GLOBAL EXPONENTIAL STABILITY OF BAM ... - Semantic Scholar

J. Appl. Math. & Informatics Vol. 29(2011), No. 1 - 2, pp. 103 - 117

Website: http://www.kcam.biz

GLOBAL EXPONENTIAL STABILITY OF BAM NEURAL NETWORKS WITH IMPULSES AND DISTRIBUTED DELAYS YUANFU SHAO∗ AND ZHENGUO LUO

Abstract. By using an important lemma, some analysis techniques and Lyapunov functional method, we establish the sufficient conditions of the existence of equilibrium solution of a class of BAM neural network with impulses and distributed delays. Finally, applications and an example are given to illustrate the effectiveness of the main results. AMS Mathematics Subject Classification : 34D05, 34D20, 93D05. Key words and phrases : Exponential stability, Equilibrium, Lyapunov functional, Delays, Impulses.

1. Introduction Recently, BAM neural networks have attracted the attention of many researchers due to its applications in many fields such as pattern recognition, automatic control and optimization, and many results for BAM neural networks have been derived [1-8]. Further, the theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulse, but also represents a more natural framework for mathematical modelling of many real world phenomena, such as population dynamics and neural networks, hence, the impulsive differential equations have been extensively studied recently [5,6,9-19]. On the other hand, in practice, it is preferable and desirable that neural networks not only converge to equilibrium points but also admit a convergence rate which is as fast as possible. Since the exponential stability gives a fast convergence rate to the equilibrium point, it is necessary to study the exponential stability and to estimate the exponential convergence rate, see [9,10,20-27]. Therefore, it is necessary and important for scholars to study the existence and exponential stability of equilibrium points for impulsive neural networks Received March 28, 2010. Revised May 20, 2010. Accepted June 21, 2010. author. c 2011 Korean SIGCAM and KSCAM. ° 103

∗

Corresponding

104

Yuanfu Shao and Zhenguo Luo

with delays [9,10,23-26]. For example, Zhou [10] investigated the following BAM neural networks:  m P   x0i (t) = −ai xi (t) + hji gj (yj (t))    j=1  m  P R∞    + lji 0 kji (s)fj (yj (t − τji − s))ds + bi , t 6= tk   j=1   R tk  ∆xi (t)

= Iik (xi (t)) = Bik xi (tk ) + n P ¯

tk −1

Cik (s)xi (s)ds + αik , t = tk

  = −¯ yj0 (t) aj yj (t) + hij g¯i (xi (t))    i=1   n P¯ R∞¯    + lij 0 kij (s)f¯i (xi (t − σij − s))ds + ¯bj , t 6= tk    i=1   ∆y (t) = J (y (t)) = B¯ y (t ) + R tk C¯ (s)y (s)ds + α ¯ jk , t = tk j j jk j jk j k t −1 jk

(1.1)

k

By using the contraction mapping principle and Lyapunov functional, the sufficient conditions ensuring global exponential stability of the equilibrium points of (1.1) are established. Motivated by above discussion, in this paper, we shall establish a class of impulsive BAM neural network with distributed delays as follows:  m P   x0i (t) = −ai ei (xi (t)) + bji fj (yj (t))    j=1   m Rτ P    + l k (s)gj (yj (t − τji − s))ds + Ii , t 6= tk ji  0 ji   j=1   ∆xi (t) = xi (t+ ) − xi (t− ) = I˜ik (xi (t)), t = tk n P  0  (1.2) (t) = −c h (y (t)) + y dij pi (xi (t))  j j j j   i=1   n  Rσ P˜   + lij (t) 0 k˜ij (s)qi (xi (t − σij − s))ds + Jj , t 6= tk    i=1   ∆yj (t) = yj (t+ ) − yj (t− ) = J˜jk (yj (t)), t = tk with initial values ˜ ≤ s ≤ 0, h = σ+ xi (s) = φxi (s), −h ≤ s ≤ 0, yj (s) = φyj (s), −h ˜=τ+ h

max

max

{σij },

1≤i≤n,1≤j≤m

˜ 0], R). {τji }, φxi ∈ C([−h, 0], R), φyj ∈ C([−h,

1≤i≤n,1≤j≤m

where xi (t) and yj (t) are the states of the ith neuron and the jth neuron at time t, t ∈ R+ = [0, +∞), respectively. ai , cj denote the neuron charging times. bji , lji , dij and ˜lij (t) are the weights of the neuron interconnections. Ii and Jj are the external inputs on the neurons. ∆xi (t) and ∆yj (t) are the impulses at moments t = tk and t1 < t2 < · · · is a strictly increasing sequence such that limk→∞ tk = ∞, i = 1, 2, · · · , n, j = 1, 2, · · · , m. τ > 0, σ > 0 are constants. As usual in the theory of impulsive differential equations, at the points of discontinuity tk of the solution z(t) = (x1 (t), x2 (t), · · · , xn (t), y1 (t), y2 (t), · · · , ym (t))T , − we assume that z(t+ k ) exists, and z(tk ) = z(tk ). It is clear that there exist the 0 − 0 + 0 − limits z (tk ), z (tk ) such that z (tk ) = z 0 (tk ).

