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Stability of Bidirectional Associative Memory networks with impulses
Applied Mathematics and Computation 218 (2011) 1658–1667
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Stability of Bidirectional Associative Memory networks with impulses q Zhiguo Luo a, Jianli Li a,⇑, Jianhua Shen b a b
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, PR China Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, PR China
a r t i c l e
i n f o
a b s t r a c t By using Lyapunov functional and some analysis technique, sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a Bidirectional Associative Memory (BAM) neutral network with Lipschitzian activation functions without assuming their boundedness, monotonicity or differentiability and subjected to impulsive state displacements at fixed instants of time. The sufficient conditions are in terms of the parameters of the network only and are easy to verify. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Neural network Impulsive Lyapunov function Asymptotic stability
1. Introduction In recent years, a class of neural network related to Bidirectional Associative Memory (BAM) has been proposed. These models generalize the single-layer auto-associative circuit. Therefore, this class of network possesses good application prospects in the area of pattern recognition, signal and image process etc. [1,2]. The research on stability of BAM network has many nice works [3–7]. The dynamical characteristics of the network are assumed to be governed by the dynamics of the following system of ordinary differential equations
X dxi ðtÞ ¼ ai xi ðtÞ þ cij fj ðyj ðtÞÞ þ ci ; t > t 0 ; dt j¼1 p
m X dyj ðtÞ dji g i ðxi ðtÞÞ þ ej ; t > t 0 ; ¼ bj yj ðtÞ þ dt i¼1
ð1:1Þ
in which i = 1, 2, . . . , m, j = 1, 2, . . . , p; xi(t), yj(t) denote the potential (or voltage) of the cell i and j at time t respectively, ai, bj are positive constants, they denote the rate with which the cell i and j reset their potential to the resting state when isolated from the other cells and inputs; the connection weights cij, dji are real numbers, they denote the strengths of connectivity between the cells j and i at time t respectively; ci, ej denote the ith and jth component of an external input source introduced from outside the network to the cell i and j respectively, the functions fj, gi: R ? R represent the response of the jth and ith cell to its membrane potential and are known as activation functions; we denote x(t) = (x1(t), . . . , xm(t))T 2 Rm, y(t) = (y1(t), . . . , yp(t))T 2 Rp. Dynamical systems are often classified into two categories of either continuous-time or discrete-time systems. Recently there has been a somewhat a new category of dynamical systems, which is neither purely continuous-time nor purely discrete-time ones; these are called dynamical systems with impulses. It is known that the theory of impulsive differential equations provides a natural framework for mathematical modeling of many real word phenomena. Significant progress has been made in the theory of impulsive differential equation in recent years [8–16,18]. q This work is supported by the NNSF of China (Nos. 10871063 and 10871062) and supported by Hunan Provincial Natural Science Foundation of China (No. 10JJ6002). ⇑ Corresponding author. E-mail address: [email protected] (J. Li).
0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.06.045
Z. Luo et al. / Applied Mathematics and Computation 218 (2011) 1658–1667
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In this paper, inspired by Refs.[3,8,17,18], we consider the system (1.1) subjected to certain impulsive state displacements at fixed moments of time:
9 > > t > t0 ; t – tk ; > > > > j¼1 > > > m > P > dyj ðtÞ > > ¼ b y ðtÞ þ d g ðx ðtÞÞ þ e ; t > t ; t – t ; j ji i j 0 k > j i dt = i¼1 dxi ðtÞ dt
¼ ai xi ðtÞ þ
p P
xðt þ0 Þ ¼ x0 2 Rm ; xi ðt þk Þ yj ðtþk Þ
cij fj ðyj ðtÞÞ þ ci ;
yðtþ0 Þ ¼ y0 2 Rp ;
xi ðtk Þ
¼
Ik ðxi ðt k ÞÞ;
k ¼ 1; 2; . . . ;
yj ðtk Þ
¼ eI k ðyj ðtk ÞÞ;
k ¼ 1; 2; . . . ;
> > > > > > > > > > > > > > ;
t0 < t 1 < t 2 < < t k ! 1 as k ! 1;
ð1:2Þ
þ e where xi ðtþ k Þ ¼ xi ðt k Þ; yj ðt k Þ ¼ yj ðt k Þ, the functions I k ðÞ; I k ðÞ : R ! R are assumed to be continuous. The present paper is organized as follows. In Section 2, we obtain a set of sufficient conditions for the existence of a unique equilibrium of the system. We then study, in Section 3, the stability of this equilibrium solution. Finally we work out an example.
2. Existence of equilibria In this section we consider network with globally Lipschitz activation functions without requiring them to be bounded, monotonic or differentiable. This is a significant advance in the area of BAM networks. Here we establish a number of easily verifiable sufficient conditions for the existence of unique equilibrium states. An equilibrium solution of (1.2) is a constant vector x ¼ ðx1 ; x2 ; . . . ; xm ÞT 2 Rm and y ¼ ðy1 ; y2 ; . . . ; yp ÞT 2 Rp which satisfy the system
ai xi ¼
p X
cij fj ðyj Þ þ ci ;
i ¼ 1; . . . ; m;
j¼1
bj yj
¼
m X
ð2:1Þ dji g j ðxi Þ
þ ej ;
j ¼ 1; . . . ; p;
i¼1
when the impulsive jumps Ik() and eI k ðÞ are assumed to satisfy Ik ðxi Þ ¼ 0; eI k ðyj Þ ¼ 0; k ¼ 1; 2; . . .. In order to derive sufficient conditions for the existence and stability of equilibria on the coefficients and the activation functions in (1.2), we recall some elementary facts about the norms of vectors. If x = (x1, x2, . . . , xn)T 2 Rn, then we have a choice of vectors norms in Rn, for instance kxk1, kxk2, kxk1 are the commonly used norms where
kxk1 ¼
n X
jxi j;
kxk2 ¼
( n X
i¼1
)1=2
jxi j2
;
kxk1 ¼ max jxi j: 16i6n
i¼1
We are now ready to consider the existence of equilibrium states of Eq. (2.1). Lemma 2.1. Let ai(i = 1, . . . ,m), bj(j = 1, . . . ,p) be positive numbers. Suppose that functions fj(), gi(): R ? R, i = 1, . . . ,m, j = 1, . . . , p satisfy
jfj ðuj Þ fj ðv j Þj 6 Lj juj v j j;
j ¼ 1; . . . ; p;
jg i ðui Þ g i ðv i Þj 6 Li jui v i j;
ð2:2Þ
i ¼ 1; . . . ; m:
Suppose further that ai ; cij ; Lj ; bj ; dji ; Li are such that
ai > Li
p X j¼1
jdji j;
bj > Lj
m X
jcij j:
ð2:3Þ
i¼1
Then there exists a unique solution of the system (2.1). Proof. F: Rm+p ? Rm defined by
0
1 F 1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ B C .. C Fðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ B . @ A F m ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ
where
F i ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼
p X j¼1
cij fj
yj þ ci ; bj
i ¼ 1; 2; . . . ; m:
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Defined H:Rm+p ? Rp by
