Neural networks with discrete and distributed time-varying delays: A general stability analysis

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Chaos, Solitons and Fractals 37 (2008) 1538–1547 www.elsevier.com/locate/chaos

Neural networks with discrete and distributed time-varying delays: A general stability analysis Qiankun Song a, Zidong Wang b

b,*

a Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Accepted 23 October 2006

Communicated by Prof. M.S. El Naschie

Abstract In this paper, the global asymptotic and exponential stability are investigated for a class of neural networks with both the discrete and distributed time-varying delays. By using appropriate Lyapunov–Krasovskii functional and linear matrix inequality (LMI) technique, several delay-dependent sufficient conditions are obtained to guarantee the global asymptotic and exponential stability of the addressed neural networks. These conditions are expressed in terms of LMIs, and are dependent on both the discrete and distributed time delays. Therefore, the stability of the neural networks can be checked readily by resorting to the Matlab LMI toolbox. In addition, the proposed stability criteria do not require the monotonicity of the activation functions and the differentiability of the discrete and distributed time-varying delays, which means that our results generalize and further improve those in the earlier publications. A simulation example is given to show the effectiveness and less conservatism of the obtained conditions.  2006 Elsevier Ltd. All rights reserved.

1. Introduction Time delays inevitably exist in neural networks due to various reasons. For example, time delays can be caused by the finite switching speed of amplifier circuits in neural networks [1] or deliberately introduced to achieve tasks of dealing with motion-related problems such as moving image processing [2]. The existence of time delay may lead to some complex dynamic behaviors such as oscillation, divergence, chaos, instability or other poor performance of the neural networks [3]. Therefore, stability analysis for neural networks with delays has been an attractive subject of research in the past few years. Various sufficient conditions, either delay-dependent or delay-independent, have been proposed to guarantee the global asymptotic or exponential stability for neural networks with constant and time-varying delays, for example, see [2–16] and references therein. On the other hand, as pointed out in [17–20], neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, and hence there is a distribution of propagation *

Corresponding author. Tel./fax: +44 1895 251686. E-mail addresses: [email protected] (Q. Song), [email protected] (Z. Wang).

0960-0779/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.10.044

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547

1539

delays over a period of time. In [17], a neural circuit has been designed with distributed delays, which solves a general problem of recognized patterns in time-dependent signal. In [18–20], the global stability, periodic solutions and boundedness have been investigated for neural networks with distributed delays. It is noticed that, although the signal propagation is sometimes instantaneous and can be modelled with discrete delays, it may also be distributed during a certain time period so that the distributed delays should be incorporated in the model. In other words, it is often the case that the neural network model possesses both discrete and distributed delays [21]. In view of the importance of both discrete and distributed delays in modelling neural networks, the dynamics analysis problem for neural networks with discrete and distributed delays has received some initial research attention [21– 26], and most results have been concerned with constant time delays. In [21], a two-neuron network model with multiple discrete and distributed delays has been studied, and local stability of the steady-state solutions and the oscillation around the steady-state solutions have been investigated. However, the results in [21] cannot be directly applied for general neural networks. In [22,23], the authors have considered the neural networks with discrete and distributed constant delays, and obtained several sufficient conditions to ensure the existence and global asymptotic stability of equilibrium point for the neural networks. In [24], the global exponential stability problem has been investigated for the neural networks with discrete and distributed constant delays, and two sufficient conditions have been given. In [25], the robust stability has been discussed for neural networks with discrete and distributed constant delays, and several sufficient conditions have been derived to ensure the existence of equilibrium point and also guarantee the global robust stability for the neural networks. Very recently, in [26], the authors have considered cellular neural networks with discrete and distributed time-varying delays. Unfortunately, the main results obtained in [26] have been based on the following assumptions: (1) the timevarying delays are continuously differentiable, (2) the derivatives of time delays are bounded, and (3) the activation functions are bounded and monotonically nondecreasing. It should be pointed out that, time delays can occur in an irregular fashion, and sometimes the time-varying delays are not differentiable. In this case, the methods developed in [21–26] may be difficult to be applied, and it is therefore necessary to further investigate the stability problem of neural networks with discrete and distributed time-varying delays under milder assumptions. Motivated by the above discussions, the objective of this paper is to study the asymptotic and exponential stability of neural networks with discrete and distributed time-varying delays by employing a new Lyapunov–Krasovskii functional. The obtained sufficient conditions do not require the differentiability of time-varying delays and are expressed in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. A simulation example is given to show the effectiveness and less conservatism of the proposed criteria. 1.1. Notations The notations are quite standard. Throughout this paper, Rn and Rn·m denote, respectively, the n-dimensional Euclidean space and the set of all n · m real matrices. The superscript ‘‘T’’ denotes matrix transposition. The notation X P Y (respectively, X > Y) means that X and Y are symmetric matrices, and that X  Y is positive semidefinite (respectively, positive definite). k Æ q k is the Euclidean norm in Rn. If A is a matrix, denote by kAk its operator norm, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i.e., kAk ¼ supfkAxk : kxk ¼ 1g ¼ kmax ðAT AÞ, where kmax(A) (respectively, kmin(A)) means the largest (respectively, smallest) eigenvalue of A. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