Global exponential stability of BAM neural networks with impulses

105

Our aim is, under the generalized r-norm (r > 1), by using an important lemma and constructing suitable Lyapunov functional, to obtain the sufficient conditions ensuring the existence and globally exponential stability of equilibrium solution of (1.2). The rest of this paper is organized as follows. In section 2, definitions and lemmas are introduced. In section 3, by using Forti and Tesi’s theorem, the sufficient conditions of the existence of equilibrium solution are established. In section 4, the conditions ensuring the globally exponential stability of the equilibrium point are derived. Finally in section 5, applications and an illustrative example are given to show the usefulness of the main results. 2. Preliminaries First we make some preparation and introduce some elementary definitions and lemmas. ˜ 0])T → (Rn , Rm )T Let P C be a class of function φ = (φx , φy )T : ([−h, 0], [−h, satisfying: (i) φ is piecewise continuous with first kind discontinuity at point tk , and is left-continuous at tk , k = 1, 2, · · · , p. (ii) ∆xi (tk ) = I˜ik (xi (tk )), ∆yj (tk ) = J˜jk (yj (tk )) for i = 1, 2, · · · , n, j = 1, 2, · · · , m, k = 1, 2, · · · . For each φ = (φTx , φTy )T ∈ P C, z(t) ∈ Rn+m , we define  r1  n m X X kφk =  sup |φxi (s)|r + sup |φyj (s)|r  , ˜ j=1 s∈[−h,0]

i=1 s∈[−h,0]

 kz(t)k = 

n X

|xi (t)|r +

i=1

m X

 r1 |yj (t)|r  ,

j=1

where r > 1 is a constant, z(t) = (x1 (t), x2 (t), · · · , xn (t), y1 (t), · · · , ym (t))T , φx = (φx1 , φx2 , · · · , φxn )T and φy = (φy1 , φy2 , · · · , φym )T . ∗ T ) is said to Definition 2.1. A constant vector z ∗ = (x∗1 , x∗2 , · · · , x∗n , y1∗ , · · · , ym be an equilibrium solution of impulsive system (1.2) if  Z τ m m X X  ∗ ∗ ∗  a e (x ) = b f (y ) + l g (y ) kji (s)ds + Ii  i i ji j ji j i j j   0 j=1 j=1 (i) Z σ n n  X X   ∗ ∗ ∗ ˜  dij pi (xi ) + lij qi (xi ) k˜ij (s)ds + Jj cj hj (yj ) = (2.1) i=1

i=1

0

(ii) I˜ik (x∗i ) = 0, J˜jk (yj∗ ) = 0. ∗ T Definition 2.2. The unique equilibrium z ∗ = (x∗1 , x∗2 , · · · , x∗n , y1∗ , · · · , ym ) of system (1.2) is said to be globally exponentially stable if there exists constant

106


α > 0, M ≥ 1 such that for all t > 0,   r1 n m  X X |xi (t) − x∗i |r + |yj (t) − yj∗ |r ≤ M e−αt kφ − z ∗ k,   i=1

where kφ−z ∗ k =

j=1

nP n

∗ r i=1 sups∈[−h,0] |φxi (s) − xi | +

Pm

|φyj (s) − yj∗ |r ˜ j=1 sups∈[−h,0]

o r1

.

Definition 2.3 [28]. A real matrix A = (aij )n×n is said to be an M-matrix if aii > 0, aij ≤ 0(i, j = 1, 2, · · · , n, i 6= j) and successive principle minors of A are positive. Lemma 2.1 [29]. Let Q be an n × n matrix with non-positive off-diagonal elements. Then Q is an M-matrix if and only if one of the following conditions holds: (i) There exists a vector ξ > 0 such that Qξ > 0; (ii) There exists a vector ξ > 0 such that ξ T Q > 0. Lemma 2.2 [30]. (Young inequality) Assume that a, b, p, q > 0, p + q = 1, then ap bq ≤ pa + qb. Lemma 2.3 [31]. (Forti and Tesi’ theorem) If H(x) ∈ C 0 satisfies the following conditions: (i) H(x) is injective on Rn+m , (ii) kH(x)k → +∞ as kxk → +∞, then H(x) is homeomorphism of Rn+m onto itself. Throughout this paper, we always assume that: (A1 ) ai > 0, cj > 0, bji , dij , lji , ˜lij , Ii and Jj are constants for i = 1, 2, · · · , n, j = 1, 2, · · · , m. (A2 ) ei , hj : R → R are differentiable function satisfying 0 < %i ≤ e0i (u), ei (0) = 0 and 0 < %˜j ≤ h0j (v), hj (0) = 0 for any u, v ∈ R, i = 1, 2, · · · , n, j = 1, 2, · · · , m. (A3 ) Functions fj (u), gj (u), pi (u), qi (u) satisfy the Lipschitz conditions, i.e., there exist positive constants Fj , Gj , Pi , Qi such that |fj (u) − fj (v)| ≤ Fj |u − v|,

