On exponential stability of bidirectional associative memory neural networks with time-varying delays ppt

On exponential stability of bidirectional associative memory
neural networks with time-varying delays
Ju H. Park
a,
*
, S.M. Lee
b
, O.M. Kwon
c
a
Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea
b
Platform Verification Division, BcN Business Unit, KT Co. Ltd., Daejeon, Republic of Korea
c
School of Electrical and Computer Engineering, Chungbuk National University, Cheongju 361-763, Republic of Korea
Accepted 19 April 2007
Abstract
For bidirectional associate memory neural networks with time-varying delays, the problems of determining the expo-
nential stability and estimating the exponential convergence rate are investigated by employing the Lyapunov func-
tional method and linear matrix inequality (LMI) technique. A novel criterion for the stability, which give
information on the delay-dependent property, is derived. A numerical example is given to demonstrate the effectiveness
of the obtained results.
Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction
As an extension of the unidirectional autoassociator of Hopfield [1], Kosko [2] has proposed a series of neural net-
works related to bidirectional associative memory (BAM). This class of networks has good application in the area of
pattern recognition and artificial intelligence. Therefore, the BAM neural networks has been one of the most interesting
research topics and has attracted the attention of many researchers. For instance, refer to Refs. [3–10]. Also, time delay
will inevitably occur in the communication and response of neurons owing to the unavoidable finite switching speed of
amplifiers in the electronic implementation of analog neural networks, so it is more in accordance with this fact to study
the BAM neural networks with time delays. The existence of time delay is frequently a source of oscillation and insta-

bility [11–19]. Therefore, the study of the stability and convergent dynamics of BAM with delays has raised considerable
interest in recent years, see for example [20–22] and the references cited therein.
In this paper, the problem of exponential stability for BAM with time-varying delays is considered. When it comes to
design a neural network, one concerns not only on the stability of the system but also on the convergence rate, that is to
say, one usually desires a fast response in the network, so it is important to determine the exponential stability and to
estimate the exponential convergence rate [23–28]. Based on the Lyapunov theory and linear matrix inequality frame-
work, a novel less conservative criterion is given in terms of LMI. The advantage of the proposed approach is that
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2007.05.003
*
Corresponding author.
E-mail address: (J.H. Park).
Chaos, Solitons and Fractals xxx (2007) xxx–xxx
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resulting stability criterion can be performed efficiently via existing numerical convex optimization algorithms such as
the interior-point algorithms for solving the linear matrix inequality inequalities [30].
The rest of this paper is organized as follows: in Section 2, we formulate the problem and state the well-known facts
and lemmas which would be used later; in Section 3, a new stability criterion for exponential stability of BAM with
time-varying delays will be established; in Section 4, some conclusions are drawn.
Notations: Throughout the paper, R
n
denotes the n dimensional Euclidean space, and R
nÂm
is the set of all n  m real
matrices. I denotes the identity matrix with appropriate dimensions. q denotes the elements below the main diagonal of
a symmetric block matrix. diagfÁÁÁg denotes the diagonal matrix. For symmetric matrices X and Y, the notation
X > Y(respectively, X P Y ) means that the matrix X–Y is positive definite, (respectively, nonnegative). k

M
ðÁÞ and
k
m
ðÁÞ denote the largest and smallest eigenvalue of given square matrix, respectively.
2. Problem statement
Consider the following BAM neural networks with time-varying delays:
_
u
i
ðtÞ¼Àa
i
u
i
ðtÞþ
X
m
j¼1
w
ji
g
j
ðv
j
ðt ÀsðtÞÞÞ þ I
i
; i ¼ 1; 2; ; n;
_
v
j

ðtÞ¼Àb
j
v
j
ðtÞþ
X
n
i¼1
v
ij

g
i
ðu
i
ðt À hðtÞÞÞ þ J
j
; j ¼ 1; 2; ; m;
ð1Þ
in which u ¼ðu
1
; u
2
; ; u
n
Þ
T
2 R
n
and v ¼ðv