1 H1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ C B .. C Hðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ B . A @ Hp ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ 0
where
Hj ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼
m X
dji g i
i¼1
It follows that
N¼
F H
xi þ ej ; ai
j ¼ 1; 2; . . . ; p:
: Rmþp ! Rmþp :
We show that N is a global contraction on Rm+p endowed with the norm kk1. For u ¼ ðx1 ; . . . ; xm ; y1 ; . . . ; yp ÞT ; 1 ; . . . ; y p ÞT 2 Rmþp , we have y
kNðuÞ Nðv Þk1 ¼
v ¼ ðx1 ; . . . ; xm ;
X p p X m X m X j xi yj y xi fj gi cij fj dji g i þ b b a a j j i i i¼1 j¼1 j¼1 i¼1
p p X p p m X m m m X X X X jcij jLj jdji jLi Lj X Li X j j þ j j þ jyj y jxi xi j 6 jcij jjyj y jdji jjxi xi j bj ai bj i¼1 ai j¼1 i¼1 j¼1 i¼1 j¼1 i¼1 j¼1 ! p m X X j j ¼ b1 ku v k1 ; 6 b1 jxi xi j þ jyj y
6
i¼1
j¼1
where
( b1 ¼ max max 16j6p
) p m Lj X Li X jcij j; max jdji j : 16i6m ai bj i¼1 j¼1
By the hypothesis b1 < 1 and the contraction mapping principle, N has a unique fixed point which is a unique solution of (2.1). h In the next lemma we derive a similar result by using a different norm on Rm+p. Lemma 2.2. Suppose fj, gi satisfy (2.2) and ai ; bj ; cij ; dji ; Lj ; Li are such that
ai >
p X
jcij jLj ;
bj >
m X
i ¼ 1; . . . ; m;
jdij jLi ;
j ¼ 1; . . . ; p:
i¼1
j¼1
Then the system (2.1) has a unique solution. Proof. First we note that (2.4) implies p 1 X jcij jLj < 1; ai j¼1
m 1 X jdij jLi < 1; bj i¼1
and hence
max
16i6m
p 1 X jcij jLj ai j¼1
! < 1;
max 16j6p
m 1 X jdji jLi bj i¼1
! < 1:
We consider a map T: Rm+p ? Rm+p defined by
1 F 1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ C B .. C B . C B C B B F m ðx1 ; . . . ; xm ; y ; . . . ; y Þ C 1 p C B Tðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ B C B H1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ C C B C B C B ... A @ Hp ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ 0
ð2:4Þ
Z. Luo et al. / Applied Mathematics and Computation 218 (2011) 1658–1667
1661
where
" # p 1 X F i ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ cij fj ðyj Þ þ ci ; i ¼ 1; 2; . . . ; m; ai j¼1 " # m 1 X dji g i ðxi Þ þ ej ; j ¼ 1; . . . ; p: Hj ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ bj i¼1 It follows that for u ¼ ðx1 ; . . . ; xm ; y1 ; . . . ; yp ÞT ;
v ¼ ðx1 ; . . . ; xm ; y1 ; . . . ; yp ÞT 2 Rmþp
1 ; . . . ; y p Þ; max Hj ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ kTðuÞ Tðv Þk1 ¼ max max F i ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ F i ðx1 ; . . . ; xm ; y 16i6m
16j6p
) p 1 X 1 X m 1 ; . . . ; y p Þ ¼ max max j Þ; max cij ½fj ðyj Þ fj ðy dji ½g i ðxi Þ g i ðxi Þ Hj ðx1 ; . . . ; xm ; y 16i6m ai 16j6p bj i¼1 j¼1 ( ) p m 1 X 1 X j j; max 6 max max jcij jLj jyj y jdji jLi jxi xi j 6 b2 ku v k1 ; 16i6m ai 16j6p bj i¼1 j¼1 (
where
( b2 ¼ max
max
16i6m
p m 1 X 1 X jcij jLj ; max jdji jLi 16j6p bj ai j¼1 i¼1
) < 1:
Thus it follows that T(): Rm+p ? Rm+p is a contraction and hence the map T has a unique fixed point u⁄ 2 Rm+p satisfying T(u⁄) = u⁄ and this completes the proof. h Lemma 2.3. Suppose that the fj(), gi() satisfy (2.2) and ai ; bj ; cij ; dji ; Lj ; Li such that
" 2 #1=2 p m X X cij Lj < 1; ai i¼1 j¼1
" 2 #1=2 p X m X dji Li < 1: bj j¼1 i¼1
Then system (2.1) has a unique solution. Proof.
" # " # 2 2 p p m m X X X X cij dji j j2 þ Lj jyj y Li jxi xi j2 ai bj i¼1 i¼1 j¼1 j¼1 p p p 2 m m m X X X dji 2 X X X cij j j2 þ Lj jyj y Li jxi xi j2 6 b23 ku v k22 ; 6 ai bj i¼1 j¼1 i¼1 j¼1 j¼1 i¼1
kTðuÞ Tðv Þk22 6
where
8" 9 2 #1=2 "X 2 #1=2 = p p X m X m < X cij dji b3 ¼ max L ; Li < 1: : i¼1 j¼1 ai j ; bj j¼1 i¼1 So T: Rm+p ? Rm+p is a contraction and (2.1) has a unique solution. This completes the proof. h
3. Stability of equilibria of systems with impulses In this section we derive sufficient conditions for the asymptotic stability of the equilibrium of a system when the system is subjected to impulsive displacements of the state at fixed instants of time. In particular we consider the stability of the following impulsive system:
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9 > t > t 0 ; t – t k ; i ¼ 1; . . . ; m; > > > > j¼1 > > > > m > P dyj ðtÞ > = ¼ b y ðtÞ þ d g ðx ðtÞÞ þ e ; t > t ; t – t ; j ¼ 1; . . . ; p; 0 j ji i j k j i dt dxi ðtÞ dt
¼ ai xi ðtÞ þ
p P
cij fj ðyj ðtÞÞ þ ci ;
i¼1
xðt þ0 Þ ¼ x0 2 Rm ;
ð3:1Þ
> > > > > > > > > > > ;
yðtþ0 Þ ¼ y0 2 Rp ;
xi ðt þk Þ xi ðtk Þ ¼ Ik ðxi ðt k ÞÞ; k ¼ 1; 2; . . . ; yj ðt þk Þ yj ðtk Þ ¼ eI k ðyj ðtk ÞÞ; k ¼ 1; 2; . . . ;
where Ik ; eI k : R ! R; k ¼ 1; 2; . . . are continuous and non-decreasing. We let
ui ðtÞ xi ðtÞ xi ;
ui ðtk Þ þ Ik ðui ðt k Þ þ xi Þ ¼ Gk ðui ðtk ÞÞ; i ¼ 1; 2; . . . ; m; v j ðtk Þ þ eI k ðv j ðtk Þ þ yj Þ ¼ Ge k ðv j ðtk ÞÞ; j ¼ 1; 2; . . . ; p:
v j ðtÞ yj ðtÞ yj ; Then (3.1) translate to dui ðtÞ dt
¼ ai ui ðtÞ þ
dv j ðtÞ dt
¼ bj v j ðtÞ þ
p P
cij f j ðv j ðtÞÞ;
t > t0 ; t – tk ;
dji gi ðui ðtÞÞ;
t > t0 ; t – tk ;
9 > > > > > > > > > =
j¼1 m P
uðt þ0 Þ
¼ x0 x 2 R ;
v
¼ y0 y 2 Rp ;
ðt þ0 Þ
ð3:2Þ
> > > > ¼ i ¼ 1; 2 . . . ; m; k ¼ 1; 2 . . . ; > > > > > þ e v j ðtk Þ ¼ G k ðv j ðtk ÞÞ; j ¼ 1; . . . ; m; k ¼ 1; 2; . . . ; ;
i¼1 m
ui ðt þk Þ
Gk ðui ðt k ÞÞ;
where f j ðv j ðtÞÞ ¼ fj ðv j ðtÞ þ yj Þ fj ðyj Þ; gi ðui ðtÞÞ ¼ g i ðui ðtÞ þ xi Þ g i ðxi Þ. It follows that if fj(), gi() are globally Lipschizian, then i ðÞ with the same Lipschitz constant as that of the fj() and gi(), that is so are f j ðÞ and g
f ð0Þ ¼ 0 and jf ðv Þj 6 L jv j; j j j j j
j ¼ 1; 2; . . . ; p;
gi ð0Þ ¼ 0 and jgi ðui Þj 6 Li jui j;
i ¼ 1; 2; . . . ; m:
e k ðÞ : R ! R are assumed to be continuous, non-decreasing with Gk ð0Þ ¼ G e k ð0Þ ¼ 0 for k = 1, 2, . . . . It is Furthermore Gk ðÞ; G thus sufficient to consider the stability of the trivial solutions of the translated system (3.2) in order to consider the stability of (x⁄, y⁄)T of the original system (3.1). Theorem 3.1. Suppose that ai ; bj ; cij ; dji ; Lj ; Li satisfy
min ai > 0;
min bj > 0
16i6m
16j6p
and
"
# p Li X jdji j < 1; ai j¼1
b1 ¼ max
16i6m
" b2 ¼ max 16j6p
# m Lj X jcij j < 1: bj i¼1
ð3:3Þ
e k satisfy Furthermore suppose that Gk and G
e k ðxÞj þ j G e k ðyÞj 6 G e k ðjxj þ jyjÞ for x; y 2 R: jG
jGk ðxÞj þ jGk ðyÞj 6 Gk ðjxj þ jyjÞ;
If there exist positive numbers ck, k = 1, 2, . . . such that
(
min
"
min ai Li
16i6m
p X
#
jdji j ;
j¼1
"
min bj Lj
m X
16j6p
for any positive number r and if the series
#) jcij j
ðtk t k1 Þ ln
e k ðrÞ Gk ðrÞ þ G
i¼1
P1
p X
jdji j;
bj > Lj
m X
jcij j;
i ¼ 1; . . . ; m; j ¼ 1; . . . ; p;
i¼1
j¼1
and so
" b1 ¼ min ai Li 16i6m
Let b ¼
minfb1 ; b2 g,
p X j¼1
# jdji j > 0;
! P ck > 0
ð3:5Þ
c diverges to +1, then the trival solution of (3.2) is asymptotically stable.
