Available online at www.sciencedirect.com

Chaos, Solitons and Fractals 36 (2008) 985–990

www.elsevier.com/locate/chaos

Delay-dependent exponential stability criteriafor non-autonomous cellular neural networks

with time-varying delays q

Qiang Zhang *, Xiaopeng Wei, Jin Xu

Liaoning Key Lab of Intelligent Information Processing, Dalian University, Dalian 116622, China

Accepted 10 July 2006

Abstract

Delay-dependent exponential stability of non-autonomous cellular neural networks with delays is considered in thispaper. Based on the differential inequality technique as well as a fact about Young inequality, some new sufficient con-ditions are given for global exponential stability of non-autonomous cellular neural networks with delays. The condi-tions rely on the size of time-delay. Since the results presented here do not require the differentiability of variable delay,they are less conservative than those established in the earlier references. An example is given to illustrate the applica-bility of these conditions.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Since cellular neural networks with delay (DCNNs) were first introduced in [16], they have been successfully appliedin different areas such as classification of patterns and processing of moving images. In DCNN applications, stabilityproperty plays an important role [17]. The stability of DCNNs has attracted wide interests from many authors in recentyears, see for example, [1–8,10–14,16–24] and references cited therein. To the best of our knowledge, few discussionshave been held on the stability of non-autonomous cellular neural networks with delay. In this paper, based on a delaydifferential inequality, several new criteria for global exponential stability of non-autonomous cellular neural networkswith delay are obtained. The criteria are related on the size of delay. Compared with the earlier results, our results areless restrictive. An example is illustrated to show the efficiency of the criteria.

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.07.034

q This work is supported by the National Natural Science Foundation of China (Grant Nos. 60403001, 50575026), by the Programfor Liaoning Excellent Talents in University, by the Program for Study of Science of the Educational Department of LiaoningProvince, by the Program for Dalian Science and Technology and by Dalian Youth Foundation.

* Corresponding author.E-mail address: zhangq30@yahoo.com (Q. Zhang).

986 Q. Zhang et al. / Chaos, Solitons and Fractals 36 (2008) 985–990

2. Preliminaries

The dynamic behavior of a continuous time non-autonomous neural networks with delay can be described by thefollowing state equations:

x0iðtÞ ¼ �ciðtÞxiðtÞ þXn

j¼1

aijðtÞfjðxjðtÞÞ þXn

j¼1

bijðtÞfjðxjðt � sjðtÞÞÞ þ I iðtÞ; i ¼ 1; 2; . . . ; n ð1Þ

or

x0ðtÞ ¼ �CðtÞxðtÞ þ AðtÞf ðxðtÞÞ þ BðtÞf ðxðt � sðtÞÞÞ þ IðtÞ;

where n corresponds to the number of units in a neural networks; xi(t) corresponds to the state vector at time t;f(x(t)) = [f1(x1(t)), . . . , fn(xn(t))]T 2 Rn denotes the activation function of the neurons; A(t) = [aij(t)]n·n is referred to asthe feedback matrix, B(t) = [bij(t)]n·n represents the delayed feedback matrix, while Ii(t) is a external bias vector at timet, sj(t) is the transmission delay along the axon of the jth unit and satisfies 0 6 si(t) 6 s.

Throughout this paper, we will assume that the real valued functions ci(t) > 0, aij(t), bij(t), Ii(t) are continuous func-tions. Denote D+ as the upper right Dini derivative. For any continuous function f : R! R, the upper right Dini deriv-ative of f(t) is defined as

Dþf ðtÞ ¼ limd!0þ

supf ðt þ dÞ � f ðtÞ

d:

The activation functions fi, i = 1,2, . . . ,n are assumed to satisfy the following conditions:

ðHÞ jfiðn1Þ � fiðn2Þj 6 Lijn1 � n2j; 8n1; n2:

This type of activation functions is clearly more general than both the usual sigmoid activation functions and the piece-wise linear function (PWL): fiðxÞ ¼ 1

2ðjxþ 1j � jx� 1jÞ which is used in [7].