1
; v
2
; ; v
m
Þ
T
2 R
m
are the activations of the ith neurons and the jth neu-
rons, respectively, w
ji
and v
ij
are the connection weights at the time t, I
i
and J
j
denote the external inputs, sðtÞ > 0 and
hðtÞ > 0 are positive time-varying delays which correspond to the finite speed of axonal signal transmission satisfying
sðtÞ <

s,
_
sðtÞ 6 s
d
< 1 and hðtÞ <

h,
_

hðtÞ 6 h
d
< 1, respectively, sðtÞ
Ã
¼ maxf

h;

sg, and a
i
> 0; b
j
> 0.
In this paper, it is assumed that the activate functions g
i
and

g
i
possess the following properties:
(A1) g
i
and

g
i
are nondecreasing and bounded on R; i ¼ 1; 2; ; maxfm; ng.
(A2) There exist real numbers k
1i
> 0 and k

2i
> 0 such that
0 6
g
i
ðn
1
ÞÀg
i
ðn
2
Þ
n
1
À n
2
6 k
1i
; i ¼ 1; 2; ; m;
0 6

g
i
ðn
1
ÞÀ

g
i
ðn

2
Þ
n
1
À n
2
6 k
2i
; i ¼ 1; 2; ; n:
ð2Þ
It is clear that under the assumptions (A1) and (A2), system (1) has at least one equilibrium. Assume that
u
Ã
¼ðu
Ã
1
; u
Ã
2
; ; u
Ã
n
Þ
T
and v
Ã
¼ðv
Ã
1
; v

Ã
2
; ; v
Ã
m
Þ
T
are the equilibrium point of the system, then we will shift the equilibrium
points to the origin by the transformation x
i
ðtÞ¼u
i
ðtÞÀu
Ã
i
, y
j
ðtÞ¼v
j
ðtÞÀv
Ã
j
,

f
i
ðx
i
ðtÞÞ ¼


g
i
ðu
i
ðtÞÞ À

g
i
ðu
Ã
i
Þ, and
f
j
ðy
j
ðtÞÞ ¼ g
j
ðv
j
ðtÞÞ Àg
j
ðv
Ã
j
Þ. Then, the transformed system is as follows:
_
x
i
ðtÞ¼Àa

i
x
i
ðtÞþ
X
m
j¼1
w
ji
f
j
ðy
j
ðt ÀsðtÞÞÞ; i ¼ 1; 2; ; n;
_
y
j
ðtÞ¼Àb
j
y
j
ðtÞþ
X
n
i¼1
v
ij

f
i

ðx
i
ðt ÀhðtÞÞÞ; j ¼ 1; 2; ; m;
x
i
ðsÞ¼/
i
ðsÞ; y
j
ðsÞ¼w
j
ðsÞ; s 2½Às
Ã
; 0; i ¼ 1; 2; ; n; j ¼ 1; 2; ; m;
ð3Þ
where the activate functions f
i
and

f
i
satisfy the following properties:
(H1) f
i
and

f
i
are bounded on R; i ¼ 1; 2; ; maxfm; ng,
(H2) There exist real numbers k

1i
> 0 and k
2i
> 0 such that
0 6
f
i
ðn
1
ÞÀf
i
ðn
2
Þ
n
1
À n
2
6 k
1i
; i ¼ 1; 2; ; m;
0 6

f
i
ðn
1
ÞÀ

f

i
ðn
2
Þ
n
1
À n
2
6 k
2i
; i ¼ 1; 2; ; n;
ð4Þ
(H3) f
i
ð0Þ¼0;

f
i
ð0Þ¼0; 8i.
For convenience, we can rewrite Eq. (3) in the form
2 J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx
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_
xðtÞ¼ÀAxðtÞþWf ðyðt ÀsðtÞÞÞ;
_
yðtÞ¼ÀByðtÞþV

f ðxðt ÀhðtÞÞÞ;

ð5Þ
where xðtÞ¼ðx
1
ðtÞ; x
2
ðtÞ; ; x
n
ðtÞÞ
T
; yðtÞ¼ðy
1
ðtÞ; y
2
ðtÞ; ; y
m
ðtÞÞ
T
, A ¼ diagða
1
; a
2
; ; a
n
Þ, B ¼ diagðb
1
; b
2
; ; b
m
Þ,