k¼1 k
Proof. Since b1 < 1, b2 < 1, we have from (3.3) that
ai > Li
r
ð3:4Þ
" b2 ¼ min bj Lj 16j6p
m X
# jcij j > 0:
i¼1
then b > 0, we consider a Lyapunov function V(t) defined by
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VðtÞ ¼
m X
p X
jui ðtÞj þ
i¼1
jv j ðtÞj:
j¼1 dþ VðtÞ dt
For t – tk, we find the upper right derivative þ
d VðtÞ 6 dt 6
m X
j¼1
i¼1
"
m X
ai jui ðtÞj þ
i¼1
¼
ai jui ðtÞj þ Li
i¼1
6
p X
jcij jLj jv j ðtÞj þ
p X j¼1
"
m X
p X
ai Li
i¼1
j¼1
#
j¼1
"
m X
along the solutions of (3.2) given by
" # X p m X p ai jui ðtÞj þ m jcij jjf j ðv j ðtÞÞj þ bj jv j ðtÞj þ jdji jjg i ðui ðtÞÞj
j¼1
# jdji jjui ðtÞj þ
p X
i¼1
"
m X
bj jv j ðtÞj þ
jui ðtÞj þ
j¼1
p X
# jdji jLi jui ðtÞj
i¼1
"
bj jv j ðtÞj þ Lj
m X
# jcij jjv j ðtÞj
i¼1
j¼1
!#
jdji j
p X
"
bj L j
m X
!#
jcij j
jv j ðtÞj 6 b1
i¼1
j¼1
m X
jui ðtÞj b2
i¼1
p X
jv j ðtÞj 6 bVðtÞ:
j¼1
It follows that on each interval of the form (tk, tk1), k = 0, 1, 2, . . . , V(t) is non-increasing. Suppose now that t0 2 [tn1, tn) for e n ðdÞ < e=2; n ¼ 1; 2; . . ., this is possible since some n 2 {1, 2, . . . }. Given any e > 0, choose a d > 0 such that Gn(d) < e/2 and G e n ðÞ are continuous and Gn ð0Þ ¼ G e n ð0Þ ¼ 0; n ¼ 1; 2; . . .. We note that if V(t0) < d then Gn() and G
Vðt n Þ 6 Vðt0 Þ 6 Gn ðVðt 0 ÞÞ 6 Gn ðdÞ < e n ðVðt0 ÞÞ 6 G e n ðdÞ < Vðt n Þ 6 Vðt0 Þ 6 G
e 2
;
e
2
:
Also
Vðt þn Þ
¼
m X
jui ðtþn Þj
þ
i¼1
p X
jv
þ j ðt n Þj
¼
m X
jGn ðui ðt n ÞÞj
þ
p X
i¼1
j¼1
6 Gn
m X
jui ðt n Þj þ
p X
i¼1
!
en jv j ðtn Þj þ G
e n ðv j ðt ÞÞj 6 Gn jG n
j¼1 m X
jui ðt n Þj þ
i¼1
j¼1
p X
! jv j ðtn Þj
m X i¼1
! jui ðt n Þj
en þG
p X
! jv
j ðt n Þj
j¼1
e
¼ Gn Vðt n Þ þ G n Vðt n Þ
j¼1
e n ðVðt0 ÞÞ 6 Gn ðdÞ þ G e n ðdÞ < e: 6 Gn ðVðt 0 ÞÞ þ G If we show that the sequence fVðt þ k Þg is non-increasing for k P n, it will then follow that V(t) < e for t P tn. Suppose that k P n + 1 for each l = n + 1, n + 2, . . . , k 1, we have that
Vðt þl Þ 6 Vðt þlþ1 Þ 6 6 Vðt þn Þ < e: From þ
d VðtÞ 6 bVðtÞ; dt
t – tk ;
we have
Z
t k
tþ k1
dV 6 V
Z
tk
bdt; t k1
or
Z
Vðt Þ k
Vðt þ Þ k1
ds 6 bðtk tk1 Þ: s
We also have that
Z
Vðtþ Þ k
Vðt Þ k
ds 6 s
Z
Gk ðVðt ÞÞþe G k ðVðt ÞÞ k k
Vðt Þ k
ds : s
Thus
Z
Vðt Þ k
Vðt þ Þ k1
ds þ s
Z
Vðt þ Þ k
Þ Vðt k
ds ¼ s
and hence by hypothesis
Z
Vðt þ Þ k
Vðt þ Þ k1
ds 6 bðtk tk1 Þ þ s
Z
Gk ðVðt ÞÞþe G k ðVðt ÞÞ k k Þ Vðt k
ds ; s
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Z
! e k Vðt Þ Gk Vðt k Þ þ G ds k 6 bðtk tk1 Þ þ ln 6 ck < 0: s Vðt k Þ
Vðtþ Þ k
Vðt þ Þ k1
Hence we have
Vðtþk Þ < Vðtþk1 Þ;
k ¼ 1; 2; . . . ;
which implies that the sequence fVðt þ k Þg is non-increasing. We can now conclude that
VðtÞ 6 e;
t P t0 :
This completes the proof of the stability of the trivial solution of the systems (3.2) since we have from this that m X
jui ðt0 Þj þ
i¼1
p X
jv j ðt 0 Þj < d implies
m X
jui ðtÞj þ
i¼1
j¼1
p X
jv j ðtÞj < e;
t P t0 :
j¼1
þ Next we show that the sequence fVðtþ k Þg ! 0 as k ? 1. If this is not true, then there exists g > 0 such that Vðt k Þ P g for k P n, then we have
g 6 Vðtþk Þ 6 Vðtþk1 Þ and so we have
ck 6
Z
Vðt þ Þ k1
Vðt þ Þ k
ds Vðt þk1 Þ Vðt þk Þ 6 ; s g
or equivalently
Vðtþk Þ 6 Vðt þk1 Þ ck g: By induction, we have
Vðtþnþl Þ 6 Vðtþn Þ g
nþl X
ck ;
k¼nþ1 þ which implies that Vðtþ nþl Þ ! 1 as l ? 1, which is a contradiction with V(t) P 0. Hence Vðt k Þ ! 0 as k ? 1. The asymptotic stability of the trivial solution of (3.2) follows from V(t) ? 0 as t ? +1 from the definition of the Lyapunov function V(t) since we have already established stability of the trivial solution.