|gj (u) − gj (v)| ≤ Gj |u − v|,

|pi (u) − pi (v)| ≤ Pi |u − v|,

|qi (u) − qi (v)| ≤ Qi |u − v|

with fj (0) = gj (0) = 0, pi (0) = qi (0) = 0 for any u, v ∈ R, i = 1, 2, · · · , n, j = 1, 2, · · · , m. (A4 ) Functions kji (t) and k˜ij (t) are positive piecewise continuous and satisfy Z τ Z σ ˜ σ), eηt kji (t)dt = ψ(η, τ ), eηt k˜ij (t)dt = ψ(η, 0

0

˜ σ) are continuous in η. When τ = ∞, σ = ∞, ψ(η, τ ) ≡ where ψ(η, τ ) and ψ(η, ˜ ϕ(η), ψ(η, σ) ≡ ϕ(η) with ϕ(0) = ϕ(0) ˜ = 1.


107

3. Existence of equilibrium solution In this section, employing the Forti and Tesi’s theorem, we will establish the sufficient conditions of the existence of equilibrium solution of system (1.2). Theorem 3.1. Assume that (A1 )−(A4 ) hold. Further, if there exists a constant r > 1 suchµthat the following condition holds.¶ ˜ rA − (r − 1)G −P˜ (A5 ) Γ = is a nonsingular M-matrix, ˜ −F˜ rC − (r − 1)Q where A = diag(a1 %1 , a2 %2 , · · · , an %n ), C = diag(c1 %˜1 , c2 %˜2 , · · · , cm %˜m ), ˜ = diag(G ˜1, G ˜2, · · · , G ˜ ), Q ˜ = diag(Q ˜1, Q ˜2, · · · , Q ˜ m ), P˜ = (˜ G pij )n×m , Rτ Pmn ˜ ˜ ˜ F = (fji )m×n , Gi = j=1 |bji |Fj + |lji |Gj 0 kji (s)ds, R R ˜ j = Pn |dij |Pi + |˜lij |Qi σ k˜ij (s)ds, p˜ij = Pi |dij | + Qi |˜lij | σ k˜ij (s)ds, Q i=1 0 0 Rτ f˜ji = |bji |Fj + |lji |Gj 0 kji (s)ds. Then system (1.2) admits exactly one equilib∗ T rium solution z ∗ = (x∗1 , · · · , x∗n , y1∗ , · · · , ym ) . Proof. For z = (x1 , x2 , · · · , xn , y1 , · · · , ym ) ∈ Rn+m , define a mapping ψ : Rn+m → Rn+m as follows:  m m R P P τ  bji fj (yj ) − k (s)gj (yj )ds − Ii  ψi (z) = ai ei (xi ) − 0 ji j=1 j=1 (3.1) n n P PRσ   ψn+j (z) = cj hj (yj ) − kij (s)qi (xi )ds − Jj , dij pi (xi ) − i=1

i=1

0

where ψ(z) = (ψ1 (z), ψ2 (z), · · · , ψn (z), ψn+1 (z), · · · , ψn+m (z))T ∈ Rn+m . Firstly, we demonstrate that the mapping ψ is injective, i.e., ψ(z) = ψ(˜ z) implies that z = z˜ for any z, z˜ ∈ Rn+m . It is clear that ψ(z) = ψ(˜ z ) means:  a (e (x ) − e (˜x )) − Pm b (f (y ) − f (˜y )) j j i i i i i  R τ j=1 ji j j P  m kji (s)(gj (yj ) − gj (˜ yj ))ds = 0 − j=1 lji 0 P (3.2) n cj (hj (yj ) − hj (˜ yj )) −R i=1 dij (pi (xi ) − pi (˜ xi ))   Pn σ −

˜

l i=1 ji

0

kij (s)(qi (xi ) − qi (˜ xi ))ds = 0

Then from (A2 ) − (A4 ) and (3.2), we derive that ¢ Rτ ½ Pm ¡ ai %i |xi − x ˜i | ≤ |b |F + |lji |Gj k (s)ds |yj − y˜j |, ¡ ji j ¢ R 0σ ji Pj=1 n ˜ ˜ cj %˜j |yj − y˜j | ≤

i=1

|dij |Pi + |lij |Qi

0

kij (s)ds |xi − x ˜i |.