W ¼ðw
ij
Þ
mÂn
, V ¼ðv
ij
Þ
nÂm
, f ¼ðf
1
; f
2
; ; f
m
Þ
T
, and

f ¼ðf
1
; f
2
; ; f
n
Þ
T
.
The following facts, definition, and lemmas will be used for deriving main result.
Fact 1 (Schur complement). Given constant symmetric matrices R
1

; R
2
; R
3
where R
1
¼ R
T
1
and 0 < R
2
¼ R
T
2
, then
R
1
þ R
T
3
R
À1
2
R
3
< 0 if and only if
R
1
R
T

3
R
3
ÀR
2
"#
< 0; or
ÀR
2
R
3
R
T
3
R
1

< 0:
Fact 2.
For any z; y 2 R
nÂm
, and any positive definite matrix X 2 R
nÂn
, the following inequality:
2z
T
y 6 z
T
X
À1

z þ y
T
Xy
holds.
Definition 1.
For system defined by (5), if there exist the positive constants k and l > 1 such that
kxðtÞk þ kyðtÞk 6 le
Àkt
sup
Às
Ã
6h60
kxðhÞk þ sup
Às
Ã
6h60
kyðhÞk

8t > 0;
then, the trivial solution of the system (5) is exponentially stable where k is called the convergence rate (or degree) of
exponential stability.
Lemma 1. [29] Suppose that (2) holds, then
Z
u
v
½g
i
ðsÞÀg
i
ðvÞds 6 ½u À v½g

i
ðuÞÀg
i
ðvÞ; i ¼ 1; 2; ; n:
Lemma 2 [32]. For any constant matrix R 2 R
nÂn
, R ¼ R
T
> 0, scalar c > 0, vector function x : ½0; c!R
n
such that the
integrations concerned are well defined, then
Z
c
0
xðsÞds

T
R
Z
c
0
xðsÞds

6 c
Z
c
0
x
T

ðsÞRxðsÞds: ð6Þ
3. Main result
In this section, we present a stability criterion for exponential stability of system (1) using the Lyapunov stability
theory and linear matrix inequality approach.
Now the following theorem gives a new criterion for the stability of system (1).
Theorem 1. For given positive matrices K
1
¼ diagfk
11
; k
12
; ; k
1n
g, K
2
¼ diagfk
21
; k
22
; ; k
2n
g, positive scalars

h and

s,
the equilibrium point of system (1) is globally exponentially stable with convergence rate k if there exist two positive
diagonal matrices D ¼ diagfd
1
; d

2
; ; d
n
g and E ¼ diagfe
1
; e
2
; ; e
n
g, positive definite matrices P, Q, R
1
,R
2
,Z
1
,Z
2
,
L
1
, L
2
and any matrices N
i
ði ¼ 1; 2; ; 10Þ satisfying the following LMI:
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P ¼

U
1
þ N
1
A þ AN
T
1
2kD þ A
T
N
T
2
A
T
N
T
3
N
1
þ A
T
N
T
4
A
T
N
T
5
I R

1
À 2DAK
À1
2
0 N
2
0
IIU
2
N
3
0
IIIN
4
þ N
T
4
þ

hL
1
N
T
5
IIIIÀ

h
À1
e
À2k


h
L
1
IIIII
IIIII
IIIII
IIIII
IIIII
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 PW ÀN