Theorem 3.2. Suppose that the parameters ai ; bj ; cij ; dji ; Li ; Lj ði ¼ 1; . . . ; m; j ¼ 1; . . . ; pÞ are such that p X m X jdji jLi ai j¼1 i¼1
!2 < 1;
2 p m X X jcij jLj < 1: bj i¼1 j¼1
e k ðxÞ ¼ ~ Suppose further that Gk ðxÞ ¼ hk x; G hk x; 1 < hk ; ~ hk <
ð3:6Þ p ffiffiffi 3 4 and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 !ffi v !3 u m u p 2 2 2 2 p m uX X uX X jc jd j L 1 4 ij j Lj ji i 5ðt k t k1 Þ lnðhk þ ~hk Þ > c ; t k 2t k 2 2 a2i bj i¼1 i¼1 j¼1 j¼1 for k = 1, 2, . . . , here k = min{min16i6mai, min16j6pbj}. If the series (3.2) is exponentially asymptotically stable. Proof. By (3.6), there exists a positive number a such that
" 2 #1=2 p m X X jcij jLj a > > 0; b 2k j i¼1 j¼1 2 !2 31=2 p X m X jdji jLi 5 a 4 1 > > 0: a 2k i j¼1 i¼1
1
Defined
VðtÞ ¼ then
" # p m X 1 at X 1 2 1 2 e ui ðtÞ þ v j ðtÞ ; 2 ai bj i¼1 j¼1
P1
c ¼ þ1, then the trivial solution of the impulsive system
k¼1 k
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" # p m X dVðtÞ 1 dui ðtÞ X 1 dv ðtÞ ¼ aVðtÞ þ eat þ ui ðtÞ v j ðtÞ j dt ai dt bj dt i¼1 j¼1 ( " # " #) p p m m X X X X 1 1 at 2 2 ¼ aVðtÞ þ e ui ðtÞ þ ui ðtÞ cij f j ðv j ðtÞÞ þ v j ðtÞ þ v j ðtÞ dji gi ðui ðtÞÞ ai bj i¼1 i¼1 j¼1 j¼1 ( " # " #) p p m m X X X X 1 1 at 2 2 ui ðtÞ þ jui ðtÞj jcij jLj jv j ðtÞj þ v j ðtÞ þ jv j ðtÞj jdji jLi jui ðtÞj 6 aVðtÞ þ e ai bj i¼1 i¼1 j¼1 j¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u m u m uX p p m uX X X X uX u p 2 jcij j at 2 at t at 2 at t t 2 ui ðtÞ þ e ui ðtÞ Lj jv j ðtÞj e v j ðtÞ þ e v j ðtÞ 6 aVðtÞ e ai i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u p m uX X jdji j t Li jui ðtÞj bj i¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u m u m uX p p p 2 m X X X X X X u u u p 2 jc j ij 2 at 2 at t 2 e 6 aVðtÞ eat u2i ðtÞ þ eat t u2i ðtÞ t L v ðtÞ v ðtÞ þ e v j ðtÞ j j j 2 ai i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u p m m X uX X jdji j2 2 t L u2i ðtÞ i 2 b i¼1 i¼1 j¼1 j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v ! v !ffi3 v u m u p uX p p p 2 2 m m X X X X X X X u u u m 2 jc j jd j ij ji 2 at 2 at 2 at 4t 2 5 t t þ 6 aVðtÞ e ui ðtÞ e v j ðtÞ þ e L L u ðtÞ v 2j ðtÞ j i i 2 a2i bj i¼1 i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 j¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v ! v !ffi3 u m u p p p 2 2 m m X X X X X X u u 1 jc j jd j ij ji 2 6 aVðtÞ eat u2i ðtÞ eat v 2j ðtÞ þ eat 4t L þt L2i 5 2 2 a2i j bj i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 ! p m X X u2i ðtÞ þ v 2j ðtÞ i¼1
j¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi3 u m u p p 2 2 m m X X X X X u u 1 jc j 1 jd j ij ji L2j t L2i 5 u2i ðtÞ 6 aVðtÞ eat 41 t 2 2 2 i¼1 j¼1 ai 2 j¼1 i¼1 bj i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p p 2 2 m X X X uX X 1u jc j 1 jd j ij ji 2 at 4 e 1 t Lj t L2i 5 v 2j ðtÞ 2 2 2 i¼1 j¼1 ai 2 j¼1 i¼1 bj j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p m m X X X X 1u jcij j2 2 1u jdji j2 2 5 X 1 2 at 4 t t 6 aVðtÞ ke 1 L L u ðtÞ 2 i¼1 j¼1 a2i j 2 j¼1 i¼1 b2j i ai i i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p p 2 2 m X X X X X u u 1 jc j 1 jd j 1 2 ij ji keat 41 t L2j t L2i 5 v j ðtÞ 2 2 2 i¼1 j¼1 ai 2 j¼1 i¼1 bj b j j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p 2 2 m uX X uX X 1 jc j 1 jd j ij ji 2 ¼ aVðtÞ k41 t L t L2 5VðtÞ ¼ bVðtÞ; 2 i¼1 j¼1 a2i j 2 j¼1 i¼1 b2j i 2
t – tk ;
where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ! v !ffi u m u p p 2 2 m X X uX X u jc j jd j a ij ji b ¼ k42 t L2 t L2i 5 > 0: 2 k a2i j bj j¼1 j¼1 i¼1 i¼1 2
The remaining details of proof are similar to those of the previous theorem and hence we omit them here.
h
Theorem 3.3. Suppose that the parameters ai ; bj ; cij ; dji ; Lj ; Li are such that
" b1 ¼ min ai 16i6m
p X j¼1
# Li jdji j > 0;
" b2 ¼ min bj Lj 16j6p
m X i¼1
# jcij j > 0:
ð3:7Þ
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e k satisfy Furthermore suppose that Gk and G
e k ðjxjÞeet 6 G e k ðjxjeet Þ Gk ðjxjÞeet 6 Gk ðjxjeet Þ; G for e > 0, x 2 R, and there exist positive numbers ck satisfying
bðt k t k1 Þ ln
e k ðrÞ Gk ðrÞ þ G
!
r
where b = min{b1, b2}, and
P1
P ck ;
k ¼ 1; 2; . . . ;
c ¼ þ1. Then the trivial solution of (3.2) is exponentially asymptotically stable.
k¼1 k
Proof. By (3.7), there exists an e > 0 such that
b e ¼ g > 0: We consider a Lyapunov function V(t) defined by
VðtÞ ¼ eet kuðtÞk1 þ kv ðtÞk1 ¼ eet max jui ðtÞj þ max jv j ðtÞj ¼ jui0 ðtÞj þ jv j0 ðtÞj eet ; 16i6m
16j6p
t P t0 ;
where i0 2 {1, 2, . . . , m}, j0 = {1, 2, . . . , p}. For t – tk, we find the upper right derivation d+V(t)/dt along the solutions (3.2) given by
" # þ p m X X d VðtÞ 6 eVðtÞ þ eet ai0 jui0 ðtÞj þ jci0 j jjf j ðv j ðtÞÞj bj0 jv j0 ðtÞj þ jdj0 i jjgi ðui ðtÞÞj dt j¼1 i¼1 " # p m X X et jci0 j jLj jv j0 ðtÞj bj0 jv j0 ðtÞj þ jdj0 i jLi jui0 ðtÞj 6 eVðtÞ þ e ai0 jui0 ðtÞj þ j¼1
" 6 eVðtÞ þ e
et
ai0 þ
m X
!
jdj0 i jLi jui0 ðtÞj þ bj0 þ
i¼1
i¼1 p X
!
#
jci0 j jLj jv j0 ðtÞj 6 eVðtÞ bVðtÞ ¼ ðb eÞVðtÞ
j¼1
¼ gVðtÞ: Again the remaining details of proof are similar to those of Theorem 3.1 and we omit them to avoid redundancy. h
4. An illustrative example Let m = p = 1, fj(x) = gi(x) 2jxj. Consider the BAM neural networks with impulses:
dx ¼ 3xðtÞ þ f ðyðtÞÞ þ 5:5; t – t k ; dt dy ¼ 3yðtÞ þ f ðxðtÞÞ þ 4:5; t – tk ; dt 1 xðtþk Þ xðtk Þ ¼ ðxðt k Þ 1:5Þ; 2 1 þ yðt k Þ yðtk Þ ¼ ðyðtk Þ 0:5Þ: 2
ð4:1Þ
It is easy to check the condition (3.7) satisfies and b1 = b2 = 1, so if min{tk tk1} P 2, k = 1, 2, . . . , then
2 ln and
P1
e k ðrÞ Gk ðrÞ þ G
r
¼ 2 ln 3 ¼ ck > 0;
c ¼ þ1, here Gk ðrÞ ¼ Gk ðrÞ ¼ 32 r. By Theorem 3.3, the equilibrium is exponentially asymptotically state.
k¼1 k
References [1] B. Kosko, Adaptive bi-directional associative memories, Appl. Opt. 26 (1987) 4947–4960. [2] B. Kosko, Bi-directional associative memories, IEEE Trans. Syst. Man Cybernet. 18 (1988) 49–60. [3] J. Liang, J. Cao, Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays, Chaos Solitons Fract. 22 (2004) 773–785. [4] K. Gopalsamy, X. He, Delay-independant stability in bi-directional associative memory networks, IEEE Trans. Neural Netw. 5 (1994) 998–1002. [5] J. Cao, L. Wang, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Trans. Neural Netw. 13 (2002) 457–463. [6] Y. Zhang, Y. Yang, Global stability analysis of bi-directional associative memory neural networks with time delay, Int. J. Circ. Theor. Appl. 29 (2001) 185–196. [7] J. Cao, Global asymptotic stability of delayed bidirectional associative memory neural networks, Appl. Math. Comput. 142 (2003) 333–339. [8] K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput. 154 (2004) 783–813.