(3.3)

On the other hand, we obtain from (A5 ) and Lemma 2.1 that, there exists ξ = (ξ1 , ξ2 , · · · , ξn , ξn+1 , · · · , ξn+m )T > 0 such that  m Rτ P  rξi ai %i − ξi (r − 1) (|bji |Fj + |lji |Gj k (s)ds)   0 ji  j=1   m  Rσ P  ˜ij (s)ds)) > 0  ξn+j (|dij |Pi + |˜ lij |Qi − k  0 j=1

n Rσ P  ˜ij (s)ds)  rξn+j cj %˜j − ξn+j (r − 1) (|dij |Pi + |˜ lij |Qi k  0   i=1   n Rτ P    ξi (|bji |Fj + |lji |Gj − kji (s)ds) > 0. i=1

0

(3.4)

108


Further, by Lemma 2.2, it follows from (3.3) that n X

ξi ai %i |xi − x ˜ i |r

i=1 n

m µ X X

≤

ξi

Z

|bji |Fj + |lji |Gj

i=1

j=1

n

m

ξi i=1

|yj − y˜j ||xi − x ˜i |r−1

kji (s)ds 0

X Xµ ≤

¶

τ

Z

(3.5)

¶

τ

|bji |Fj + |lji |Gj

kji (s)ds

³ ×

0

j=1

1 r−1 |xi − x ˜i |r + |yj − y˜j |r r r

´ ,

and m X

ξn+j cj %˜j |yj − y˜j |r

j=1

n µ X

m X

≤ j=1

i=1

Z

(3.5) plus (3.6) lead to Ã n m X ξi (r − 1) X ξi ai %˜i −

Z

r

¶

σ

+|˜ lij |Qi

˜ij (s)ds) k

+|˜ lij |Qi

˜ij (s)ds k

³ ×

Z

kji (s)ds) − 0

j=1

|xi − x ˜i |r +

X ξi i=1

r

m X ξn+j j=1

m ³ X

ξn+j cj %˜j −

Z

n

σ

1 r−1 |yj − y˜j |r + |xi − x ˜i |r r r

τ

(|bji |Fj + |lji |Gj

j=1

˜ij (s)ds) − k 0

(3.6)

¶

σ

0

Z

|yj − y˜j |r−1 |xi − x ˜i |

0

i=1

i=1

˜ij (s)ds k

|dij |Pi + |˜ lij |Qi

ξn+j j=1

¶

τ 0

n µ X

m X

≤

Z


ξn+j

(|bji |Fj + |lji |Gj

ξn+j (r − 1) r

!

τ

kji (s)ds)

r

´ .

(|dij |Pi

n X

(|dij |Pi i=1

(3.7)

|yj − y˜j |r ≤ 0

0

Substituting (3.4) into (3.7), we have |xi − x ˜i |r − 0, |yj − y˜j |r = 0. That is, xi = x ˜i , yj = y˜j for i = 1, 2, · · · , n, j = 1, 2, · · · , m, namely, z = z˜, which means ψ ∈ C 0 is injective on Rn+m . Next we demonstrate the property kψ(z)k → ∞ as kzk → ∞. Consider e mapping ψ(z) = ψ(z) − ψ(0), i.e., Z τ m m X X kji (s)gj (yj )ds, ψ˜i (z) = ai ei (xi ) − bji fj (yj ) − lji j=1 n X

ψen+j (z) = cj hj (yj ) −

i=1

j=1 n X

dij pi (xi ) −

i=1

0

Z ˜lij

σ

k˜ij (s)qi (xi )ds

0

for z = (x1 , x2 , · · · , xn , y1 , · · · , ym )T ∈ Rn+m , i = 1, 2, · · · , n, j = 1, 2, · · · , m. It e is enough to show that kψ(z)k → ∞ as kzk → ∞. Using the Young inequality,


109

we have n X

ei (z)) rξi |xi |r−1 sgn(xi )(ai ei (xi ) − ψ

i=1

=

n X

Ã rξi |xi |

r−1

sgn(xi )

i=1

≤

n m X X

≤

Z lji

gj (yj )kji (s)ds 0

j=1

Z

!