1
W 00
00ÀN
2
W þ DW 00
V
T
Q À V
T
N
T
6
V
T
E ÀV
T
N
T
7
ÀN
3
W À V
T
N
T
8
ÀV
T
N
T

9
ÀV
T
N
T
10
00ÀN
4
W 00
00ÀN
5
W 00
U
3
þ N
6
B þ BN
T
6
2kE þ B
T
N
T
7
B
T
N
T
8
N

6
þ B
T
N
T
9
B
T
N
T
10
I R
2
À 2EBK
À1
1
0 N
7
0
IIU
4
N
8
0
IIIN
9
þ N
T
9
þ


sL
2
N
T
10
III IÀs
À1
e
À2k

s
L
2
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7

7
7
7
5
< 0;
ð7Þ
where
U
1
¼ 2kP À 2PA þZ
1
;
U
2
¼Àe
À2k

h
ð1 À h
d
ÞR
1
À e
À2k

h
ð1 À h
d
ÞK
À1

2
Z
1
K
À1
2
;
U
3
¼ 2kQ À 2QB þ Z
2
;
U
4
¼Àe
À2ks
ð1 À s
d
ÞR
2
À e
À2ks
ð1 À s
d
ÞK
À1
1
Z
2
K

À1
1
:
Proof. Consider a Lyapunov function candidate as
V ¼ V
1
þ V
2
þ V
3
þ V
4
þ V
5
; ð8Þ
where
V
1
¼ e
2kt
x
T
ðtÞPxðtÞþe
2kt
y
T
ðtÞQyðtÞ;
V
2
¼ 2

X
n
i¼1
d
i
e
2kt
Z
x
i
ðtÞ
0

f
i
ðsÞds þ 2
X
m
i¼1
e
i
e
2kt
Z
y
i
ðtÞ
0
f
i

ðsÞds;
V
3
¼
Z
t
tÀhðtÞ
e
2ks

f
T
ðxðsÞÞR
1

f ðxðsÞÞds þ
Z
t
tÀsðtÞ
e
2ks
f
T
ðyðsÞÞR
2
f ðyðsÞÞds;
V
4
¼
Z

t
tÀhðtÞ
e
2ks
x
T
ðsÞZ
1
xðsÞds þ
Z
t
tÀsðtÞ
e
2ks
y
T
ðsÞZ
2
yðsÞds;
V
5
¼
Z
t
tÀ

h
Z
t
s

e
2ku
_
x
T
ðuÞL
1
_
xðuÞdu ds þ
Z
t
tÀ

s
Z
t
s
e
2ku
_
y
T
ðuÞL
2
_
yðuÞdu ds:
Now, let us calculate the time derivative of V
i
along the trajectory of (5). First the derivative of V
1

is
_
V
1
¼ e
2kt
f2kx
T
ðtÞPxðtÞþ2x
T
ðtÞP
_
xðtÞþ2ky
T
ðtÞQyðtÞþ2y
T
ðtÞQ
_
yðtÞg
¼ e
2kt
f2kx
T
ðtÞPxðtÞþ2x
T
ðtÞP ðÀAxðtÞþWf ðyðt ÀsðtÞÞÞÞ þ2ky
T
ðtÞQyðtÞ
þ 2y
T

ðtÞQðÀByðtÞþV

f ðxðt À hðtÞÞÞÞg: ð9Þ
Second, we get the bound of
_
V
2
as
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_
V
2
¼ 2
X
n
i¼1
d
i
e
2kt
ð2k
Z
x
i
ðtÞ
0


f
i
ðsÞds þ

f
i
ðx
i
ðtÞÞ
_
x
i
ðtÞÞ þ2
X
m
i¼1
e
i
e
2kt
ð2k
Z
y
i
ðtÞ
0
f
i
ðsÞds þ f
i

ðy
i
ðtÞÞ
_
y
i
ðtÞÞ
¼
X
n
i¼1
4kd
i
e
2kt
Z
x
i
ðtÞ
0

f
i
ðsÞds þ 2e
kt

f
T
ðxðtÞÞD
_

xðtÞþ
X
m
i¼1
4ke
i
e
2kt
Z
y
i
ðtÞ
0
f
i
ðsÞds þ 2e
kt
f
T
ðyðtÞÞE
_
yðtÞ
6 e
2kt
f4k

f
T
ðxðtÞÞDxðtÞÀ2


f
T
ðxðtÞÞDAxðtÞþ2

f
T
ðxðtÞÞDWf ðyðt ÀsðtÞÞÞg þ e
2kt
f4kf
T
ðyðtÞÞEyðtÞ
À 2f
T
ðyðtÞÞEByðtÞþ2f
T
ðyðtÞÞEV

