on global exponential stability of delayed cellular neural networks with time-varying delays

Applied Mathematics and Computation 162 (2005) 679–686

www.elsevier.com/locate/amc

On global exponential stabilityof delayed cellular neural networks

with time-varying delays

Qiang Zhang a,b,*, Xiaopeng Wei a, Jin Xu a

a Advanced Design Technology Center, Dalian University, Dalian 116622, Chinab School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China

Abstract

A new sufficient condition has been presented ensuring the global exponential sta-

bility of cellular neural networks with time-varying delays by using an approach based

on delay differential inequality combining with Young inequality. The results estab-

lished here extend those earlier given in the literature. Compared with the method of

Lyapunov functionals as in most previous studies, our method is simpler and more

effective for stability analysis.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Global exponential stability; Cellular neural networks; Time-varying delays; Delay

differential inequality; Lyapunov functionals

1. Introduction

Cellular neural networks (CNNs) were introduced by Chua and Yang in

1988 [1]. CNNs have been extensively discussed in the past decade. CNNs with

time delay (DCNNs) proposed in [2] have found applications in many areasincluding classification of patterns and processing of moving images. Such

applications rely on the existence of an equilibrium point or of a unique

* Corresponding author. Address: Advanced Design Technology Center, Dalian University,

Dalian 116622, China.

E-mail address: [email protected] (Q. Zhang).

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.01.004

mail to: [email protected]

680 Q. Zhang et al. / Appl. Math. Comput. 162 (2005) 679–686

equilibrium point, and its stability. To the best of our knowledge, cellular

neural networks with time-varying delays are seldom considered [3–17].However, in practice, time delays are usually variable, and sometimes vary

violently with time. Therefore, the studies of neural networks with time-varying

delays are more important and actual than those with constant delays. Fur-

thermore, in designing a neural network, one is not only interested in the

stability of the network, but also in performance. In particular, it is often

desired that a neural network converges fast enough in order to achieve fast

response. The aim of this brief is to provide some new results on global

exponential stability for cellular neural networks with time-varying delays. Theresults in the earlier references emerge as special cases of the main results given

here.

2. Global exponential convergence analysis

The dynamic behavior of a continuous time DCNN can be described by the

following state equations:

x0iðtÞ ¼ �cixiðtÞ þXn

j¼1

aijfjðxjðtÞÞ þXn

j¼1

bijfjðxjðt � sjðtÞÞÞ þ Ji;

i ¼ 1; 2; . . . ; n ð1Þ

or equivalently

x0ðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt � sðtÞÞÞ þ J ; ð2Þ

where n corresponds to the number of units in a neural network;

xðtÞ ¼ ½x1ðtÞ; . . . ; xnðtÞ�T 2 Rn corresponds to the state vector at time t;f ðxðtÞÞ ¼ ½f1ðx1ðtÞÞ; . . . ; fnðxnðtÞÞ�T 2 Rn denotes the activation function of the

neurons; f ðxðt� sðtÞÞÞ ¼ ½f1ðx1ðt� s1ðtÞÞÞ; . . . ; fnðxnðt� snðtÞÞÞ�T 2 Rn;C;A;B; Jare constant matrices; C ¼ diagðci > 0Þ (a positive diagonal matrix) represents

the rate with which the ith unit will reset its potential to the resting state in

isolation when disconnected from the network and external inputs. A ¼ faijg isreferred to as the feedback matrix, B ¼ fbijg represents the delayed feedback

matrix, while J ¼ ½J1; . . . ; Jn�T is an external bias vector, sjðtÞ is the transmis-

sion delay along the axon of the jth unit and satisfies 06 sjðtÞ6 s (s is a

constant). The activation function fi (i ¼ 1; 2; . . . ; n) satisfies the following

condition

(H) Each fi is bounded continuous and satisfies

jfiðn1Þ � fiðn2Þj6 Lijn1 � n2j

for each n1; n2 2 R, n1 6¼ n2.

Q. Zhang et al. / Appl. Math. Comput. 162 (2005) 679–686 681

This type of activation functions is clearly more general than both the usual

sigmoid activation functions in Hopfield networks and the piecewise linearfunction (PWL): fiðxÞ ¼ 1

2ðjxþ 1j � jx� 1jÞ in standard cellular networks [1].

Suppose that the system (1) is supplemented with initial conditions of the

form

xiðsÞ ¼ /iðsÞ; s 2 ½�s; 0�; i ¼ 1; 2; . . . ; n

in which /iðsÞ is continuous for s 2 ½�s; 0�, and Eq. (1) has an equilibrium

point x� ¼ ðx�1; x�2; . . . ; x�nÞ. We denote

k/ � x�k ¼ sup�s6 s6 0

Xn

j¼1

j/jðsÞ"

� x�j jr

#1=r

:

We say that an equilibrium point x� ¼ ðx�1; x�2; . . . ; x�nÞ is globally exponen-

tially stable if there exist constants � > 0 and M P 1 such that

kxðtÞ � x�k6Mk/ � x�ke��t; tP 0:

In order to simplify the proofs and compare our results, we present some

lemmas as follows.