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[9] J. Zhen, Z. Ma, M. Han, The existence of periodic solutions of the n-species Lotka–Volterra competition systems with impulsive, Chaos Solitons Fract. 22 (2004) 181–188. [10] W. Ding, M. Han, J. Mi, Periodic boundary value problem for the second order impulsive functional equations, Comput. Math. Appl. 50 (2005) 491–507. [11] D. Qian, X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl. 303 (2005) 288–303. [12] J. Nieto, R. Rodriguez-Lopez, Periodic boundary value problem for non-Lipschitzan impulsive functional differential equations, J. Math. Anal. Appl. 318 (2006) 593–610. [13] J. Belley, M. Virgilio, Periodic Duffing delay equations with state dependent impulses, J. Math. Anal. Appl. 306 (2005) 646–662. [14] J. Li, J. Shen, Periodic boundary value problems for delay differential equations with impulses, J. Comput. Appl. Math. 193 (2006) 563–573. [15] W. Li, H. Huo, Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, J. Comput. Appl. Math. 174 (2005) 227–238. [16] S. Saker, Oscillation and global attractivity of impulsive periodic delay respiratory dynamics model, Chinese Ann. Math. Ser. B 26 (2005) 511–522. [17] J. Cao, J. Wang, Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays, Neural Netw. 17 (2004) 379–390. [18] J. Shen, Z. Jing, Stability analysis for systems with impulses effects, Int. J. Theor. Phys. 45 (2006) 1703–1717.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability of Bidirectional Associative Memory networks with impulses q Zhiguo Luo a, Jianli Li a,⇑, Jianhua Shen b a b
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, PR China Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, PR China
a r t i c l e
i n f o
a b s t r a c t By using Lyapunov functional and some analysis technique, sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a Bidirectional Associative Memory (BAM) neutral network with Lipschitzian activation functions without assuming their boundedness, monotonicity or differentiability and subjected to impulsive state displacements at fixed instants of time. The sufficient conditions are in terms of the parameters of the network only and are easy to verify. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Neural network Impulsive Lyapunov function Asymptotic stability
1. Introduction In recent years, a class of neural network related to Bidirectional Associative Memory (BAM) has been proposed. These models generalize the single-layer auto-associative circuit. Therefore, this class of network possesses good application prospects in the area of pattern recognition, signal and image process etc. [1,2]. The research on stability of BAM network has many nice works [3–7]. The dynamical characteristics of the network are assumed to be governed by the dynamics of the following system of ordinary differential equations
X dxi ðtÞ ¼ ai xi ðtÞ þ cij fj ðyj ðtÞÞ þ ci ; t > t 0 ; dt j¼1 p
m X dyj ðtÞ dji g i ðxi ðtÞÞ þ ej ; t > t 0 ; ¼ bj yj ðtÞ þ dt i¼1
ð1:1Þ
in which i = 1, 2, . . . , m, j = 1, 2, . . . , p; xi(t), yj(t) denote the potential (or voltage) of the cell i and j at time t respectively, ai, bj are positive constants, they denote the rate with which the cell i and j reset their potential to the resting state when isolated from the other cells and inputs; the connection weights cij, dji are real numbers, they denote the strengths of connectivity between the cells j and i at time t respectively; ci, ej denote the ith and jth component of an external input source introduced from outside the network to the cell i and j respectively, the functions fj, gi: R ? R represent the response of the jth and ith cell to its membrane potential and are known as activation functions; we denote x(t) = (x1(t), . . . , xm(t))T 2 Rm, y(t) = (y1(t), . . . , yp(t))T 2 Rp. Dynamical systems are often classified into two categories of either continuous-time or discrete-time systems. Recently there has been a somewhat a new category of dynamical systems, which is neither purely continuous-time nor purely discrete-time ones; these are called dynamical systems with impulses. It is known that the theory of impulsive differential equations provides a natural framework for mathematical modeling of many real word phenomena. Significant progress has been made in the theory of impulsive differential equation in recent years [8–16,18]. q This work is supported by the NNSF of China (Nos. 10871063 and 10871062) and supported by Hunan Provincial Natural Science Foundation of China (No. 10JJ6002). ⇑ Corresponding author. E-mail address: [email protected] (J. Li).
0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.06.045
Z. Luo et al. / Applied Mathematics and Computation 218 (2011) 1658–1667
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In this paper, inspired by Refs.[3,8,17,18], we consider the system (1.1) subjected to certain impulsive state displacements at fixed moments of time:
9 > > t > t0 ; t – tk ; > > > > j¼1 > > > m > P > dyj ðtÞ > > ¼ b y ðtÞ þ d g ðx ðtÞÞ þ e ; t > t ; t – t ; j ji i j 0 k > j i dt = i¼1 dxi ðtÞ dt
¼ ai xi ðtÞ þ
p P
xðt þ0 Þ ¼ x0 2 Rm ; xi ðt þk Þ yj ðtþk Þ
cij fj ðyj ðtÞÞ þ ci ;
yðtþ0 Þ ¼ y0 2 Rp ;
xi ðtk Þ
¼
Ik ðxi ðt k ÞÞ;
k ¼ 1; 2; . . . ;
yj ðtk Þ
¼ eI k ðyj ðtk ÞÞ;
k ¼ 1; 2; . . . ;
> > > > > > > > > > > > > > ;
t0 < t 1 < t 2 < < t k ! 1 as k ! 1;
ð1:2Þ
þ e where xi ðtþ k Þ ¼ xi ðt k Þ; yj ðt k Þ ¼ yj ðt k Þ, the functions I k ðÞ; I k ðÞ : R ! R are assumed to be continuous. The present paper is organized as follows. In Section 2, we obtain a set of sufficient conditions for the existence of a unique equilibrium of the system. We then study, in Section 3, the stability of this equilibrium solution. Finally we work out an example.
2. Existence of equilibria In this section we consider network with globally Lipschitz activation functions without requiring them to be bounded, monotonic or differentiable. This is a significant advance in the area of BAM networks. Here we establish a number of easily verifiable sufficient conditions for the existence of unique equilibrium states. An equilibrium solution of (1.2) is a constant vector x ¼ ðx1 ; x2 ; . . . ; xm ÞT 2 Rm and y ¼ ðy1 ; y2 ; . . . ; yp ÞT 2 Rp which satisfy the system
ai xi ¼
p X
cij fj ðyj Þ þ ci ;
i ¼ 1; . . . ; m;
j¼1
bj yj
¼
m X
ð2:1Þ dji g j ðxi Þ
þ ej ;
j ¼ 1; . . . ; p;
i¼1
when the impulsive jumps Ik() and eI k ðÞ are assumed to satisfy Ik ðxi Þ ¼ 0; eI k ðyj Þ ¼ 0; k ¼ 1; 2; . . .. In order to derive sufficient conditions for the existence and stability of equilibria on the coefficients and the activation functions in (1.2), we recall some elementary facts about the norms of vectors. If x = (x1, x2, . . . , xn)T 2 Rn, then we have a choice of vectors norms in Rn, for instance kxk1, kxk2, kxk1 are the commonly used norms where
kxk1 ¼
n X
jxi j;
kxk2 ¼
( n X
i¼1
)1=2
jxi j2
;
kxk1 ¼ max jxi j: 16i6n
i¼1
We are now ready to consider the existence of equilibrium states of Eq. (2.1). Lemma 2.1. Let ai(i = 1, . . . ,m), bj(j = 1, . . . ,p) be positive numbers. Suppose that functions fj(), gi(): R ? R, i = 1, . . . ,m, j = 1, . . . , p satisfy
jfj ðuj Þ fj ðv j Þj 6 Lj juj v j j;
j ¼ 1; . . . ; p;
jg i ðui Þ g i ðv i Þj 6 Li jui v i j;