f ðxðt ÀhðtÞÞÞg ð10Þ
where Lemma 1 is utilized.
Third, the bound of
_
V
3
is as follows:
_
V
3
6 e
2kt


f
T
ðxðtÞÞR
1

f ðxðtÞÞ À e
2kðtÀ

hÞ
ð1 À h
d
Þ

f
T
ðxðt ÀhðtÞÞÞR
1

f ðxðt ÀhðtÞÞÞ þ e
2kt
f
T
ðyðtÞÞR
2
f ðyðtÞÞ
À e
2kðtÀsÞ
ð1 À s
d
Þf

T
ðyðt ÀsðtÞÞÞR
2
f ðyðt ÀsðtÞÞÞ: ð11Þ
Next, we obtain the followings:
_
V
4
6e
2kt
x
T
ðtÞZ
1
xðtÞÀe
2kðtÀ

hÞ
ð1 À h
d
Þx
T
ðt ÀhðtÞÞZ
1
xðt À hðtÞÞ þ e
2kt
y
T
ðtÞZ
2

yðtÞÀe
2kðtÀsÞ
ð1 À s
d
Þy
T
Âðt ÀsðtÞÞZ
2
yðt ÀsðtÞÞ: ð12Þ
Finally, we have
_
V
5
¼ e
2kt

h_x
T
ðtÞL
1
_xðtÞÀ
Z
t
tÀ

h
e
2ks
_x
T

ðsÞL
1
_xðsÞds þ e
2kt
s_y
T
ðtÞL
2
_yðtÞÀ
Z
t
tÀs
e
2ks
_y
T
ðsÞL
2
_yðsÞds
6 e
2kt

h
_
x
T
ðtÞL
1
_
xðtÞÀe

2kðtÀ

hÞ
Z
t
tÀ

h
_
x
T
ðsÞL
1
_
xðsÞds þ e
2kt

s
_
y
T
ðtÞL
2
_
yðtÞÀe
2kðtÀ

sÞ
Z
t

tÀs
_
y
T
ðsÞL
2
_
yðsÞds
6 e
2kt

h
_
x
T
ðtÞL
1
_
xðtÞÀe
À2k

h

h
À1
Z
t
tÀ

h

_
xðsÞds

Þ
T
L
1
Z
t
tÀ

h
_
xðsÞds

þ

s
_
y
T
ðtÞL
2
_
yðtÞ

Àe
À2k

s


s
À1
Z
t
tÀ

s
_
yðsÞds

T
L
2
Z
t
tÀ

s
_
yðsÞds

)
6 e
2kt

h
_
x
T

ðtÞL
1
_
xðtÞÀe
À2k

h

h
À1
Z
t
tÀhðtÞ
_
xðsÞds
!
T
L
1
Z
t
tÀhðtÞ
_
xðsÞds
!
þ

s
_
y

T
ðtÞL
2
_
yðtÞ
8
<
:
Àe
À2k

s

s
À1
Z
t
tÀsðtÞ
_
yðsÞds
!
T
L
2
Z
t
tÀsðtÞ
_
yðsÞds
!

9
=
;
; ð13Þ
where Lemma 2 is used in the second inequality.
Thus, it follows that:
_
V 6 e
2kt
(
2kx
T
ðtÞPxðtÞþ2x
T
ðtÞP ðÀAxðtÞþWf ðyðt ÀsðtÞÞÞÞþ2ky
T
ðtÞQyðtÞþ2y
T
ðtÞQðÀByðtÞþV

