427
0.9 0
,
0 0.9
A
§
·
¨
¸
©
¹
0.5 0
0 0.5
B
§
·
¨
¸
©
¹
,
1
2 ( )
tan ( ),
1, 2,
i
i
i
s x
x i
S
and
1
h
.
Using the LMI Toolbox in MATLAB, we found that the LMIs in Theorem 3.1 are feasible and
0.5340 0.0241
0.0241 0.6009
P
§
·
¨
¸
©
¹
,
0.0937 0.0209
0.0209 0.3326
W
§
·
¨
¸
©
¹
are set of solutions to the LMIs equation (6).
By a straightforward, we have
0 8365 0
0
0 2694
.
.
\
§
·
¨
¸
©
¹
.
The eigenvalues are -0.8365 and -0.2694, respectively. This implies the matrix
0
\
. It follows from Lemma
2.1 that the zero solution of delay-difference control system of Hopfield neural networks is asymptotically stable.
Applications
In this section, we apply the main result of this paper, which provides a sufficient condition for the
asymptotic stability of delay-difference control system of Hopfield neural networks with multiple delays in terms
of certain matrix inequalities.
We consider delay-difference control system of Hopfield neural networks with multiple delays of the form
1
( 1)
( )
( (
))
( ) ,
m
i
i
i
v k
Av k
B S v k h Cu k f
¦
(8)
where
( )
n
v k
:
R
is the neuron state vector,
1
0
m
h
h
d d d
,
1
{ , , }
n
A diag a a
,
0
i
a
t
,
1, 2,...,
i
n
is the
n n
u
constant relaxation matrix,
i
B
,
1, 2,...,
i
n
are the
n n
u
constant weight
matrices,
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