ð2:2Þ
i ¼ 1; . . . ; m:
Suppose further that ai ; cij ; Lj ; bj ; dji ; Li are such that
ai > Li
p X j¼1
jdji j;
bj > Lj
m X
jcij j:
ð2:3Þ
i¼1
Then there exists a unique solution of the system (2.1). Proof. F: Rm+p ? Rm defined by
0
1 F 1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ B C .. C Fðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ B . @ A F m ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ
where
F i ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼
p X j¼1
cij fj
yj þ ci ; bj
i ¼ 1; 2; . . . ; m:
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Defined H:Rm+p ? Rp by
1 H1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ C B .. C Hðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ B . A @ Hp ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ 0
where
Hj ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼
m X
dji g i
i¼1
It follows that
N¼
F H
xi þ ej ; ai
j ¼ 1; 2; . . . ; p:
: Rmþp ! Rmþp :
We show that N is a global contraction on Rm+p endowed with the norm kk1. For u ¼ ðx1 ; . . . ; xm ; y1 ; . . . ; yp ÞT ; 1 ; . . . ; y p ÞT 2 Rmþp , we have y
kNðuÞ Nðv Þk1 ¼
v ¼ ðx1 ; . . . ; xm ;
X p p X m X m X j xi yj y xi fj gi cij fj dji g i þ b b a a j j i i i¼1 j¼1 j¼1 i¼1
p p X p p m X m m m X X X X jcij jLj jdji jLi Lj X Li X j j þ j j þ jyj y jxi xi j 6 jcij jjyj y jdji jjxi xi j bj ai bj i¼1 ai j¼1 i¼1 j¼1 i¼1 j¼1 i¼1 j¼1 ! p m X X j j ¼ b1 ku v k1 ; 6 b1 jxi xi j þ jyj y
6
i¼1
j¼1
where
( b1 ¼ max max 16j6p
) p m Lj X Li X jcij j; max jdji j : 16i6m ai bj i¼1 j¼1
By the hypothesis b1 < 1 and the contraction mapping principle, N has a unique fixed point which is a unique solution of (2.1). h In the next lemma we derive a similar result by using a different norm on Rm+p. Lemma 2.2. Suppose fj, gi satisfy (2.2) and ai ; bj ; cij ; dji ; Lj ; Li are such that
ai >
p X
jcij jLj ;
bj >
m X
i ¼ 1; . . . ; m;
jdij jLi ;
j ¼ 1; . . . ; p:
i¼1
j¼1
Then the system (2.1) has a unique solution. Proof. First we note that (2.4) implies p 1 X jcij jLj < 1; ai j¼1
m 1 X jdij jLi < 1; bj i¼1
and hence
max
16i6m
p 1 X jcij jLj ai j¼1
! < 1;
max 16j6p
m 1 X jdji jLi bj i¼1
! < 1:
We consider a map T: Rm+p ? Rm+p defined by
1 F 1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ C B .. C B . C B C B B F m ðx1 ; . . . ; xm ; y ; . . . ; y Þ C 1 p C B Tðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ B C B H1 ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ C C B C B C B ... A @ Hp ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ 0
ð2:4Þ
Z. Luo et al. / Applied Mathematics and Computation 218 (2011) 1658–1667
1661
where
" # p 1 X F i ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ cij fj ðyj Þ þ ci ; i ¼ 1; 2; . . . ; m; ai j¼1 " # m 1 X dji g i ðxi Þ þ ej ; j ¼ 1; . . . ; p: Hj ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ ¼ bj i¼1 It follows that for u ¼ ðx1 ; . . . ; xm ; y1 ; . . . ; yp ÞT ;
v ¼ ðx1 ; . . . ; xm ; y1 ; . . . ; yp ÞT 2 Rmþp
1 ; . . . ; y p Þ; max Hj ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ kTðuÞ Tðv Þk1 ¼ max max F i ðx1 ; . . . ; xm ; y1 ; . . . ; yp Þ F i ðx1 ; . . . ; xm ; y 16i6m
16j6p
) p 1 X 1 X m 1 ; . . . ; y p Þ ¼ max max j Þ; max cij ½fj ðyj Þ fj ðy dji ½g i ðxi Þ g i ðxi Þ Hj ðx1 ; . . . ; xm ; y 16i6m ai 16j6p bj i¼1 j¼1 ( ) p m 1 X 1 X j j; max 6 max max jcij jLj jyj y jdji jLi jxi xi j 6 b2 ku v k1 ; 16i6m ai 16j6p bj i¼1 j¼1 (
where
( b2 ¼ max
max
16i6m
p m 1 X 1 X jcij jLj ; max jdji jLi 16j6p bj ai j¼1 i¼1
) < 1:
Thus it follows that T(): Rm+p ? Rm+p is a contraction and hence the map T has a unique fixed point u⁄ 2 Rm+p satisfying T(u⁄) = u⁄ and this completes the proof. h Lemma 2.3. Suppose that the fj(), gi() satisfy (2.2) and ai ; bj ; cij ; dji ; Lj ; Li such that
" 2 #1=2 p m X X cij Lj < 1; ai i¼1 j¼1
" 2 #1=2 p X m X dji Li < 1: bj j¼1 i¼1
Then system (2.1) has a unique solution. Proof.
" # " # 2 2 p p m m X X X X cij dji j j2 þ Lj jyj y Li jxi xi j2 ai bj i¼1 i¼1 j¼1 j¼1 p p p 2 m m m X X X dji 2 X X X cij j j2 þ Lj jyj y Li jxi xi j2 6 b23 ku v k22 ; 6 ai bj i¼1 j¼1 i¼1 j¼1 j¼1 i¼1
kTðuÞ Tðv Þk22 6
where
8" 9 2 #1=2 "X 2 #1=2 = p p X m X m < X cij dji b3 ¼ max L ; Li < 1: : i¼1 j¼1 ai j ; bj j¼1 i¼1 So T: Rm+p ? Rm+p is a contraction and (2.1) has a unique solution. This completes the proof. h
3. Stability of equilibria of systems with impulses In this section we derive sufficient conditions for the asymptotic stability of the equilibrium of a system when the system is subjected to impulsive displacements of the state at fixed instants of time. In particular we consider the stability of the following impulsive system:
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9 > t > t 0 ; t – t k ; i ¼ 1; . . . ; m; > > > > j¼1 > > > > m > P dyj ðtÞ > = ¼ b y ðtÞ þ d g ðx ðtÞÞ þ e ; t > t ; t – t ; j ¼ 1; . . . ; p; 0 j ji i j k j i dt dxi ðtÞ dt
¼ ai xi ðtÞ þ
p P
cij fj ðyj ðtÞÞ þ ci ;
i¼1
xðt þ0 Þ ¼ x0 2 Rm ;
ð3:1Þ
> > > > > > > > > > > ;
yðtþ0 Þ ¼ y0 2 Rp ;
xi ðt þk Þ xi ðtk Þ ¼ Ik ðxi ðt k ÞÞ; k ¼ 1; 2; . . . ; yj ðt þk Þ yj ðtk Þ ¼ eI k ðyj ðtk ÞÞ; k ¼ 1; 2; . . . ;
where Ik ; eI k : R ! R; k ¼ 1; 2; . . . are continuous and non-decreasing. We let
ui ðtÞ xi ðtÞ xi ;
ui ðtk Þ þ Ik ðui ðt k Þ þ xi Þ ¼ Gk ðui ðtk ÞÞ; i ¼ 1; 2; . . . ; m; v j ðtk Þ þ eI k ðv j ðtk Þ þ yj Þ ¼ Ge k ðv j ðtk ÞÞ; j ¼ 1; 2; . . . ; p:
v j ðtÞ yj ðtÞ yj ; Then (3.1) translate to dui ðtÞ dt
¼ ai ui ðtÞ þ
dv j ðtÞ dt
¼ bj v j ðtÞ þ
p P
cij f j ðv j ðtÞÞ;
t > t0 ; t – tk ;
dji gi ðui ðtÞÞ;
t > t0 ; t – tk ;
9 > > > > > > > > > =
j¼1 m P
uðt þ0 Þ
¼ x0 x 2 R ;
v
¼ y0 y 2 Rp ;
ðt þ0 Þ
ð3:2Þ
> > > > ¼ i ¼ 1; 2 . . . ; m; k ¼ 1; 2 . . . ; > > > > > þ e v j ðtk Þ ¼ G k ðv j ðtk ÞÞ; j ¼ 1; . . . ; m; k ¼ 1; 2; . . . ; ;
i¼1 m
ui ðt þk Þ
Gk ðui ðt k ÞÞ;
where f j ðv j ðtÞÞ ¼ fj ðv j ðtÞ þ yj Þ fj ðyj Þ; gi ðui ðtÞÞ ¼ g i ðui ðtÞ þ xi Þ g i ðxi Þ. It follows that if fj(), gi() are globally Lipschizian, then i ðÞ with the same Lipschitz constant as that of the fj() and gi(), that is so are f j ðÞ and g
f ð0Þ ¼ 0 and jf ðv Þj 6 L jv j; j j j j j
j ¼ 1; 2; . . . ; p;
gi ð0Þ ¼ 0 and jgi ðui Þj 6 Li jui j;
i ¼ 1; 2; . . . ; m:
e k ðÞ : R ! R are assumed to be continuous, non-decreasing with Gk ð0Þ ¼ G e k ð0Þ ¼ 0 for k = 1, 2, . . . . It is Furthermore Gk ðÞ; G thus sufficient to consider the stability of the trivial solutions of the translated system (3.2) in order to consider the stability of (x⁄, y⁄)T of the original system (3.1). Theorem 3.1. Suppose that ai ; bj ; cij ; dji ; Lj ; Li satisfy
min ai > 0;
min bj > 0
16i6m
16j6p
and
"
# p Li X jdji j < 1; ai j¼1
b1 ¼ max
16i6m
" b2 ¼ max 16j6p
# m Lj X jcij j < 1: bj i¼1
ð3:3Þ
e k satisfy Furthermore suppose that Gk and G
e k ðxÞj þ j G e k ðyÞj 6 G e k ðjxj þ jyjÞ for x; y 2 R: jG
jGk ðxÞj þ jGk ðyÞj 6 Gk ðjxj þ jyjÞ;
If there exist positive numbers ck, k = 1, 2, . . . such that
(
min
"
min ai Li
16i6m
p X
#
jdji j ;
j¼1
"
min bj Lj
m X
16j6p
for any positive number r and if the series
#) jcij j
ðtk t k1 Þ ln
e k ðrÞ Gk ðrÞ þ G
i¼1
P1
p X
jdji j;
bj > Lj
m X
jcij j;
i ¼ 1; . . . ; m; j ¼ 1; . . . ; p;
i¼1
j¼1
and so
" b1 ¼ min ai Li 16i6m
Let b ¼
minfb1 ; b2 g,
p X j¼1
# jdji j > 0;
! P ck > 0
ð3:5Þ
c diverges to +1, then the trival solution of (3.2) is asymptotically stable.
