425
(1,1)=
,
hG W
(2,2)=
,
W
and
(3, 3)
.
hG
Proof
Consider the Lyapunov function
1
2
( ( ))
( ( ))
( ( ))
V y k V y k V y k
, where
1
1
( ( ))
(
) ( ) ( ),
k
T
i k h
V y k
h k i x i Gx i
¦
2
1
( ( ))
( ) ( ),
k
T
i k h
V y k
x i Wx i
¦
G
and
W
being symmetric positive definite solutions of (6) and
>
@
( )
( ), (
) .
y k x k x k h
Then difference of
( ( ))
V y k
along trajectory of solution of (3) is given by
1
2
( ( ))
( ( ))
( ( ))
V y k
V y k
V y k
'
'
'
,
where
1
1
1
( ( ))
(
) ( ) ( )
( ) ( )
( ) ( ),
k
k
T
T
T
i k h
i k h
V y k
h k i x i Gx i
hx k Gx k
x i Gx i
§
·
'
'
¨
¸
©
¹
¦
¦
2
1
( ( ))
( ) ( )
( ) ( )
(
) (
)
k
T
T
T
i k h
V y k
x i Wx i
x k Wx k x k h Wx k h
§
·
'
'
¨
¸
©
¹
¦
,
(7)
Then we have
( )[
] ( )
T
V x k hG W x k
' d
(
) (
)
T
x k h Wx k h
1
( ) ( ).
k
T
i k h
x i Gx i
¦
Using Lemma 2.2, we obtain
1
1
1
1
1
( ) ( )
( ) ( )
( ) .
T
k
k
k
T
i k h
i k h
i k h
x i Gx i
x i
hG
x i
h
h
§
·
§
·
t ¨
¸
¨
¸
©
¹
©
¹
¦
¦
¦