Lemma 1 (Young inequality [18]). Assume that a > 0, b > 0, p > 1, 1p þ 1

q ¼ 1,then the following inequality:

ab61

pap þ 1

qbq

holds.

Lemma 2 (Halanay inequality[19]). Let a and b be constants with 0 < b < a.Let xðtÞ be a continuous nonnegative function on tP t0 � s satisfying inequality(3) for tP t0.

x0ðtÞ6 � axðtÞ þ b�xðtÞ; ð3Þ

where �xðtÞ¼def supt�s6 s6 tfxðsÞg. Then

xðtÞ6�xðt0Þe�rðt�t0Þ; ð4Þ

where r is a bound on the exponential convergence rate and is the unique positivesolution of

r ¼ a � bers: ð5Þ

Theorem. Assume that there exist real constants fij, gij and positive constantsci > 0, pP 1, i; j ¼ 1; 2; . . . ; n such that


min16 i6 n

pci

(�Xn

j¼1

cjciLijajijpð1�fjiÞ

�þ ðp � 1ÞLjjaijj

pfijp�1 þ ðp � 1ÞLjjbijj

pgijp�1

�)

> max16 i6 n

Xn

j¼1

cjciLijbjijpð1�gjiÞ

( ); ð6Þ

then the equilibrium point x� of system (1) is globally exponentially stable.

Proof. Let yiðtÞ ¼ xiðtÞ � x�i , define a function

V ðtÞ ¼Xn

i¼1

cijyiðtÞjp:

Calculating and estimating the upper right derivative DþV of V along the

solution of (1) as follows:

DþV ðtÞ ¼Xn

i¼1

cipjyiðtÞjp�1

signðyiðtÞÞy0iðtÞ

¼Xn

i¼1

cipjyiðtÞjp�1

signðyiðtÞÞ"� ciyiðtÞ þ

Xn

j¼1

aij fjðyjðtÞ�

þ x�j Þ � fjðx�j Þ

þXn

j¼1

bij fjðyjðt�

� sjðtÞÞ þ x�j Þ � fjðx�j Þ#

6

Xn

i¼1

cip

"� cijyiðtÞjp þ

Xn

j¼1

jaijjLjjyiðtÞjp�1jyjðtÞj

þXn

j¼1

jbijjLjjyiðtÞjp�1jyjðt � sjðtÞÞj#

¼Xn

i¼1

cip

(� cijyiðtÞjp þ

Xn

j¼1

LjjyjðtÞkaijj1�fij jaijjfijp�1jyiðtÞj

� �p�1

þXn

j¼1

Ljjyjðt � sjðtÞÞkbijj1�gij jbijjgijp�1jyiðtÞj

� p�1): ð7Þ� �

Let a ¼ jyjðtÞkaijj1�fij , b ¼ jaijjfijp�1jyiðtÞj

p�1

, by Lemma 1, we have

jyjðtÞkaijj1�fij jaijjfijp�1jyiðtÞj

� �p�1

61

pjaijjpð1�fijÞjyjðtÞjp þ

p � 1

pjaijj

pfijp�1jyiðtÞjp: ð8Þ

Similarly, let a ¼ jyjðt � sjðtÞÞkbijj1�gij , b ¼ jbijjgijp�1jyiðtÞj

� p�1

, by Lemma 1, we

get


jyjðt � sjðtÞÞkbijj1�gij jbijjgijp�1jyiðtÞj

� p�1

61

pjbijjpð1�gijÞjyjðt � sjðtÞÞjp þ

p � 1

pjbijj

pgijp�1jyiðtÞjp: ð9Þ

Substituting Eqs. (8) and (9) into Eq. (7), we obtain

DþV ðtÞ6Xn

i¼1

cip

(� cijyiðtÞjp þ

Xn

j¼1

1

pLjjaijjpð1�fijÞjyjðtÞjp

þXn

j¼1

p � 1

pLjjaijj

pfijp�1jyiðtÞjp þ

Xn

j¼1

1

pLjjbijjpð1�gijÞjyjðt � sjðtÞÞjp

þXn

j¼1

p � 1

pLjjbijj

pgijp�1jyiðtÞjp

)

¼Xn

i¼1

ci

"� pci þ

Xn

j¼1

ðp � 1ÞLjjaijjpfijp�1 þ

Xn

j¼1

ðp � 1ÞLjjbijjpgijp�1

þXn

j¼1


#jyiðtÞjp þ

Xn

i¼1

ciXn

j¼1


" #jyiðt � siðtÞÞjp

6 � min16 i6 n

pci

(�Xn

j¼1


�þ ðp � 1ÞLjjaijj

pfijp�1

þ ðp � 1ÞLjjbijjpgijp�1

�)V ðtÞ þ max

16 i6 n

Xn

j¼1


( )V ðtÞ:

According to Lemma 2, we obtain

cminkxðtÞ � x�kp 6 V ðtÞ6 V ðt0Þ expð�rðt � t0ÞÞ:

It follows from above that

kxðtÞ � x�k6 c1=pmax

c1=pmin

k/ � x�k exp

�� rpt�:

Therefore, the proof is completed. h

Corollary 1. If there exist constants ci > 0 such that

min16 i6 n

ci

(�Xn

j¼1

cjciLijajij

� �)> max

16 i6 n

Xn

j¼1

cjciLijbjij

( );


Proof. Taking p ¼ 1 in theorem above, then we can easily obtain Corollary

1. h


Corollary 2. If there exist constants ci > 0 such that

min16 i6 n

2ci �Xn

j¼1

cjciLijajij þ Lj jaijj þ jbijj

�� ( )> max

16 i6 n

Xn

j¼1

cjciLijbjij

( );


Proof. It is easy to check that the inequality (6) is satisfied by taking p ¼ 2,

fij ¼ gij ¼ 0:5, and hence the theorem implies Corollary 2. h

Remark 1. If we let ci ¼ 1, i ¼ 1; 2; . . . ; n, then Corollarys 1 and 2 correspondto theorems 1 and 2 in [6], respectively. That is, our theorem includes the main

results in [6] as special cases.

Remark 2. For the pure-delay network model of (1) (i.e., aij ¼ 0; 8i; j), if

Li ¼ ci ¼ 1, it follows from Corollary 1 above that the inequalityPnj¼1

cjcijbjij < 1, i ¼ 1; 2; . . . ; n ensures global exponential stability of the

equilibrium point of (1). This coincides with the result in [7]. Furthermore,

according to the property of M-matrix [8], we can conclude that the conditionPnj¼1

cjcijbjij < 1 yields qðjBjÞ < 1, where qð Þ denotes spectral radius and jBj

denotes absolute-value matrix given by jBj ¼ ðjbijjÞn�n. This is consistent with

the main result in [9].

Remark 3. In [10–12], by constructing Lyapunov functional, some results onthe global asymptotic stability of (1) are presented. Different from our results,

all of those results require that the delay function sjðtÞ be differentiable. Thus,

conditions given in [10–12] are more restrictive and conservative.

Remark 4. Young inequality was first used by Cao [13]. However, the results in

[13] only hold for constant delays.

3. Examples

In this section, we will give some examples to show the validity of our re-

sults.

Example 1. Consider a delayed neural network with bounded time-varying

delays

x0ðtÞ ¼ �xðtÞ þ Af ðxðtÞÞ þ Bf ðxðt � sðtÞÞÞ; ð10Þ


where

C ¼ I ; A ¼ 0:1 �0:10:1 0:1

�; B ¼ 0:5 �0:1

0:1 0:5

�
and the activation function is PWL: fiðxÞ ¼ 1
2ðjxþ 1j � jx� 1jÞ. Clearly, fi

satisfies the assumption (H) with Li ¼ 1, i ¼ 1; 2. One can easily check that

min16 i6 n

ci

(�Xn

j¼1

cjciLijajij

� �)¼ 0:8 > max

16 i6 n

Xn

j¼1

cjciLijbjij

( )¼ 0:6:

Therefore, by Corollary 1, the equilibrium point of (10) is globally exponen-tially stable. By a simple computation, we can easily seen that the matrix

Aþ AT ¼ 0:2 0

0 0:2

�is not negative semidefinite. Thus the condition in [14]

does not hold. For this example, our result is less restrictive than that given in

[14].

Example 2. Consider the following delayed neural networks

x0ðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt � sðtÞÞÞ; ð11Þ
where
C ¼ 1:5 0

0 1:5

�; A ¼ 0:1 �0:1

0:2 0:1

�; B ¼ 1 0:2

�0:3 0:2

�

and the activation function is described by: fiðxÞ ¼ tanhðxÞ. Clearly, fi satisfiesthe assumption (H) with Li ¼ 1, i ¼ 1; 2. One can check that

min16 i6 n

ci

(�Xn

j¼1

cjciLijajij

� �)¼ 1:2 < max

16 i6 n

Xn

j¼1

cjciLijbjij

( )¼ 1:3;

min16 i6 n

2ci

(�Xn

j¼1

cjciLijajij

�þ Ljðjaijj þ jbijjÞ

�)

¼ max16 i6 n

Xn

j¼1

cjciLijbjij

( )¼ 1:3:

Therefore, the main results in [6] do not hold for this example. On the other

hand, if we let p ¼ 3, fij ¼ gij ¼ 0:5, c1 ¼ c2 ¼ 1 in theorem above, one can

verify that

min16 i6 n

3ci

(�Xn

j¼1

jajij

þ 2ðjaijj þ jbijjÞ�)

¼ 1:4 > max16 i6 n

Xn

j¼1

jbjij( )

¼ 1:3:

Hence, the equilibrium point of Eq. (11) is globally exponentially stable.


Acknowledgements

The project was supported by the National Natural Science Foundation of

China (Grant nos. 60174037, 50275013).

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on global exponential stability of delayed cellular neural networks with time-varying delays

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