2. Model description and preliminaries In this paper, we consider the following model dxðtÞ ¼ DxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt  sðtÞÞÞ þ C dt

Z

t

f ðxðsÞÞ ds þ J

ð1Þ

trðtÞ

for t P 0, where x(t) = (x1(t), x2(t), . . . , xn(t))T 2 Rn is the state vector of the network at time t, n corresponds to the number of neurons, D = diag(d1, d2, . . . ,dn) > 0 is a positive diagonal matrix, A = (aij)n·n, B = (bij)n·n and C = (cij)n·n represent the connection weight matrix, the discretely delayed connection weight matrix and the distributively delayed connection weight matrix, respectively. f(x(t)) = (f1(x1(t)), f2(x2(t)), . . . ,fn(xn(t)))T denotes the neuron activation at time t. J = (J1, J2, . . . ,Jn)T 2 Rn is a constant external input vector. s(t) > 0 and r(t) > 0 denote the discrete time-varying delay and the distributed time-varying delay, respectively, and are assumed to satisfy 0 6 s(t) 6 s, 0 6 r(t) 6 r, where s and r are constants.

1540

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547

The initial condition associated with model (1) is xi ðsÞ ¼ ui ðsÞ;

i ¼ 1; 2; . . . n;

ð2Þ

where ui(s) is bounded and continuously differential on [q, 0], q = max{s, r}. Throughout this paper, we make the following assumptions: (H1) The activation functions are bounded. (H2) There exists a positive diagonal matrix F = diag(F1, F2, . . . , Fn) such that jfj ðv1 Þ  fj ðv2 Þj 6 F j jv1  v2 j for all v1, v2 2 R, j = 1, 2, . . . , n. Since activation functions are bounded, by employing the well-known Brouwer’s fixed point theorem, one can easily prove that there exists an equilibrium point for model (1). In the sequel we shall analyze the global asymptotic and exponential stability of the equilibrium point, which in turn implies the uniqueness of the equilibrium point. To simplify the stability analysis of model (1), we let x* be the equilibrium point of model (1), and shift the intended equilibrium point x* to the origin by letting y = x  x*, and then model (1) can be transformed into: Z t dyðtÞ ¼ DyðtÞ þ AgðyðtÞÞ þ Bgðyðt  sðtÞÞÞ þ C gðyðsÞÞ ds ð3Þ dt trðtÞ for t P 0, where gj ðy j ðtÞÞ ¼ fj ðy j ðtÞ þ xj Þ  fj ðxj Þ. It follows from assumption (H2) that jgj ðy j ðtÞÞj 6 F j jy j ðtÞj;

j ¼ 1; 2; . . . ; n:

Thus, for any positive diagonal matrix K, we can have gT ðyðtÞÞKgðyðtÞÞ 6 y T ðtÞF KFyðtÞ:

ð4Þ

Definition 1. [22,23] The equilibrium point 0 of model (3) is said to be globally asymptotically stable if it is locally stable in the sense of Lyapunov and globally attractive, where global attractivity means that every trajectory tends to the equilibrium point as t ! 1. Definition 2. The equilibrium point 0 of model (3) is said to be globally exponentially stable, if there exist two positive constants e > 0 and M > 0 such that every solution y(t) of model (3) satisfies   _ kyðtÞk 6 Meet sup kyðsÞk þ sup kyðsÞk q6s60

q6s60

for all t P 0. To obtain our main results, the following lemmas are necessary. Lemma 1. [25] Let a,b 2 Rn, P be a positive definite matrix, then 2aT b 6 aT P  1a + bT Pb. Lemma 2. [22] For any constant matrix W 2 Rm·m, WT = W > 0, scalar h > 0, vector function x:[0,h] ! Rm such that the integrations concerned are well defined, then Z h T Z h  Z h xðsÞ ds W xðsÞ ds 6 h xT ðsÞW xðsÞ ds: 0

0

0

Lemma 3. [22] Given constant matrices P, Q and R, where PT = P, QT = Q, then the linear matrix inequality (LMI) 

P RT

R Q

 <0

is equivalent to the following conditions Q > 0;

P þ RQ1 RT < 0:

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547

1541

3. Main results Let us first establish the delay-dependent criterion for the asymptotic stability of neural networks with discrete and distributed time-varying delays by using an LMI approach. Theorem 1. Under assumptions (H1) and (H2), the equilibrium point 0 of model (3) is globally asymptotically stable if there exist a symmetric positive definite matrix P, six positive diagonal matrices Yi > 0 (i = 1, 2, 3, 4, 5, 6), and matrices Q1, Q2, Xij (i, j = 1, 2, 3) such that the following two LMIs hold: 0 1 X 11 X 12 X 13 B T C X ¼ @ X 12 X 22 X 23 A > 0; ð5Þ 0

X T13

X T23

X1

B T B X2 B B XT B 3 B T B A Q1 B B X ¼ B BT Q1 B B CTQ B 1 B B 0 B B @ 0

X 33 X2

X3

QT1 A

X4

0

0

0

0

X5

0

0

0

0

Y 1

0

0

0

0

0

0

0

Y 2

0

0

0

0

0

0

QT2 A

QT2 B

0

0

0

0

0

0

0

Y 3

0

0

AT Q2

0

0

0

0

Y 4

0

B Q2

0

0

0

0

0

Y 5

C T Q2

0

0

0

0

0

0

T

0

QT1 B QT1 C

0

1

C QT2 C C C 0 C C C 0 C C C 0 C < 0; C 0 C C C 0 C C C 0 A

ð6Þ

Y 6

QT1 D

 DQ1 þ F ðY 1 þ r2 Y 3 þ Y 4 þ r2 Y 6 ÞF þ sX 11 þ X 13 þ X T13 , X2 ¼ P  QT1  DQ2 , X3 ¼ sX 12  X 13 þ where X1 ¼ T T X 23 , X4 ¼ Q2  Q2 þ sX 33 , and X5 ¼ F ðY 2 þ Y 5 ÞF þ sX 22  X 23  X T23 . Proof. Consider the following Lyapunov–Krasovskii functional candidate for model (3) as V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ; where V 1 ðtÞ ¼ y T ðtÞPyðtÞ; Z t Z t V 2 ðtÞ ¼ y_ T ðsÞX 33 y_ ðsÞ ds dn; V 3 ðtÞ ¼ r V 4 ðtÞ ¼

Z

t

t 0

Z

t

gT ðyðsÞÞðY 3 þ Y 6 ÞgðyðsÞÞ ds dn;

ð9Þ

n

tr

Z

ð8Þ

n

ts

Z

ð7Þ

n

uT Xu ds dn;

ð10Þ

nsðnÞ

where u ¼ ðy T ðnÞ y T ðn  sðnÞÞ y_ T ðsÞÞT . Evaluating the time derivative of V1(t) along the trajectories of model (3), we obtain dV 1 ðtÞ _ ¼ 2y T ðtÞP yðtÞ dt

 ¼ 2y T ðtÞP yðtÞ _ þ 2ðy T ðtÞQT1 þ y_ T ðtÞQT2 Þ _y ðtÞ  DyðtÞ þ AgðyðtÞÞ: Z t  þBgðyðt  sðtÞÞÞ þ C gðyðsÞÞ ds trðtÞ

_  2y T ðtÞQT1 DyðtÞ þ 2y T ðtÞQT1 AgðyðtÞÞ _  2y T ðtÞQT1 yðtÞ ¼ 2y T ðtÞP yðtÞ Z t gðyðsÞÞ ds þ 2y T ðtÞQT1 Bgðyðt  sðtÞÞÞ þ 2y T ðtÞQT1 C trðtÞ

