smallest and largest element of a set K={t1,t2,,tm} by minK and maxK, respectively. λ(A) denotes the spectrum set of matrix A. denotes some norm of vector and matrix. In particular, 2

Existence and uniqueness of the nonnegative equilibrium

In this section, we discuss the existence and uniqueness of the nonnegative equilibrium in the NCP sense.

Theorem 2 Existence and Uniqueness of the Nonnegative Equilibrium

Suppose a()A2 , d()D , and g()G . Let D=diag{D1,,Dn} and G=diag{G1,,Gn} . If there exists a positive definite diagonal matrix P=diag{P1,P2,,Pn} such that{P[DG1(A+B)]}s>0holds, then for each IRn , there exists a unique nonnegative equilibrium of the system(3)in the NCP sense.

Proof

Let fi(x)=di(xi)j=1n(aij+bij)gj(xj),i=1,,n,f(x)=(f1(x),,fn(x))F(x)=f(x+)+x where x+ and x are

R+n-global asymptotic stability of the nonnegative equilibrium

In this section, we discuss the R+n-global asymptotic stability of the nonnegative equilibrium defined in the previous section. Let x be the nonnegative equilibrium of the system (3) in the NCP sense and y(t)=x(t)x. Thus, the system (2) can be rewritten as dyi(t)dt=ai(yi(t))[di(yi(t))+j=1naijgj(yj(t))+j=1nbijgj(yj(tτ))+Ji] or in matrix form dy(t)dt=a(y(t))[d(y(t))+Ag(y(t))+Bg(y(tτ))+J] where for i=1,,n, ai(s)=ai(s+xi),a(y)=diag{a1(y1),,an(yn)}di(s)=di(s+xi)di(xi),d

Comparison and numerical example

In Chen and Rong (2003), the authors investigated global stability of delayed Cohen–Grossberg neural networks where amplifier functions are assumed strictly positive. In this way there was not of much difference between dealing with Cohen–Grossberg neural networks and cellular or Hopfield neural networks which do not contain any amplifier functions. The same linear matrix inequality as (16) was presented to guarantee the existence, uniqueness, and global stability of the equilibrium. In this

Conclusions

In this paper, we investigate the dynamics of the positive solutions of the Cohen–Grossberg neural networks with a time delay. Based on the theory of NCP and the LMI technique, a condition is obtained guaranteeing the existence, uniqueness, and global stability of the nonnegative equilibrium in the NCP sense. If the equilibrium is positive, then the stability is globally exponential. Numerical examples verify the viability of the theoretical results.

References (17)

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This work is supported by the National Science Foundation of China 60074005, 60374018.

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