τ

¶

τ

(3.8)

kji (s)ds |yj |

|bji |Fj + |lji |Gj 0

i=1 j=1 n X m X

m X

bji fj (yj ) +

j=1

µ rξi |xi |r−1

m X

µ ξi

Z

¶

τ

kji (s)ds ((r − 1)|xi |r + |yj |r )

|bji |Fj + |lji |Gj 0

i=1 j=1

and m X

ej (z)) rξn+j |yj |r−1 sgn(yj )(cj hj (yj ) − ψ

j=1

=

m X

rξn+j |yj |

r−1

sgn(yj )

j=1

≤

m n X X

µ

XX

≤

dij pi (xi ) +

i=1

Z


r−1

rξn+j |yj |

Z

n X

˜ lij

! ˜ij (s)ds qi (xi )k

0

i=1 σ

τ

¶

˜ij (s)ds |xi | k

(3.9)

0

j=1 i=1 m

Ã n X

µ

n

Z

σ


ξn+j

¶ ˜ij (s)ds ((r − 1)|yj |r + |xi |r ) k

0

j=1 i=1

(3.8) plus (3.9), then n X

ei (z)) + rξi |xi |r−1 sgn(xi )(ai ei (xi ) − ψ

i=1

n X m µ X

≤

µ

ξi (r − 1)

Z

µ

˜ij (s)ds k

µ

+ξi

|xi |r

0

µ Z

|bji |Fj + |lji |Gj

Z


ξn+j (r − 1) j=1 i=1

τ

¶¶ kji (s)ds

0

¶

kji (s)ds

¶¶

σ

|dij |Pi + |˜ l|ij Qi

+

j=1

0

Z

m X n µ X

ej (z)) rξn+j |yj |r−1 sgn(yj )(cj hj (yj ) − ψ

τ

|bji |Fj + |lji |Gj

i=1 j=1

+ξn+j

m X

σ

¶ ˜ij (s)ds k

0

|yj |r .

110


That is, n X

(

m µ X

rξi ai %i − i=1

Z

¶¶¾

σ


+ξn+j

˜ij (s)ds k

n µ X

rξn+j cj %˜j −

µ

+ξi

0

Z

|bji |Fj + |lji |Gj

˜ij (s)ds k 0

¶¶¾

τ

¶

σ


ξn+j (r − 1)

Z

kji (s)ds

µ

i=1

j=1

¶

τ

|xi |r

0

(

+

Z

|bji |Fj + |lji |Gj

j=1

µ

m X

µ ξi (r − 1)

|yj |r

kji (s)ds 0

n X

ei (z)|xi |r−1 + ξi rψ

≤ i=1

ϑ

!

m X

r

|xi | + i=1

r

|yj |

≤ rξ

+

Ã n X

j=1

(

ϑ = min

ej (z)|yj |r−1 . ξn+j rψ

j=1

Therefore, Ã n X

where

m X

i=1

Ã min

1≤i≤n

Z Z

m X

Z

τ

((r − 1)ξi (|bji |Fj + |lji |Gj j=1

˜ij (s)ds)) k 0

, min

1≤j≤m

kji (s)ds) 0

Ã

¶

σ

! ej (z)|yj |r−1 ψ

j=1

m X

rξi ai %i −

+ξn+j (|dij |Pi + |˜ lij Qi +|˜ lij |Qi

ei (z)|xi |r−1 + ψ

n X

rξn+j cj %˜j −

Z

σ

(ξn+j (|dij |Pi i=1

¶¾

τ

˜ij (s)ds)(r − 1) + ξi (|bji|Fj + |lji |Gj k

kji (s)ds))

0

> 0,

0

ξ + = max{ξ1 , ξ2 , · · · , ξn , ξn+1 , · · · , ξn+m }.

By applying H¨older inequality, we have Ã n m n m X X X rξ + X |xi |r +

i=1

|yj |r ≤

j=1

|xi |r +

ϑ

i=1

1 s

|yj |r

! 1s Ã n X

j=1

i=1

|xi | +

|yj |

j=1

≤

ϑ

r

|ψi (z)| +

i=1

! r1 ej (z)|r |ψ

j=1

1 r

where s > 0, r > 0 such that + = 1. That is, Ã n ! r1 Ã n m X X rξ + X e r r i=1

ei (z)|r + |ψ

m X

m X

! r1 ej (z)| |ψ

r

,

j=1

+ e e from which we assert that kψ(z)k → ∞ as kzk → ∞. i.e., kzk ≤ rξϑ kψ(z)k, 0 By Lemma 2.3, we conclude that ψ ∈ C is a homeomorphism on Rn+m , which guarantees the existence of a unique solution z ∗ ∈ Rn+m of the algebraic system (2.1) which defines the unique equilibrium state of the impulsive network (1.2). This completes the proof.


