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Introduction
In recent decades, Hopfield neural networks have been extensively studied in many aspects and
successfully applied to many fields such as pattern identifying, voice recognizing, system controlling, signal
processing systems, static image treatment, and solving nonlinear algebraic system, etc. Such applications are
based on the existence of equilibrium points, and qualitative properties of systems. In electronic implementation,
time delays occur due to some reasons such as circuit integration, switching delays of the amplifiers and
communication delays, etc. Therefore, the study of the asymptotic stability of Hopfield neural networks with
delays is of particular importance to manufacturing high quality microelectronic Hopfield neural networks. While
stability analysis of continuous-time neural networks can employ the stability theory of differential system (Liu
et
al.
2003), it is much harder to study the stability of discrete-time neural networks (Elaydi and Peterson 1990) with
time delays (Arik 2005) or impulses (Gubta and Jin 1996). The techniques currently available in the literature for
discrete-time systems are mostly based on the construction Lyapunov second method (Hale 1977). For Lyapunov
second method, it is well known that no general rule exists to guide the construction of a proper Lyapunov
function for a given system. In fact, the construction of the Lyapunov function becomes a very difficult task.
In this paper, we consider delay-difference control system of Hopfield neural networks of the form
( 1)
( )
( (
))
( )
v k
Av k BS v k h Cu k f
,
(1)
where
( )
n
v k
:
R
is the neuron state vector,
0,
h
t
1
{ , , }
n
A diag a a
,
0
i
a
t
,
1, 2,...,
i
n
is the
n n
u
constant relaxation matrix,
B
is the
n n
u
constant weight matrix,
C
is
n m
u
constant matrix,
( )
m
u k
R
is the control vector,
1
( , , )
n
n
f
f
f
R
is the constant external input vector and
1 1
( ) [ ( ), , ( )]
T
n n
S z s z
s z
with
>
@
1
, ( 1,1)
i
s C
R
where
i
s
is the neuron activations and
monotonically increasing for each
1, 2,...,
i
n
.
The asymptotic stability of the zero solution of the delay-differential system of Hopfield neural networks
has been developed during the past several years. We refer to monographs by Burton (Burton 1993) and Ye (Ye
1944) and the references cited therein. Much less is known regarding the asymptotic stability of the zero solution
of the delay-difference control system of Hopfield neural networks. Therefore, the purpose of this paper is to
establish sufficient condition for the asymptotic stability of the zero solution of equation (1) in terms of certain
matrix inequalities.