þ 2y_ ðtÞQT2 AgðyðtÞÞ Z t þ 2y_ T ðtÞQT2 Bgðyðt  sðtÞÞÞ þ 2y_ T ðtÞQT2 C gðyðsÞÞ ds

 2y_

T

_ ðtÞQT2 yðtÞ

 2y_

T

ðtÞQT2 DyðtÞ

T

trðtÞ

1542

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547 T _  2y T ðtÞQT1 DyðtÞ þ y T ðtÞQT1 AY 1 6 2y T ðtÞðP  QT1  DQ2 ÞyðtÞ 1 A Q1 yðtÞ T þ gT ðyðtÞÞY 1 gðyðtÞÞ þ y T ðtÞQT1 BY 1 2 B Q1 yðtÞ T þ gT ðyðt  sðtÞÞÞY 2 gðyðt  sðtÞÞÞ þ y T ðtÞQT1 CY 1 3 C Q1 yðtÞ ! ! T Z t Z t T _ þ y_ T ðtÞQT2 AY 1 þ gðyðsÞÞ ds Y 3 gðyðsÞÞ ds  2y_ T ðtÞQT2 yðtÞ 4 A Q2 y_ ðtÞ trðtÞ

trðtÞ

T T þ gT ðyðtÞÞY 4 gðyðtÞÞ þ y_ T ðtÞQT2 BY 1 5 B Q2 y_ ðtÞ þ g ðyðt  sðtÞÞÞY 5 gðyðt  sðtÞÞÞ !T ! Z t Z t T _ þ y_ T ðtÞQT2 CY 1 yðtÞ þ C Q gðyðsÞÞ ds Y gðyðsÞÞ ds 6 2 6 trðtÞ

6y

trðtÞ

T T 1 T ðtÞð2QT1 D þ QT1 AY 1 1 A Q1 þ FY 1 F þ Q1 BY 2 B Q1 T T T QT1 CY 1 y ðtÞ 3 C Q1 þ FY 4 F ÞyðtÞ þ y ðtÞð2P  2Q1  2DQ2 Þ_ T T 1 T T T T 1 T _ y_ ðtÞð2Q2 þ Q2 AY 4 A Q2 þ Q2 BY 5 B Q2 þ QT2 CY 1 6 C Q2 ÞyðtÞ Z t gT ðyðsÞÞðY 3 þ Y 6 ÞgðyðsÞÞ ds: y T ðt  sðtÞÞðFY 2 F þ FY 5 F Þyðt  sðtÞÞ þ r trðtÞ

T

þ þ þ

In deriving the above inequalities, we have made use of Lemma 1, inequality (4) and Lemma 2. Calculating the time derivatives of V2(t), we get Z t dV 2 ðtÞ _ ds: ¼ sy_ T ðtÞX 33 y_ ðtÞ  y_ T ðsÞX 33 yðsÞ dt ts Similarly, computing the time derivatives of V3(t) and V4(t), we have Z t Z t dV 3 ðtÞ ¼r gT ðyðtÞÞðY 3 þ Y 6 ÞgðyðtÞÞ dn  r gT ðyðsÞÞðY 3 þ Y 6 ÞgðyðsÞÞ ds dt tr tr Z t 6 r2 y T ðtÞF ðY 3 þ Y 6 ÞFyðtÞ  r gT ðyðsÞÞðY 3 þ Y 6 ÞgðyðsÞÞ ds: dV 4 ðtÞ ¼ dt

Z

ð11Þ

ð12Þ

ð13Þ

trðtÞ t

ðy T ðtÞ y T ðt  sðtÞÞ tsðtÞ



¼ sðtÞ

yðtÞ yðt  sðtÞÞ

T 

y_ T ðsÞÞX ðy T ðtÞ

X 11 X T12

X 12 X 22

 2y T ðt  sðtÞÞX 23 yðt  sðtÞÞ þ

 Z

y T ðt  sðtÞÞ 

yðtÞ yðt  sðtÞÞ

y_ T ðsÞÞT ds

þ 2y T ðtÞX 13 yðtÞ

t

y_ T ðsÞX 33 y_ ðsÞ ds

tsðtÞ

6 y T ðtÞðsX 11 þ 2X 13 ÞyðtÞ þ 2y T ðtÞðsX 12  X 13 þ X T23 Þyðt  sðtÞÞ Z t þ y T ðt  sðtÞÞðsX 22  2X 23 Þyðt  sðtÞÞ þ y_ T ðsÞX 33 y_ ðsÞ ds:

ð14Þ

ts

It follows from inequalities (11)–(14) that dV ðtÞ T T 1 T 6 y T ðtÞð2QT1 D þ QT1 AY 1 1 A Q1 þ FY 1 F þ Q1 BY 2 B Q1 dt T 2 þ QT1 CY 1 3 C Q1 þ FY 4 F þ r F ðY 3 þ Y 6 ÞF þ sX 11 þ 2X 13 ÞyðtÞ   _ þ 2y T ðtÞ P  QT1  DQ2 yðtÞ   T T T 1 T 1 T y_ T ðtÞ 2QT2 þ QT2 AY 1 A Q 2 þ Q2 BY 5 B Q2 þ Q2 CY 6 C Q2 þ sX 33 y_ ðtÞ 4 þ 2y T ðtÞðsX 12  X 13 þ X T23 Þyðt  sðtÞÞ þ y T ðt  sðtÞÞðFY 2 F þ FY 5 F þ sX 22  2X 23 Þyðt  sðtÞÞ ¼ ðy T ðtÞ y_ T ðtÞ y T ðt  sðtÞÞÞX ðy T ðtÞ y_ T ðtÞ y T ðt  sðtÞÞÞT ; where

0

X1 B  X ¼ @ P  Q1  QT2 D sX T12  X T13 þ X 23

P  QT1  DQ2 X2

sX 12  X 13 þ X T23 0

0

F ðY 2 þ Y 5 ÞF þ sX 22  X 23  X T23

1 C A;

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547

1543

with T T T 1 T 1 T X1 ¼ QT1 D  DQ1 þ QT1 AY 1 1 A Q1 þ FY 1 F þ Q1 BY 2 B Q1 þ Q1 CY 3 C Q1

þ FY 4 F þ r2 F ðY 3 þ Y 6 ÞF þ sX 11 þ X 13 þ X T13 ; T T T 1 T 1 T X2 ¼ Q2  QT2 þ QT2 AY 1 4 A Q2 þ Q2 BY 5 B Q2 þ Q2 CY 6 C Q2 þ sX 33 :

It is easy to verify the equivalence of X < 0 and X* < 0 by using Lemma 3. Thus, from condition (6), we get dV ðtÞ <0 dt for all y(t) 5 0, which implies that the origin of model (3) is globally asymptotically stable. The proof is then completed. h Next, we are now in a position to discuss the exponential stability of model (3) as follows. Theorem 2. Under the conditions of Theorem 1, model (3) is globally exponentially stable and the exponential convergence rate index e can be estimated from the inequality ð15Þ

P < 0; where

0

P  QT1  ðD  eIÞQ2 P1 B P ¼ @ P  Q1  QT2 ðD  eIÞ P2 sX T12  X T13 þ X 23 0

1 sX 12  X 13 þ X T23 C 0 A; T 2se e F ðY 2 þ Y 5 ÞF þ sX 22  X 23  X 23

with T T T 1 T 1 T P1 ¼ ðD  eIÞQ1  QT1 ðD  eIÞ þ QT1 AY 1 1 A Q1 þ FY 1 F þ Q1 BY 2 B Q1 þ Q1 CY 3 C Q1

þ FY 4 F þ r2 e2re F ðY 3 þ Y 6 ÞF þ sX 11 þ X 13 þ X T13 ; T T T 1 T 1 T P2 ¼ Q2  QT2 þ QT2 AY 1 4 A Q2 þ Q2 BY 5 B Q2 þ Q2 CY 6 C Q2 þ sX 33 :

Proof. From X < 0, we have X* < 0. Thus, we can choose a sufficiently small constant e > 0 such that P < 0. Letting z(t) = eety(t), then model (3) can be transformed into the following model: Z t dzðtÞ ¼ ðD  eIÞzðtÞ þ eet Agðeet yðtÞÞ þ eet BgðeeðtsðtÞÞ zðt  sðtÞÞÞ þ eet C gðees zðsÞÞ ds dt trðtÞ