111

Remark 3.1. The proof of the existence of equilibrium point of (1.2) is different from those [8-10], and by applications in section 5, one can see that the results here improve or extend the corresponding results [8-10, 20]. In Theorem 3.1, if r → 1, then we have Corollaryµ3.1. Assume¶that (A1 ) − (A4 ) hold. Further, A −P˜ (A6 ) Γ0 = is a nonsingular M-matrix, −F˜ C − − − ˜m ), ˜1 , c− ˜2 , · · · , c− where A = diag(a− m% 2% 1 %1 , a2 %2 , · · · , an %n ), C = diag(c1R % σ ˜ ˜ ˜ ˜ ˜ P = (˜ pij )n×m , F = (fji )m×n , p˜ij = Pi |dij | + Qi |lij | 0 kij (s)ds, Rτ f˜ji = |bji |Fj +|lji |Gj 0 kji (s)ds. Then system (1.2) has at least one equilibrium. 4. Globally exponential stability Theorem 4.1. Assume that (A1 ) − (A5 ) hold. Further, (A7 ) I˜ik (xi (tk )) = −βik (xi (tk ) − x∗i ), J˜jk (yj (tk )) = −γjk (yj (tk ) − yj∗ ), |1 − βik |r − 1 ≤ 0, |1 − γjk |r − 1 ≤ 0 for i = 1, 2, · · · , n, j = 1, 2, · · · , m, k = 1, 2, · · · . Then the equilibrium solution z ∗ of (1.2) is globally exponentially stable. Proof. By Theorem 3.1, there exists a unique equilibrium solution ∗ T ) of (1.2). z ∗ = (x∗1 , x∗2 , · · · , x∗n , y1∗ , · · · , ym T T T Let z(t) = (x (t), y (t)) = (x1 (t), x2 (t), · · · , xn (t), y1 (t), · · · , ym (t))T be an arbitrary solution of (1.2), then we have                       

d|xi (t)−x∗ i| dt

d|yj (t)−yj∗ | dt

Pn

|b ||fj (yj (t)) − fj (yj∗ )| ≤ −ai |ei (xi (t)) − ei (x∗i )| + i=1 ji Rτ Pm + j=1 |lji | k (s)|gj (t − τji − s) − gj (yj∗ )|ds 0 ji Pn ∗ ∗ ≤ −ai %i |xi (t) − xi | + R τ i=1 |bji |Fj |yj (t) − yj | ∗ Pn kji (s)|yj (t − τji − s) − yj |ds + i=1 |lji |Gj 0

Pm

|d ||pi (xi (t)) − pi (x∗i )| ≤ −cj |hj (yj (t)) − hj (yj∗ )| + j=1 ij Rτ Pm ˜ij (s)|qi (xi (t − σij − s)) − qi (x∗ )|ds lij | + j=1 |˜ k i 0 Pm ≤ −cj %˜j |yj (t) − yj∗ | + Pi |dij ||xi (t) − x∗i | j=1 R Pm ˜ τ ˜ |lij | + kij (s)Qi |xi (t − σij − s) − x∗ |ds j=1

0

(4.1)

i

for t > 0, t 6= tk , i = 1, 2, · · · , n, j = 1, 2, · · · , m. On the other hand, according to condition (A5 ) and Lemma 2.1, there exist a vector (ξ1 , ξ2 , · · · , ξn , ξn+1 , · · · , ξn+m )T such that ³ Rτ Pm ξi rai %i − (r − 1) j=1 (|bji |Fj + |lji |Gj 0 kji (s)ds) ´ Rσ Pm − j=1 ξn+j (Pi |dij | + Qi |˜lij | 0 k˜ij (s)ds) > 0, ³ Rσ Pn ξn+j rcj %˜j − (r − 1) i=1 (|dij |Pi + |˜lij |Qi 0 k˜ij (s)ds) ¢ Rτ Pn − i=1 ξi (Fj |bji | + Gj |lji | 0 kji (s)ds) > 0.

112


Let ´ ³  Rτ Pm  k (s)ds) (|b |F + |l |G χ ji i (ε) = ξi ε − rai %i + (r − 1) ji j ji j  j=1 0   ¡ ¢ R P σ ˜ m εs + j=1 ξn+j Pi |dij | + Qi eεσij |˜ lij | 0 k ij (s)e ds ¡ ¢ Rσ P n  ˜ ˜ij (s)ds) κj (ε) = ξn+j ε −¡rcj %˜j + (r − 1) i=1 (dij Pi + ¢lij Qi 0 k   R Pn  τ + i=1 ξi Fj bji + Gj eετji lji 0 kji (s)ds

