426
From the above inequality it follows that:
( )[
] ( )
T
V x k hG W x k
' d
(
) (
)
T
x k h Wx k h
1
1
1
1
( ) ( )
( )
T
k
k
i k h
i k h
x i
hG
x i
h
h
§
·
§
·
¨
¸
¨
¸
©
¹
©
¹
¦
¦
1
(1,1)
0 0
1
( ), (
), (
( ))
0 (2, 2) 0
0 0 (3,3)
k
T
T
T
i k h
x k x k h
x i
h
§
·
§
· ¨
¸
¨
¸ ¨
¸
©
¹ ¨
¸
©
¹
¦
1
( )
(
)
1(
( ))
k
i k h
x k
x k h
x i
h
§
·
¨
¸
¨
¸
¨
¸
¨
¸
¨
¸
¨
¸
©
¹
¦
( ) ( )
T
y k y k
\
where
(1,1)=
,
hG W
(2,2)= ,
W
(3,3)
hG
,
and
1
( )
( )
(
)
1(
( ))
k
i k h
x k
y k x k h
x i
h
§
·
¨
¸
¨
¸
¨
¸
¨
¸
¨
¸
¨
¸
©
¹
¦
.
By the condition (6),
( ( ))
V y k
'
is negative definite, namely there is a number
0
E
!
such that
2
( ( ))
( ) ,
V y k
y k
E
'
d
and hence, the asymptotic stability of the system immediately follows from
Lemma 2.1. This completes the proof.
Example 3.1
Let us consider a delay-difference control system of Hopfield neural networks (3), given by the
system
( 1)
( )
( (
))
( ),
x k
Ax k BS x k h CKx k
where the matrices are