The initial conditions associated with system (1) are of the form

xiðsÞ ¼ /iðsÞ; s 2 ½�s; 0�;

in which /i(s) are continuous for s 2 [�s, 0].

Lemma 1 (Young inequality [9]). Assume that a > 0, b > 0, p > 0, q > 0 p + q = 1, then the following inequality:

apbq6 paþ qb;

holds.

Lemma 2. Let ui(t) be a continuous nonnegative functions on t P t0 � s satisfying inequality (2) for t P t0.

DþuiðtÞ 6Xn

j¼1

aijðtÞupi ðtÞuq

j ðtÞ þXn

j¼1

bijðtÞupi ðtÞuj

qðtÞ; i ¼ 1; 2; . . . ; n; ð2Þ

where aii(t) is a continuous function, aij(t)(i 5 j) and bij(t) are nonnegative continuous functions. �ujðtÞ ¼def

supt�s6s6tfujðsÞg.p and q are nonnegative constants satisfying p + q = 1. If there exists a continuous nonnegative function c(t) such that

cðtÞ þ aiiðtÞ þXn

j¼1;j6¼i

aijðtÞ þ pXn

j¼1

bijðtÞ þ qXn

j¼1

bijðtÞ exp

Z t

t�scðsÞds

� �< 0;

for all i and t P t0, then, we have uiðtÞ 6 �uðt0Þ exp �R t

t0cðsÞds

� �, "t P t0, where �uðt0Þ ¼

Pni¼1supt0�s6s6t0 juiðsÞj.

Proof. The proof is similar to that in [18]. Let vðtÞ ¼ �uðt0Þ exp �R t

t0cðsÞds

� �and wi(t) = ui(t) � v(t), then we can easily

observe that

(1) vi(t) is nonincreasing on (t0,+1);(2) wi(t) 6 0 for t 2 [t0 � s, t0]

holds for all i = 1,2, . . . ,n. We claim that wi(t) 6 0 for "i = 1,2, . . . ,n, t P t0. Otherwise, due to the fact that the functionwi(t) is continuous, there must exist k 2 {1,2, . . . ,n} and t1 > t0 such that

Q. Zhang et al. / Chaos, Solitons and Fractals 36 (2008) 985–990 987

wkðt1Þ ¼ 0; ð3ÞDþwkðt1ÞP 0; ð4ÞwjðsÞ 6 0 for j ¼ 1; 2; . . . ; n; s 2 ½t0 � s; t1�: ð5Þ

On the other hand, by inequality (2) and the inequality [15] DþðF 1 þ F 2Þ ¼ DþF 1 þ F 02 where F 02 denotes the derivativeof F2, we have

Dþwkðt1Þ ¼ Dþukðt1Þ � v0ðt1Þ

6

Xn

j¼1

akjðt1Þupkðt1Þuq

j ðt1Þ þXn

j¼1

bkjðt1Þupkðt1Þ�uq

j ðt1Þ þ �uðt0Þ exp �Z t1

t0

cðsÞds� �

cðt1Þ

¼ akkðt1Þukðt1Þ þXn

j¼1;j 6¼k

akjðt1Þupkðt1Þuq

j ðt1Þ þXn

j¼1

bkjðt1Þupkðt1Þ�uq

j ðt1Þ þ vðt1Þcðt1Þ: ð6Þ

By Lemma 1, we get

Dþwkðt1Þ 6 akkðt1Þukðt1Þ þXn

j¼1;j 6¼k

akjðt1Þðpukðt1Þ þ qujðt1ÞÞ þXn

j¼1

bkjðt1Þðpukðt1Þ þ q�ujðt1ÞÞ þ vðt1Þcðt1Þ

¼ ðakkðt1Þ þ cðt1ÞÞvðt1Þ þ pXn

j¼1;j6¼k

ðakjðt1Þ þ bkjðt1ÞÞvðt1Þ þ pbkkðt1Þvðt1Þ

þ qXn

j¼1;j 6¼k

akjðt1Þujðt1Þ þXn

j¼1

bkjðt1Þ�ujðt1Þ( )