It is clear that χi (0) < 0, κj (0) < 0. Since χi (ε), κj (ε) are continuous on [0, ∞) dκj (ε) i (ε) and χi (ε), κj (ε) → +∞ as ε → +∞, and dχdε > 0, dε > 0, then there exist ∗ ∗ constant ξi , ηj such that ³  ¢´ Rτ Pm ¡ ∗ ∗  χ |b i (ξi ) = ξi ξi − rai %i + (r − 1) ji |Fj + |lji |Gj 0 kji (s)ds  j=1   ¡ ¢ Rσ Pm ∗  ˜ij (s)eξi∗ s ds = 0 + j=1 ξn+j Pi |dij | + Qi eξi σij |˜ lij | 0 k ¡ ¢¢ Rσ Pn ¡ ˜ij (s)ds κj (ηj∗ ) = ξn+j ηj∗ −³rcj %˜j + (r − 1) i=1 |dij |Pi + |˜ lij |Qi´ 0 k   (4.2)  R Pn ∗ ∗  τ  ξi Fj |bji | + Gj eηj τji |lji | kji (s)eηj s ds = 0 + i=1

0

∗ } for i = 1, 2, · · · , n, j = By choosing 0 < λ < min{ξ1∗ , ξ2∗ , · · · , ξn∗ , η1∗ , · · · , ηm 1, 2, · · · , m, we have ³  ¢´ Rτ Pm ¡  |b |F + |l |G χ k (s)ds ji j ji j i (λ) = ξi λ − rai %i + (r − 1) ji  j=1 0   ¡ ¢ R P σ ˜ m λs + j=1 ξn+j Pi |dij | + eλσij Qi |˜ lij | 0 k ij (s)e ds < 0 ¡ ¡ ¢¢ Rσ P n  ˜ ˜ |Pi + |lij |Q¢ κj (λ) = ξn+j λ − rc 1) i=1 |d  i 0 kij (s)ds  ¡j %˜j + (r −λτ R ij Pn  τ + i=1 ξi Fj |bji | + e ji Gj |lji | 0 kji (s)eλs ds < 0.

(4.3)

Let ui (t) = eλt |xi (t) − x∗i |r , vj (t) = eλt |yj (t) − yj∗ |r , from (4.1), we derive that               

d+ ui (t) dt

d+ vj (t) dt

λt ∗ r−1 ∗ ≤ λeλt |xi (t) − x∗i |r + re i (t) − xi ) Pn|xi (t) − xi | sgn(x ∗ ∗ (−a Pi %ni |xi (t) − xRiτ| + i=1 |bji |Fj |yj (t) − yj | + i=1 |lji |Gj 0 kji (s)|yj (t − τji − s) − yj∗ |ds) λt ≤¡λeλt |yj (t) − yj∗ |r + reP |yj (t) − yj∗ |r−1 sgn(yj (t) − yj∗ ) m ∗ −cj %˜j |yj (t) − yj | + j=1 Pi |dij ||xi (t) − x∗i | Pm ˜ R τ ˜ + |lij | kij (s)Qi |xi (t − σij − s) − x∗i |ds) j=1

(4.4)

0

for t > 0, t 6= tk . When t = tk , for i = 1, 2, · · · , n, j = 1, 2, · · · , m, it follows from (A7 ) that ui (t+ ) = |1 − αik |r ui (tk ) ≤ u(tk ), vj (t+ ) = |1 − βjk |r vj (tk ) ≤ vj (tk ) k k

(4.5)

Define a Lyapunov functional as follows: ³ ´ Rτ Rt Pn Pm V (t) = i=1 ξi ui (t) + j=1 |lji |eλτji Gj 0 kji (s)eλs t−τji −s vj (z)dzds ³ ´ Rσ Rt Pn Pn + i=1 ξn+j vj (t) + i=1 |˜lij |eλσij Qi 0 k˜ij (s)eλs t−σij −s ui (z)dzds


113

By calculating the derivative of V (t) along the solution of (1.2) and from (4.3), (4.4) and Lemma 2.2, we have d+ V (t)

=

³

Pdt n

ξ i=1 i

−

Pm

j=1

d+ ui (t) dt

Pm

+

j=1

|lji |Gj eλτji

³

Pm

Rτ

Rτ

|lji |eλτji Gj

kji (s)eλs vj (t)ds

´

0

kji (s)eλs vj (t − τji − s)ds

0

R

P

+

σ ˜ d vj (t) n λs + i=1 |˜ ξn+j lij |Qi eλσij 0 k ij (s)e ui (t)ds dt ¢ R Pj=1 σ n ˜ij (s)eλs ui (t − σij − s)ds − i=1 ³ |˜ lij |Qi eλσij 0 k

+ ≤

Pn

P

m ξ (λ − rai %i )|ui (t)|r + reλt j=1 |bji |Fj |yj (t) − yj∗ ||xi (t) − i=1 i τ m +reλt j=1 |lji |Gj 0 kji (s)|xi (t) − x∗i |r−1 |yj (t − τji − s) − yj∗ |ds τ m λτji + j=1 e |lji |Gj 0 kji (s)eλs dsvj (t)