ð16Þ

for t P 0. Consider the following Lyapunov–Krasovskii functional candidate for model (16) as V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ; where V 1 ðtÞ ¼ zT ðtÞPzðtÞ; Z t Z t z_ T ðsÞX 33 z_ ðsÞ ds dn; V 2 ðtÞ ¼ n

ts

V 3 ðtÞ ¼ re2re V 4 ðtÞ ¼

Z

t 0

Z

Z

t

tr n

Z

ð17Þ ð18Þ

t

zT ðsÞF ðY 3 þ Y 6 ÞFzðsÞ ds dn;

ð19Þ

n

uT Xu ds dn;

ð20Þ

nsðnÞ

with u ¼ ðzT ðnÞ zT ðn  sðnÞÞ z_ T ðsÞÞT . Along the trajectories of model (16), we can obtain the time derivative of V1(t) as follows: dV 1 ðtÞ ¼ 2zT ðtÞP z_ ðtÞ dt  ¼ 2zT ðtÞP z_ ðtÞ þ 2ðzT ðtÞQT1 þ z_ T ðtÞQT2 Þ  z_ ðtÞ  ðD  eIÞzðtÞ þ eet Agðeet zðtÞÞ Z t  gðees zðsÞÞ ds þ eet BgðeeðtsðtÞÞ zðt  sðtÞÞÞ þ eet C trðtÞ

1544

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547

¼ 2zT ðtÞP z_ ðtÞ  2zT ðtÞQT1 z_ ðtÞ  2zT ðtÞQT1 ðD  eIÞzðtÞ þ 2eet zT ðtÞQT1 Agðeet zðtÞÞ Z t þ 2eet zT ðtÞQT1 Bgðeet zðt  sðtÞÞÞ þ 2eet zT ðtÞQT1 C gðees zðsÞÞ ds trðtÞ

 eIÞzðtÞ þ 2e z_ ðtÞQT2 Agðeet zðtÞÞ Z t þ 2eet z_ T ðtÞQT2 BgðeeðtsðtÞÞ zðt  sðtÞÞÞ þ 2eet z_ T ðtÞQT2 C gðees zðsÞÞ ds

 2_z

T

ðtÞQT2 z_ ðtÞ

 2_z

T

ðtÞQT2 ðD

et T

trðtÞ T 6 2zT ðtÞðP  QT1  ðD  eIÞQ2 Þ_zðtÞ  2zT ðtÞQT1 ðD  eIÞzðtÞ þ zT ðtÞQT1 AY 1 1 A Q1 zðtÞ T þ e2et gT ðeet zðtÞÞY 1 gðeet zðtÞÞ þ zT ðtÞQT1 BY 1 2 B Q1 zðtÞ T þ e2et gT ðeeðtsðtÞÞ zðt  sðtÞÞÞY 2 gðeeðtsðtÞÞ zðt  sðtÞÞÞ þ zT ðtÞQT1 CY 1 3 C Q1 zðtÞ ! ! T Z t Z t þ e2et gðees zðsÞÞ ds Y 3 gðees zðsÞÞ ds  2_zT ðtÞQT2 z_ ðtÞ trðtÞ

þ

trðtÞ

T z_ ðtÞQT2 AY 1 _ ðtÞ 4 A Q2 z 2et T eðtsðtÞÞ T

et

T þ e g ðe zðtÞÞY 4 gðeet zðtÞÞ þ z_ T ðtÞQT2 BY 1 _ ðtÞ 5 B Q2 z 2et T

zðt  sðtÞÞÞY 5 gðeeðtsðtÞÞ zðt  sðtÞÞÞ !T Z t Z T 1 T T 2et es þ z_ ðtÞQ2 CY 6 C Q2 z_ ðtÞ þ e gðe zðsÞÞ ds Y 6

þ e g ðe

trðtÞ

!