f ðxðt ÀhðtÞÞÞÞ
þ4k

f
T
ðxðtÞÞDxðtÞÀ2

f
T
ðxðtÞÞDAxðtÞþ2


f
T
ðxðtÞÞDWf ðyðt ÀsðtÞÞþ4kf
T
ðyðtÞÞEyðtÞÀ2f
T
ðyðtÞÞEByðtÞ
þ2f
T
ðyðtÞÞEV

f ðxðt ÀhðtÞÞþ

f
T
ðxðtÞÞR
1

f ðxðtÞÞÀe
À2k

h
ð1Àh
d
Þ

f
T
ðxðt ÀhðtÞÞÞR

1

f ðxðt ÀhðtÞÞÞ
þf
T
ðyðtÞÞR
2
f ðyðtÞÞÀe
À2k

s
ð1Às
d
Þf
T
ðyðt ÀsðtÞÞÞR
2
f ðyðt ÀsðtÞÞÞþx
T
ðtÞZ
1
xðtÞ
Àe
À2k

h
ð1Àh
d
Þx
T

ðt ÀhðtÞÞZ
1
xðt ÀhðtÞÞþy
T
ðtÞZ
2
yðtÞÀe
À2k

s
ð1Às
d
Þy
T
ðt ÀsðtÞÞZ
2
yðt ÀsðtÞÞþ

h
_
x
T
ðtÞL
1
_
xðtÞ
Àe
À2k

h


h
À1
Z
t
tÀhðtÞ
_
xðsÞds
!!
T
L
1
Z
t
tÀhðtÞ
_
xðsÞds
!!
þ

s
_
y
T
ðtÞL
2
_
yðtÞÀe
À2k


s

s
À1
Z
t
tÀsðtÞ
_
yðsÞds
!
T
L
2
Z
t
tÀsðtÞ
_
yðsÞds
!
9
=
;
:
ð14Þ
Here note that
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À 2


f
T
ðxðtÞÞDAxðtÞ 6 À2

f
T
ðxðtÞÞDAK
À1
2

f ðxðtÞÞ;
À 2f
T
ðyðtÞÞEByðtÞ 6 À2f
T
ðyðtÞÞEBK
À1
1
f ðyðtÞÞ;
À e
À2k

h
x
T
ðt ÀhðtÞÞZ
1
xðt À hðtÞÞ 6 Àe
À2k


h

f
T
ðxðt ÀhðtÞÞÞK
À1
2
Z
1
K
À1
2

f ðxðt ÀhðtÞÞÞ;
À e
À2k

s
y
T
ðt ÀsðtÞÞZ
2
yðt ÀsðtÞÞ 6 Àe
À2k

s
f
T
ðyðt ÀsðtÞÞÞK

À1
1
Z
2
K
À1
1
f ðyðt ÀsðtÞÞÞ;
ð15Þ
where the property H2 and Fact 2 are used to derive the inequalities.
For any appropriate dimensional matrices N
i
ði ¼ 1; 2; ; 10Þ, the following equations hold:
2
"
x
T
ðtÞN
1
þ

f
T
ðxðtÞÞN
2
þ

f
T
ðxðt À hðtÞÞÞN

3
þ
_
x
T
ðtÞN
4
:
þ
Z
t
tÀhðtÞ
_
xðsÞdsÞ
!
T
N
5
3
5
½
_
xðtÞþAxðtÞÀWf ðyðt ÀsðtÞÞÞ ¼ 0;
2½y
T
ðtÞN
6
þ f
T
ðyðtÞÞN

7
þ f
T
ðyðt ÀsðtÞÞÞN
8
þ
_
y
T
ðtÞN
9
þ
Z
t
tÀsðtÞ
_
yðsÞdsÞ
!
T
N
10
#
½
_
yðtÞþByðtÞÀV

f ðxðt ÀhðtÞÞÞ ¼ 0:
ð16Þ
Substituting Eq. (15) into Eq. (14) and utilizing the relationship (16) gives that
_

V 6 e
2kt
z
T
ðtÞPzðtÞ; ð17Þ
where
zðtÞ¼ x
T
ðtÞ

f
T
ðxðtÞÞ

f
T
ðxðt ÀhðtÞÞÞ _x
T
ðtÞ
Z
t
tÀhðtÞ
_xðsÞdsÞ
!
T
y
T
ðtÞ f
T
ðyðtÞÞ f

T
ðyðt ÀsðtÞÞÞ _y
T
ðtÞ
Z
t
tÀsðtÞ
_yðsÞdsÞ
!
T
2
4
3
5
T
:
Since the matrix P given in Theorem 1 is the negative definite matrix, we have
_
V 6 0, it follows that V 6 V ð0Þ. Then we
have the followings:
V ð0Þ¼x
T
ð0ÞPxð0Þþ2
X
n
i¼1
d
i
Z
x

