430
2
1
1
( ( ))
( ) ( )
i
m k
T
i
j k h
i
V y k
x j W x j
¦¦
,
i
G
and
i
W
,
1, 2,...,
i
m
being symmetric positive definite solutions of (11) and
>
@
1
( )
( ), (
), , (
) .
m
y k x k x k h x k h
Then difference of
( ( ))
V y k
along trajectory of solution of (10) is given by
1
2
( ( ))
( ( ))
( ( ))
V y k
V y k
V y k
'
'
'
,
where
1
1
1
1
1
1
( ( ))
(
) ( ) ( )
( ) ( )
( ) ( ),
i
i
m
m
m
k
k
T
T
T
i
i
i
i
i
j k h
j k h
i
i
i
V y k
h k j x j G x j
h x k G x k
x j G x j
§
·
'
'
¨
¸
©
¹
¦
¦
¦
¦
¦
2
1
1
1
1
( ( ))
( ) ( )
( ) ( )
(
) (
)
i
m
m
m
k
T
T
T
i
i
i
i
i
j k h
i
i
i
V y k
x j W x j
x k W x k
x k h W x k h
§
·
'
'
¨
¸
©
¹
¦¦
¦
¦
.
The rest of the proof is similar to that of Theorem 3.1
need hold.
Conclusions
Based on a discrete analog of the Lyapunov second method, we have established a sufficient condition for the
asymptotic stability of delay-difference control system of Hopfield neural networks in terms of certain matrix
inequalities. The result has been applied to obtain new stability condition for some class of delay-difference
control system such as delay-difference control system of Hopfield neural networks with multiple delays in terms
of certain matrix inequalities.
Ackhowledgment
This work was supported by the Thai Research Fund Grant, the Higher Education Commission and Maejo
University, Thailand. The author would like to thank the anonymous referee for his/her valuable comments and
remarks which greatly improved the final version of the paper.