t es

gðe

zðsÞÞ ds

trðtÞ

T T 1 T 6 zT ðtÞð2QT1 ðD  eIÞ þ QT1 AY 1 1 A Q1 þ FY 1 F þ Q1 BY 2 B Q1 T T T þ QT1 CY 1 zðtÞ 3 C Q1 þ FY 4 F ÞzðtÞ þ z ðtÞð2P  2Q1  2ðD  eIÞQ2 Þ_ T T T 1 T 1 T þ z_ T ðtÞð2QT2 þ QT2 AY 1 zðtÞ 4 A Q2 þ Q2 BY 5 B Q2 þ Q2 CY 6 C Q2 Þ_ Z t þ zT ðt  sðtÞÞe2es F ðY 2 þ Y 5 ÞFzðt  sðtÞÞ þ re2re zT ðsÞðY 3 þ Y 6 ÞzðsÞ ds:

ð21Þ

trðtÞ

In deriving the above inequalities, we have made use of Lemma 1, inequality (4) and Lemma 2. Calculating the time derivatives of V2(t), V3(t) and V4(t), respectively, we have Z t dV 2 ðtÞ T ¼ s_z ðtÞX 33 z_ ðtÞ  z_ T ðsÞX 33 z_ ðsÞ ds: dt ts Z t Z t dV 3 ðtÞ ¼ re2re zT ðtÞF ðY 3 þ Y 6 ÞFzðtÞ dn  zT ðsÞF ðY 3 þ Y 6 ÞFzðsÞ ds dt tr tr Z t zT ðsÞF ðY 3 þ Y 6 ÞFzðsÞ ds: 6 r2 e2re zT ðtÞF ðY 3 þ Y 6 ÞFzðtÞ  re2re

ð22Þ

ð23Þ

trðtÞ

dV 4 ðtÞ 6 zT ðtÞðsX 11 þ 2X 13 ÞzðtÞ þ 2zT ðtÞðsX 12  X 13 þ X 23 Þzðt  sðtÞÞ dt Z t þ zT ðt  sðtÞÞðsX 22  2X 23 Þzðt  sðtÞÞ þ z_ T ðsÞX 33 z_ ðsÞ ds: ts

It follows from inequalities (21)–(24) that dV ðtÞ T T 1 T 6 zT ðtÞð2QT1 ðD  eIÞ þ QT1 AY 1 1 A Q1 þ FY 1 F þ Q1 BY 2 B Q1 dt T 2 2re þ QT1 CY 1 F ðY 3 þ Y 6 ÞF þ sX 11 þ 2X 13 ÞzðtÞ 3 C Q1 þ FY 4 F þ r e   T T þ 2z ðtÞ P  Q1  ðD  eIÞQ2 z_ ðtÞ   T T T 1 T 1 T _ ðtÞ þ z_ T ðtÞ 2QT2 þ QT2 AY 1 4 A Q2 þ Q2 BY 5 B Q2 þ Q2 CY 6 C Q2 þ sX 33 z þ 2zT ðtÞðsX 12  X 13 þ X 23 Þzðt  sðtÞÞ   þ zT ðt  sðtÞÞ e2se F ðY 2 þ Y 5 ÞF þ sX 22  2X 23 zðt  sðtÞÞ ¼ ðzT ðtÞ

z_ T ðtÞ

zT ðt  sðtÞÞÞPðzT ðtÞ z_ T ðtÞ zT ðt  sðtÞÞÞT ;

which indicates from P < 0 that dV ðtÞ 60 dt

ð24Þ

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547

1545

for all t P 0. Hence, V ðtÞ 6 V ð0Þ for all t P 0. From the definitions of Vi(t) (i = 1, 2, 3, 4), it is not difficult to obtain the following inequalities: V ðtÞ P kmin ðP ÞkzðtÞk2 for all t P 0, and V 1 ð0Þ 6 kmax ðP Þkzð0Þk2 6 kmax ðP Þ sup kzðsÞk2 ; q6s60 2

V 2 ð0Þ 6 s kmax ðX 33 Þ sup k_zðsÞk 6 s2 kmax ðX 33 Þ sup k_zðsÞk2 ; 2

s6s60

q6s60

V 3 ð0Þ 6 r3 e2re kmax ðFY 3 F þ FY 6 F Þ sup kzðsÞk2 6 r3 e2re kmax ðFY 3 F þ FY 6 F Þ sup kzðsÞk2 ; r6s60

q6s60

V 4 ð0Þ ¼ 0: Thus, V ð0Þ 6 ðkmax ðP Þ þ r3 e2re kmax ðFY 3 F þ FY 6 F ÞÞ sup kzðsÞk2 þ s2 kmax ðX 33 Þ sup k_zðsÞk2 q6s60

q6s60

6 að sup kzðsÞk þ sup k_zðsÞkÞ2 ; q6s60

ð25Þ

q6s60

where a = max{kmax(P) + r3e2rekmax(FY3F + FY6F), s2kmax(X33)}. Take M ¼ ! kzðtÞk 6 M