i
ð0Þ
0

f
i
ðsÞds þ
Z
0
Àhð0Þ
e
2ks

f
T
ðxðsÞÞR
1

f ðxðsÞÞds þ
Z
0
Àhð0Þ
e
2ks
x
T
ðsÞZ
1
xðsÞds
þ y

T
ð0ÞQyð0Þþ2
X
m
i¼1
e
i
Z
y
i
ð0Þ
0
f
i
ðsÞds þ
Z
0
Àsð0Þ
e
2ks
f
T
ðyðsÞÞR
2
f ðyðsÞÞds þ
Z
0
Àsð0Þ
e
2ks

y
T
ðsÞZ
2
yðsÞds
þ
Z
0
À

h
Z
0
s
e
2ku
_
x
T
ðuÞL
1
_
xðuÞdu ds þ
Z
0
Às
Z
0
s
e

2ku
_
y
T
ðuÞL
2
_
yðuÞdu ds: ð18Þ
Also, we further get the bound of V ð0Þ as follows:
V ð0Þ 6 k
M
ðP Þk/k
2
þ 2d
M
k
1M
k/k
2
þðk
M
ðR
1
Þk
2
1M
þ k
M
ðZ
1

ÞÞ
Z
0
À

h
e
2ks
x
T
ðsÞxðsÞds þ k
M
ðQÞkwk
2
þ 2e
M
k
2M
kwk
2
þðk
M
ðR
2
Þk
2
2M
þ k
M
ðZ

2
ÞÞ
Z
0
À

s
e
2ks
y
T
ðsÞyðsÞd s þk
M
ðL
1
Þ
Z
0
À

h
Z
0
s
_
x
T
ðuÞ
_
xðuÞdu ds þ k

M
ðL
2
Þ
Z
0
À

s
Â
Z
0
s
_
y
T
ðuÞ
_
yðuÞdu ds; ð19Þ
where d
M
¼ maxðd
i
Þ, e
M
¼ maxðe
i
Þ, k
1M
¼ maxðk

1i
Þ, k
2M
¼ maxðk
2i
Þ, k/k¼sup
À

h6h60
kxðhÞk, and kwk¼sup
Às6h60
kyðhÞk.
It follows from Fact 2 that:
_
x
T
ðsÞ
_
xðsÞ 6 2x
T
ðsÞA
T
AxðsÞþ2f
T
ðyðs ÀsðsÞÞÞW
T
Wf ðyðs ÀsðsÞÞÞ
6 2k
M
ðA

T
AÞk/k
2
þ 2k
M
ðW
T
W Þk
M
ðK
2
1
Þkwk
2
_
y
T
ðsÞ
_
yðsÞ 6 2y
T
ðsÞB
T
ByðsÞþ2

f
T
ðxðs À hðsÞÞÞV
T
V


f ðxðs À hðsÞÞÞ
6 2k
M
ðB
T
BÞkwk
2
þ 2k
M
ðV
T
V Þk
M
ðK
2
2
Þk/k
2
:
ð20Þ
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From the relationship (20) and simple calculation, we further have
V ð0Þ 6 k
M
ðP Þk/k
2

þ 2d
M
k
1M
k/k
2
þðk
M
ðR
1
Þk
2
1M
þ k
M
ðZ
1
ÞÞk/k
2
1 À e
À2k

h
2k
þ k
M
ðQÞkwk
2
þ 2e
M

k
2M
kwk
2
þðk
M
ðR
2
Þk
2
2M
þ k
M
ðZ
2
ÞÞkwk
2
1 À e
À2k

s
2k
þ

h
2
k
M
ðL
1

Þð2k
M
ðA
T
AÞk/k
2
þ 2k
M
ðW
T
W Þk
M
ðK
2
1
Þkwk
2
Þ
þ

s
2
k
M
ðL
2
Þð2k
M
ðB
T

BÞkwk
2
þ 2k
M
ðV
T
V Þk
M
ðK
2
Þ
2
Þk/k
2
Þ
¼fk
M
ðPÞþ2d
M
k
1M
þðk
M
ðR
1
Þk
2
1M
þ k
M

