Global exponential stability of impulsive fuzzy Cohen–Grossberg neural
networks with mixed delays and reaction–diffusion terms
Chenhui Zhou
a
, Hongyu Zhang
a
, Hongbin Zhang
a,
n
, Chuangyin Dang
b
a
Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, PR China
b
Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong
article info
Article history:
Received 15 November 2011
Received in revised form
16 January 2012
Accepted 5 February 2012
Communicated by Y. Liu
Available online 31 March 2012
Keywords:
Fuzzy Cohen–Grossberg neural networks
(FCGNNs)
Mixed time delays
Impulsive
Reaction–diffusion
Exponential stability
abstract
This paper is concerned with the problem of exponential stability for a class of impulsive fuzzy Cohen–
Grossberg neural networks with mixed time delays and reaction–diffusion. The mixed delays include time-
varying delays and continuously distributed delays. Based on the Lyapunov method, Poincare
´
Integral
Inequality, and the linear matrix inequality (LMI) approach, we found some new sufficient conditions
ensuring the global exponential stability of equilibrium point for impulsive fuzzy Cohen–Grossberg neural
networks with mixed time delays and reaction–diffusion terms. These global exponential stability
conditions depend on the reaction–diffusion terms and time delays. The results presented in this paper
are less conservative than the existing sufficient stability conditions. Finally, some examples are given to
show the effectiveness and superiority of the theoretical results.
& 2012 Elsevier B.V. All rights reserved.
1. Introduction
Since Cohen–Grossberg neural networks (CGNNs) were first
introduced by Cohen and Grossberg [1], many researchers have
done extensive research work on this subject due to their
important applications in many areas such as parallel computa-
tion, associative memory and optimization problems (see [2–12]).
Such applications depend heavily on the dynamical behaviors of
the networks such as stability, convergence, and oscillatory
properties. In particular, stability analysis for neural networks
plays an important role in the design and applications of the
networks. Some other models, such as Hopfield neural networks,
recurrent neural networks, cellular neural networks, and bidirec-
tional associative memory neural networks, are special cases of
this model. Therefore, it is important and necessary to investigate
the stability of CGNNs.
In implementation of neural networks, time delays are unavoid-
ably encountered due to the finite switching speed of neurons and
amplifiers [46]. It has been found that the existence of time delays
may lead to instability and oscillation in a neural network. There-
fore, the study of stability for delayed neural networks is of both
theoretical and practical importance [41–44]. In recent years, some
results on stability of Cohen–Grossberg neural networks with delays
have been obtained (see [2–9,11–16]). In general, diffusion effects
cannot be avoided in neural networks when electrons are moving in
asymmetric electromagnetic fields. Therefore, it is necessary to
consider the activations varying in space as well as in time. In
[17–26], the authors considered the stability of neural networks
with time delays and reaction–diffusion terms.
However, besides delay effect and reaction–diffusion, impul-
sive effects are also likely to exist in neural networks [27,29,40].
For instance, in implementation of electronic networks, the state
of the networks is subject to instantaneous perturbations and
experiences abrupt change at certain instants, which may be
caused by switching phenomenon, frequency change or other
sudden noise, that is, it does exhibit impulsive effects. Therefore,
it is necessary to consider both impulsive effect and reaction–
diffusion terms on the stability of delayed neural networks.
Several kinds of neural networks with impulse have been inves-
tigated. For example, some results on global stability of impulsive
Cohen–Grossberg neural networks with delays were obtained in
[28,30,31]. In 2005, some new sufficient conditions for global
stability of impulsive delay model were obtained by establishing
the delay differential inequalities with impulsive initial condi-
tions in [32]. In 2007, some results on global exponential stability
of impulsive neural networks with time-varying delays and
reaction–diffusion terms were obtained in [34]. Recently, some
sufficient conditions ensuring the global exponential stability of
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journal homepage: www.elsevier.com/locate/neucom
Neurocomputing
0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.neucom.2012.02.012
n
Corresponding author.
E-mail address: zhanghb@uestc.edu.cn (H. Zhang).
Neurocomputing 91 (2012) 67–76