1=2

a kmin ðP Þ

, then

sup kzðsÞk þ sup k_zðsÞk q6s60

q6s60

for all t P 0. It follows from z(t) = eety(t) that et

kyðtÞk 6 Me

!

sup kyðsÞk þ sup k_y ðsÞk q6s60

q6s60

for all t P 0. Therefore, model (3) is globally exponentially stable and the exponential convergence rate index e can be estimated from (15). The proof is completed. h Remark 1. In [26], the sufficient conditions on the global asymptotic stability of model (1) have been obtained under the condition that s_ ðtÞ 6 h where h is constant. However, the presented results in this paper do not need the conditions that the time-varying delay is differentiable and the derivative is bounded. 4. An Example Consider a two-neuron neural network (3), where       0:9 0 1 1:7 1 0:6 ; B¼ ; D¼ ; A¼ 0:5 0:8 0 0:8 1:6 1 f1 ðxÞ ¼ f2 ðxÞ ¼ 0:1ðjx þ 1j  jx  1jÞ;

sðtÞ ¼ 0:15j sin tj;

 C¼

0:4 0:1

 0:3 ; 0:2

rðtÞ ¼ 0:1j cos tj:

Obviously, Assumptions (H1) and (H2) are satisfied with F = diag{0.2,0.2}, s = 0.15 and r = 0.1. By the Matlab LMI Control Toolbox, we find a solution to the LMIs in (5) and (6) as follows: 0 1 22:6263 0:4507 20:7321 1:2420 5:9191 0:1442 B 0:4507 22:8777 1:2422 19:8874 0:1024 5:7533 C B C B C B 20:7321 1:2422 25:4289 0:2049 10:2679 0:6423 C C; X ¼B B 1:2420 19:8874 0:2049 25:7924 0:5971 10:6097 C B C B C @ 5:9191 0:1024 10:2679 0:5971 18:7004 5:1656 A 0:1442

5:7533

0:6423

10:6097

5:1656

20:9842

1546

Q. Song, Z. Wang / Chaos, Solitons and Fractals 37 (2008) 1538–1547 1.5 y1 y2

1

y

0.5 0 0.5 1 1.5

0

5

10

15

20

25

30

t Fig. 1. State responses of y1(t) and y2(t).

 P¼

17:0549

2:6290

 ;



11:9708

1:0211

 ;



5:6441

1:8247



Q1 ¼ Q2 ¼ : 2:6290 20:2105 0:9970 14:9894 1:9911 6:5294       108:3976 0 70:9928 0 39:9180 0 ; Y2 ¼ ; Y3 ¼ ; Y1 ¼ 0 114:3013 0 70:8069 0 40:0133       29:1322 0 22:4117 0 41:1904 0 ; Y5 ¼ ; Y6 ¼ : Y4 ¼ 0 34:2721 0 24:5697 0 41:0158 Therefore, by Theorems 1 and 2, we know that model (3) is globally asymptotically and exponentially stable. Moreover, from (15), we can also get that the exponential convergence index e = 0.0937. Fig. 1 shows a numerical simulation of the network with an initial state (/1(s), /2(s))T = (sin(6s), cos(6s))T, s 2 [0.15,0]. The simulation results verifies the convergence of the network state. It should be pointed out that the condition in [26] cannot be applied to this example since it requires the differentiability of the time-varying delays.

5. Conclusions In this paper, the global asymptotic and exponential stability have been investigated for a class of neural networks with both discrete and distributed time-varying delays. Two delay-dependent sufficient conditions in LMIs form have been obtained for global asymptotic and exponential stability of such systems by using appropriate Lyapunov–Krasovskii functional and linear matrix inequality (LMI) technique. The proposed results generalize and improve the earlier publications, and does not require the monotonicity of the activation functions and the differentiability of the discrete and distributed time-varying delays. An example with simulation has been provided to demonstrate the effectiveness and less conservatism of the obtained results.

Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant 50608072.

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