ðZ
1
ÞÞ
1 À e
À2k

h
2k
þ 2

h
2
k
M
ðL
1
Þk
M
ðA
T
AÞ
þ 2

s
2
k
M
ðL
2
Þk

M
ðV
T
V Þk
M
ðK
2
Þ
2
Þgk/k
2
þ k
M
ðQÞþ2e
M
k
2M
þðk
M
ðR
2
Þk
2
2M
þ k
M
ðZ
2
ÞÞ
1 À e

À2k

s
2k

þ2

h
2
k
M
ðL
1
Þk
M
ðW
T
W Þk
M
ðK
2
1
Þþ2

s
2
k
M
ðL
2

Þk
M
ðB
T
BÞ

kwk
2
 c
1
k/k
2
þ c
2
kwk
2
: ð21Þ
Furthermore, we have
V P e
2kt
ðk
m
ðPÞkxðtÞk
2
þ k
m
ðQÞkyðtÞk
2
Þ:
Then we easily obtain

kxðtÞk þ kyðtÞk 6
ffiffiffi
2
p
ðkxðtÞk
2
þkyðtÞk
2
Þ
1=2
6 lðk/k
2
þkwk
2
Þ
1=2
e
Àkt
6 lðk

/kþk

wkÞe
Àkt
for all t P 0, where l P 1 is a constant and
k

/k¼ sup
Às
Ã

6h60
kxðhÞk; k

wk¼ sup
Às
Ã
6h60
kyðhÞk:
Thus by Definition 1, system (5) is exponentially stable and has the exponential convergence rate k. This completes the
proof. h
Remark 1. The criterion given in Theorem 1 is delay-dependent. It is well known that the delay-dependent criteria are
generally less conservative than delay-independent criteria when the delay is small.
Remark 2. The solutions of Theorem 1 can be obtained by solving the eigenvalue problem with respect to solution
variables, which is a convex optimization problem [30]. In this paper, we utilize Matlab’s LMI Toolbox [31] which
implements interior-point algorithm. This algorithm is significantly faster than classical convex optimization algo-
rithms [30].
Example 1. Consider the following BAM neural networks (5) with

f
i
ðxÞ¼
1
2
ðjx
i
þ 1jÀjx
i
À 1jÞ, f
j
ðyÞ¼

1
2
ðjy
j
þ 1jÀjy
j
À 1jÞ, s =1,h ¼ 0:5 and
A ¼ I; B ¼ 2I; W ¼
0:05 0:25 0:05
0:10:05 0:15
0:15 0:15 0:05
2
6
4
3
7
5
; V ¼
0:75 0:75 0:95
00:50:15
0:15 0:15 0:05
2
6
4
3
7
5
: ð22Þ
From the functions


f
i
ðxÞ and f
j
ðyÞ, we can easily obtain K
1
¼ K
2
¼ I. When the exponential convergence rate is taken
as k = 0.4, the criteria given in [27,28,26] cannot determine that system (22) is exponentially stable. However when our
criterion given in Theorem 1 is applied to the system (22), our maximum allowable convergence rate for guaranteeing
exponential stability of the system (22) is k = 0.57. Thus our result is less conservative than those of the existing works
[26–28]. When the time-varying delays are considered for the system (22) with hðtÞ 6 1 and sðtÞ 6 0:5, the maximum
allowable convergence rate is summarized in Table 1.
Table 1
Convergence rate k
h
d
ð¼ s
d
Þ 0.3 0.5 0.7 0.9
Maximum allowable convergence rate k 0.52 0.47 0.39 0.21
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4. Concluding remarks
A novel criterion for exponential stability of BAM neural networks with time-varying delays has been presented by
combining the Lyapunov functional method with LMI framework. The criterion is delay-dependent and expressed by
LMI. Throughout a numerical example, it is shown that our criterion is less conservative than those of existing results.

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Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003
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