R R

P

P

x∗i |r−1

´ Rτ Pm λτji λs |l |G e k (s)e v (t − τ − s)ds ji j ji j ji j=1 0 Pm Pn + j=1 ξn+j ((λ − rcj %˜j )|vj (t)|r + reλt i=1 |dij |Pi |xi (t) − x∗i ||yj (t) − yj∗ |r−1 R P σ ˜ n ∗ r−1 ∗ +reλt i=1 |˜ |xi (t − σij − s) − i 0 kij (s)|yj (t) − yj | ¢ xi |ds Rσ Pn ˜ lij |Q λσij λs ˜ |lij |Qi e kij (s)e (ui (t) − ui (t − σij − s))ds + 0 ¡ P P i=1 ³ −

≤

m n ξ (λ − rai %i )|ui (t)|r + reλt j=1 |bji |Fj r1 |yj (t) − yj∗ |r i=1 i τ m + r−1 |xi (t) − x∗i |r + reλt j=1 |lji |Gj 0 kji (s) r−1 |xi (t) − x∗i |r r r τ m λτji ∗ r λs 1 |lji | 0 kji (s)e ds|vj (t)| + r |yj (t − τji − s)yj | ds + j=1 e

¢

¢

−

Pm

j=1

¡

|lji |Gj eλτji

Rτ 0

(4.6)

¡

R

P P

R

´

kji (s)eλs vj (t − τji − s)ds +

Pm

j=1

¡1

Pn

ξn+j

¢

(λ − rcj %˜j )|vj (t)|rR+ reλt ¡i=1 |dij |Pi r |xi (t) − x∗i |r + r−1 |yj (t) − yj∗ |¢r r Pn ˜ σ ˜ λt ∗ r r−1 1 |lij |Qi R +re kij (s) r |yj (t) − yj | + r |xi (t − σ¢ij − s) − x∗i |r ds 0 Pn i=1 σ λσ ij ˜ ˜ij (s)eλs (ui (t) − ui (t − σij − s))ds + i=1 k 0 ³ |lij |Qi e

≤

Pn

ξi (λ − rai %i + (r − 1)

i=1

+

Pm ¡ j=1

¡

Pm

ξ Pi |dij | + Qi |˜ lij |eλσij j=1 n+j m ξn+j (λ − rcj %˜j + (r − 1) j=1 τ n ξ Fj |bji | + Gj |lji |eλτji 0 i=1 i

Rσ

Rτ

¢´

0

¢

kji (s)ds

˜ij (s)eλs ds k

ui (t) ¢ Rσ ˜ ˜ |d |P + | l ij i ij |Qi 0 kij (s)ds i=1 ¢¢ R kji (s)eλs ds vj (t)

P0n ¡

¡

P

+ P + 0, t 6= tk , k = 1, 2, · · · . When t = tk , we obtain from (4.5) that V (t+ k) =

Pn

i=1

+ =

µ ui (t+ k)

ξi

Pn i=1

Pn

µ

+

Pm

j=1

j=1

vj (t+ k)

ξn+j

³

ξ ui (t+ k)+ i=1 i

+

Pm

³

+

0

λs

kji (s)e

Pm

|˜ l |Qi j=1 ij

Pm

|l |Gj j=1 ji

ξn+j vj (t+ k)+

≤ V (tk ), k = 1, 2,

|lji |Gj

Rτ

Pn

i=1

Rτ 0

Rσ 0

Rσ 0

t+ −τji −s k

˜ij (s)eλs k

vj (z)dzds

R t+ k

t+ −σij −s k

R λs tk

kji (s)e

|˜ lij |Qi

¶

R t+ k

˜ij (s)e k

R

ui (z)dzds

´

v (z)dzds −s j

tk −τji λs tk

tk −σij −s

¶

ui (z)dzds

(4.7)

´

114


It follows from (4.6) and (4.7) that V (t) ≤ V (0) for all t > 0.

(4.8)

By the definition of V (t) and (4.8), we have Ã n ! m X X − ξi

ui (t) +

i=1

≤

n X

Ã

ξi

ui (0) +

i=1

+

m X

ξn+j

≤ξ

n X i=1

+ξ

+

m X

vj (0) +

Ã

≤ξ ι

Z

Z

τ

Gj |lji |eλτji

n X

Ã 1+

m X

Z Qi |˜ lij |e

1+

m X

Z

i=1

Z ˜ij (s)e k

σ

Z

ui (z)dzds −σij −s

!

˜ij (s)eλs (σij + s)ds k

!

τ λs

Gj |lji |e

kji (s)e (τji + s)ds 0

sup ui (t) +

sup ui (t) −h