k¼1 k
Proof. Since b1 < 1, b2 < 1, we have from (3.3) that
ai > Li
r
ð3:4Þ
" b2 ¼ min bj Lj 16j6p
m X
# jcij j > 0:
i¼1
then b > 0, we consider a Lyapunov function V(t) defined by
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VðtÞ ¼
m X
p X
jui ðtÞj þ
i¼1
jv j ðtÞj:
j¼1 dþ VðtÞ dt
For t – tk, we find the upper right derivative þ
d VðtÞ 6 dt 6
m X
j¼1
i¼1
"
m X
ai jui ðtÞj þ
i¼1
¼
ai jui ðtÞj þ Li
i¼1
6
p X
jcij jLj jv j ðtÞj þ
p X j¼1
"
m X
p X
ai Li
i¼1
j¼1
#
j¼1
"
m X
along the solutions of (3.2) given by
" # X p m X p ai jui ðtÞj þ m jcij jjf j ðv j ðtÞÞj þ bj jv j ðtÞj þ jdji jjg i ðui ðtÞÞj
j¼1
# jdji jjui ðtÞj þ
p X
i¼1
"
m X
bj jv j ðtÞj þ
jui ðtÞj þ
j¼1
p X
# jdji jLi jui ðtÞj
i¼1
"
bj jv j ðtÞj þ Lj
m X
# jcij jjv j ðtÞj
i¼1
j¼1
!#
jdji j
p X
"
bj L j
m X
!#
jcij j
jv j ðtÞj 6 b1
i¼1
j¼1
m X
jui ðtÞj b2
i¼1
p X
jv j ðtÞj 6 bVðtÞ:
j¼1
It follows that on each interval of the form (tk, tk1), k = 0, 1, 2, . . . , V(t) is non-increasing. Suppose now that t0 2 [tn1, tn) for e n ðdÞ < e=2; n ¼ 1; 2; . . ., this is possible since some n 2 {1, 2, . . . }. Given any e > 0, choose a d > 0 such that Gn(d) < e/2 and G e n ðÞ are continuous and Gn ð0Þ ¼ G e n ð0Þ ¼ 0; n ¼ 1; 2; . . .. We note that if V(t0) < d then Gn() and G
Vðt n Þ 6 Vðt0 Þ 6 Gn ðVðt 0 ÞÞ 6 Gn ðdÞ < e n ðVðt0 ÞÞ 6 G e n ðdÞ < Vðt n Þ 6 Vðt0 Þ 6 G
e 2
;
e
2
:
Also
Vðt þn Þ
¼
m X
jui ðtþn Þj
þ
i¼1
p X
jv
þ j ðt n Þj
¼
m X
jGn ðui ðt n ÞÞj
þ
p X
i¼1
j¼1
6 Gn
m X
jui ðt n Þj þ
p X
i¼1
!
en jv j ðtn Þj þ G
e n ðv j ðt ÞÞj 6 Gn jG n
j¼1 m X
jui ðt n Þj þ
i¼1
j¼1
p X
! jv j ðtn Þj
m X i¼1
! jui ðt n Þj
en þG
p X
! jv
j ðt n Þj
j¼1
e
¼ Gn Vðt n Þ þ G n Vðt n Þ
j¼1
e n ðVðt0 ÞÞ 6 Gn ðdÞ þ G e n ðdÞ < e: 6 Gn ðVðt 0 ÞÞ þ G If we show that the sequence fVðt þ k Þg is non-increasing for k P n, it will then follow that V(t) < e for t P tn. Suppose that k P n + 1 for each l = n + 1, n + 2, . . . , k 1, we have that
Vðt þl Þ 6 Vðt þlþ1 Þ 6 6 Vðt þn Þ < e: From þ
d VðtÞ 6 bVðtÞ; dt
t – tk ;
we have
Z
t k
tþ k1
dV 6 V
Z
tk
bdt; t k1
or
Z
Vðt Þ k
Vðt þ Þ k1
ds 6 bðtk tk1 Þ: s
We also have that
Z
Vðtþ Þ k
Vðt Þ k
ds 6 s
Z
Gk ðVðt ÞÞþe G k ðVðt ÞÞ k k
Vðt Þ k
ds : s
Thus
Z
Vðt Þ k
Vðt þ Þ k1
ds þ s
Z
Vðt þ Þ k
Þ Vðt k
ds ¼ s
and hence by hypothesis
Z
Vðt þ Þ k
Vðt þ Þ k1
ds 6 bðtk tk1 Þ þ s
Z
Gk ðVðt ÞÞþe G k ðVðt ÞÞ k k Þ Vðt k
ds ; s
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Z
! e k Vðt Þ Gk Vðt k Þ þ G ds k 6 bðtk tk1 Þ þ ln 6 ck < 0: s Vðt k Þ
Vðtþ Þ k
Vðt þ Þ k1
Hence we have
Vðtþk Þ < Vðtþk1 Þ;
k ¼ 1; 2; . . . ;
which implies that the sequence fVðt þ k Þg is non-increasing. We can now conclude that
VðtÞ 6 e;
t P t0 :
This completes the proof of the stability of the trivial solution of the systems (3.2) since we have from this that m X
jui ðt0 Þj þ
i¼1
p X
jv j ðt 0 Þj < d implies
m X
jui ðtÞj þ
i¼1
j¼1
p X
jv j ðtÞj < e;
t P t0 :
j¼1
þ Next we show that the sequence fVðtþ k Þg ! 0 as k ? 1. If this is not true, then there exists g > 0 such that Vðt k Þ P g for k P n, then we have
g 6 Vðtþk Þ 6 Vðtþk1 Þ and so we have
ck 6
Z
Vðt þ Þ k1
Vðt þ Þ k
ds Vðt þk1 Þ Vðt þk Þ 6 ; s g
or equivalently
Vðtþk Þ 6 Vðt þk1 Þ ck g: By induction, we have
Vðtþnþl Þ 6 Vðtþn Þ g
nþl X
ck ;
k¼nþ1 þ which implies that Vðtþ nþl Þ ! 1 as l ? 1, which is a contradiction with V(t) P 0. Hence Vðt k Þ ! 0 as k ? 1. The asymptotic stability of the trivial solution of (3.2) follows from V(t) ? 0 as t ? +1 from the definition of the Lyapunov function V(t) since we have already established stability of the trivial solution.
