428
We consider the sufficient condition for asymptotic stability of the zero solution
v
of equation (8) in terms of
certain matrix inequalities. Without loss of generality, we can assume that
*
0, (0) 0
v
S
and
f
=0 (for
otherwise, we let
*
x v v
and define
*
*
( ) (
) ( ))
S x S x v S v
.
The new form of equation (8) is now given by
1
( 1)
( )
( (
))
( ).
m
i
i
i
x k
Ax k
B S x k h Cu k
¦
(9)
This is a basic requirement for controller design. Now, we are interested designing a feedback controller for the
system (9) as
( )
( ),
u k Kx k
where
K
is
n m
u
constant control gain matrix.
The new form of equation (9) is now given by
1
( 1)
( )
( (
))
( ).
m
i
i
i
x k
Ax k
B S x k h CKx k
¦
(10)
Theorem 4.1
The zero solution of the delay-difference system of Hopfield neural networks with multiple delays
equation (10) is asymptotically stable if there exist symmetric positive definite matrices
P
,
i
G
,
i
W
, and
1
[ , , ] 0
n
L diag l
l
!
,
1, 2,...,
i
m
satisfying the following matrix inequalities of the form