Theorem 3.2. Suppose that the parameters ai ; bj ; cij ; dji ; Li ; Lj ði ¼ 1; . . . ; m; j ¼ 1; . . . ; pÞ are such that p X m X jdji jLi ai j¼1 i¼1
!2 < 1;
2 p m X X jcij jLj < 1: bj i¼1 j¼1
e k ðxÞ ¼ ~ Suppose further that Gk ðxÞ ¼ hk x; G hk x; 1 < hk ; ~ hk <
ð3:6Þ p ffiffiffi 3 4 and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 !ffi v !3 u m u p 2 2 2 2 p m uX X uX X jc jd j L 1 4 ij j Lj ji i 5ðt k t k1 Þ lnðhk þ ~hk Þ > c ; t k 2t k 2 2 a2i bj i¼1 i¼1 j¼1 j¼1 for k = 1, 2, . . . , here k = min{min16i6mai, min16j6pbj}. If the series (3.2) is exponentially asymptotically stable. Proof. By (3.6), there exists a positive number a such that
" 2 #1=2 p m X X jcij jLj a > > 0; b 2k j i¼1 j¼1 2 !2 31=2 p X m X jdji jLi 5 a 4 1 > > 0: a 2k i j¼1 i¼1
1
Defined
VðtÞ ¼ then
" # p m X 1 at X 1 2 1 2 e ui ðtÞ þ v j ðtÞ ; 2 ai bj i¼1 j¼1
P1
c ¼ þ1, then the trivial solution of the impulsive system
k¼1 k
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" # p m X dVðtÞ 1 dui ðtÞ X 1 dv ðtÞ ¼ aVðtÞ þ eat þ ui ðtÞ v j ðtÞ j dt ai dt bj dt i¼1 j¼1 ( " # " #) p p m m X X X X 1 1 at 2 2 ¼ aVðtÞ þ e ui ðtÞ þ ui ðtÞ cij f j ðv j ðtÞÞ þ v j ðtÞ þ v j ðtÞ dji gi ðui ðtÞÞ ai bj i¼1 i¼1 j¼1 j¼1 ( " # " #) p p m m X X X X 1 1 at 2 2 ui ðtÞ þ jui ðtÞj jcij jLj jv j ðtÞj þ v j ðtÞ þ jv j ðtÞj jdji jLi jui ðtÞj 6 aVðtÞ þ e ai bj i¼1 i¼1 j¼1 j¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u m u m uX p p m uX X X X uX u p 2 jcij j at 2 at t at 2 at t t 2 ui ðtÞ þ e ui ðtÞ Lj jv j ðtÞj e v j ðtÞ þ e v j ðtÞ 6 aVðtÞ e ai i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u p m uX X jdji j t Li jui ðtÞj bj i¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u m u m uX p p p 2 m X X X X X X u u u p 2 jc j ij 2 at 2 at t 2 e 6 aVðtÞ eat u2i ðtÞ þ eat t u2i ðtÞ t L v ðtÞ v ðtÞ þ e v j ðtÞ j j j 2 ai i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u p m m X uX X jdji j2 2 t L u2i ðtÞ i 2 b i¼1 i¼1 j¼1 j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v ! v !ffi3 v u m u p uX p p p 2 2 m m X X X X X X X u u u m 2 jc j jd j ij ji 2 at 2 at 2 at 4t 2 5 t t þ 6 aVðtÞ e ui ðtÞ e v j ðtÞ þ e L L u ðtÞ v 2j ðtÞ j i i 2 a2i bj i¼1 i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 j¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v ! v !ffi3 u m u p p p 2 2 m m X X X X X X u u 1 jc j jd j ij ji 2 6 aVðtÞ eat u2i ðtÞ eat v 2j ðtÞ þ eat 4t L þt L2i 5 2 2 a2i j bj i¼1 i¼1 i¼1 j¼1 j¼1 j¼1 ! p m X X u2i ðtÞ þ v 2j ðtÞ i¼1
j¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi3 u m u p p 2 2 m m X X X X X u u 1 jc j 1 jd j ij ji L2j t L2i 5 u2i ðtÞ 6 aVðtÞ eat 41 t 2 2 2 i¼1 j¼1 ai 2 j¼1 i¼1 bj i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p p 2 2 m X X X uX X 1u jc j 1 jd j ij ji 2 at 4 e 1 t Lj t L2i 5 v 2j ðtÞ 2 2 2 i¼1 j¼1 ai 2 j¼1 i¼1 bj j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p m m X X X X 1u jcij j2 2 1u jdji j2 2 5 X 1 2 at 4 t t 6 aVðtÞ ke 1 L L u ðtÞ 2 i¼1 j¼1 a2i j 2 j¼1 i¼1 b2j i ai i i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p p 2 2 m X X X X X u u 1 jc j 1 jd j 1 2 ij ji keat 41 t L2j t L2i 5 v j ðtÞ 2 2 2 i¼1 j¼1 ai 2 j¼1 i¼1 bj b j j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! !ffi3 u m u p p 2 2 m uX X uX X 1 jc j 1 jd j ij ji 2 ¼ aVðtÞ k41 t L t L2 5VðtÞ ¼ bVðtÞ; 2 i¼1 j¼1 a2i j 2 j¼1 i¼1 b2j i 2
t – tk ;
where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ! v !ffi u m u p p 2 2 m X X uX X u jc j jd j a ij ji b ¼ k42 t L2 t L2i 5 > 0: 2 k a2i j bj j¼1 j¼1 i¼1 i¼1 2
The remaining details of proof are similar to those of the previous theorem and hence we omit them here.
h
Theorem 3.3. Suppose that the parameters ai ; bj ; cij ; dji ; Lj ; Li are such that
" b1 ¼ min ai 16i6m
p X j¼1
# Li jdji j > 0;
" b2 ¼ min bj Lj 16j6p
m X i¼1
# jcij j > 0:
ð3:7Þ
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e k satisfy Furthermore suppose that Gk and G
e k ðjxjÞeet 6 G e k ðjxjeet Þ Gk ðjxjÞeet 6 Gk ðjxjeet Þ; G for e > 0, x 2 R, and there exist positive numbers ck satisfying
bðt k t k1 Þ ln
e k ðrÞ Gk ðrÞ þ G
!
r
where b = min{b1, b2}, and
P1
P ck ;
k ¼ 1; 2; . . . ;
c ¼ þ1. Then the trivial solution of (3.2) is exponentially asymptotically stable.
k¼1 k
Proof. By (3.7), there exists an e > 0 such that
b e ¼ g > 0: We consider a Lyapunov function V(t) defined by
VðtÞ ¼ eet kuðtÞk1 þ kv ðtÞk1 ¼ eet max jui ðtÞj þ max jv j ðtÞj ¼ jui0 ðtÞj þ jv j0 ðtÞj eet ; 16i6m
16j6p
t P t0 ;
where i0 2 {1, 2, . . . , m}, j0 = {1, 2, . . . , p}. For t – tk, we find the upper right derivation d+V(t)/dt along the solutions (3.2) given by
" # þ p m X X d VðtÞ 6 eVðtÞ þ eet ai0 jui0 ðtÞj þ jci0 j jjf j ðv j ðtÞÞj bj0 jv j0 ðtÞj þ jdj0 i jjgi ðui ðtÞÞj dt j¼1 i¼1 " # p m X X et jci0 j jLj jv j0 ðtÞj bj0 jv j0 ðtÞj þ jdj0 i jLi jui0 ðtÞj 6 eVðtÞ þ e ai0 jui0 ðtÞj þ j¼1
" 6 eVðtÞ þ e
et
ai0 þ
m X
!
jdj0 i jLi jui0 ðtÞj þ bj0 þ
i¼1
i¼1 p X
!
#
jci0 j jLj jv j0 ðtÞj 6 eVðtÞ bVðtÞ ¼ ðb eÞVðtÞ
j¼1
¼ gVðtÞ: Again the remaining details of proof are similar to those of Theorem 3.1 and we omit them to avoid redundancy. h
4. An illustrative example Let m = p = 1, fj(x) = gi(x) 2jxj. Consider the BAM neural networks with impulses:
dx ¼ 3xðtÞ þ f ðyðtÞÞ þ 5:5; t – t k ; dt dy ¼ 3yðtÞ þ f ðxðtÞÞ þ 4:5; t – tk ; dt 1 xðtþk Þ xðtk Þ ¼ ðxðt k Þ 1:5Þ; 2 1 þ yðt k Þ yðtk Þ ¼ ðyðtk Þ 0:5Þ: 2
ð4:1Þ
It is easy to check the condition (3.7) satisfies and b1 = b2 = 1, so if min{tk tk1} P 2, k = 1, 2, . . . , then
2 ln and
P1
e k ðrÞ Gk ðrÞ þ G
r
¼ 2 ln 3 ¼ ck > 0;
c ¼ þ1, here Gk ðrÞ ¼ Gk ðrÞ ¼ 32 r. By Theorem 3.3, the equilibrium is exponentially asymptotically state.
k¼1 k
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