6 akkðt1Þ þ cðt1Þ þ pXn

j¼1;j 6¼k

ðakjðt1Þ þ bkjðt1ÞÞ þ pbkkðt1Þ( )

vðt1Þ

þ qXn

j¼1;j6¼k

akjðt1Þvðt1Þ þXn

j¼1

bkjðt1Þ�ujðt1Þ( )

: ð7Þ

Since

�ujðt1Þ ¼ supt1�s6s6t1

ujðsÞ 6 supt1�s6s6t1

vðsÞ 6 vðt1 � sÞ ¼ �uðt0Þ exp �Z t1�s

t0

cðsÞds� �

¼ vðt1Þ exp

Z t1

t1�scðsÞds

� �; ð8Þ

substituting (8) into (7), we obtain

Dþwkðt1Þ 6 akkðt1Þ þ cðt1Þ þ pXn

j¼1;j 6¼k

ðakjðt1Þ þ bkjðt1ÞÞ þ pbkkðt1Þ( )

vðt1Þ

þ qXn

j¼1;j6¼k

akjðt1Þvðt1Þ þXn

j¼1

bkjðt1Þ exp

Z t1

t1�scðsÞds

� �vðt1Þ

( )

¼ akkðt1Þ þ cðt1Þ þ pXn

j¼1;j6¼k

ðakjðt1Þ þ bkjðt1ÞÞ þ pbkkðt1Þ(

þ qXn

j¼1;j 6¼k

akjðt1Þ þXn

j¼1

bkjðt1Þ exp

Z t1

t1�scðsÞds

� � !)vðt1Þ

< 0: � �

This contradicts (4). Thus, uiðtÞ 6 �uðt0Þ exp �

R tt0

cðsÞds , for all t P t0. This completes the proof. h

3. Global exponential stability analysis

In this section, we will use the above lemma to establish the exponential stability of system (1). Consider two solu-tions x(t) and z(t) of system (1) for t > 0 corresponding to arbitrary initial values x(s) = /(s) and z(s) = u(s) fors 2 [�s, 0]. Let yi(t) = xi(t) � zi(t), then we have

988 Q. Zhang et al. / Chaos, Solitons and Fractals 36 (2008) 985–990

y0iðtÞ ¼ �ciðtÞyiðtÞ þXn

j¼1

aijðtÞðfjðxjðtÞÞ � fjðzjðtÞÞÞ þXn

j¼1

bijðtÞðfjðxjðt � sjðtÞÞÞ � fjðzjðt � sjðtÞÞÞÞ: ð9Þ

Set gj(yj(t)) = fj(yj(t) + zj(t)) � fj(zj(t)), one can rewrite Eq. (9) as

y0iðtÞ ¼ �ciðtÞyiðtÞ þXn

j¼1

aijðtÞgjðyjðtÞÞ þXn

j¼1

bijðtÞgjðyjðt � sjðtÞÞÞ: ð10Þ

Note that the functions fj satisfy the hypothesis (H), that is,

jgiðn1Þ � giðn2Þj 6 Lijn1 � n2j; 8n1; n2;

gið0Þ ¼ 0: ð11Þ

By Eq. (10), we have

DþjyiðtÞj 6 �ciðtÞjyiðtÞj þXn

j¼1

jaijðtÞjLjjyjðtÞj þXn

j¼1

jbijðtÞjLjjyjðt � sjðtÞÞj:

Theorem 1. Eq. (1) is globally exponentially stable if there exist real constants r P 1, e > 0, di > 0 (i = 1,2, . . . , n) such that

e� rciðtÞ þXn

j¼1

rjaijðtÞjLjdi

dj

� �1r

þXn

j¼1

ðr � 1ÞjbijðtÞjLjdi

dj

� �1r

þXn

j¼1

jbijðtÞjLjdi

dj

� �1r

ees < 0: ð12Þ

Proof. Let uiðtÞ ¼ dir jyiðtÞj

r. Calculate the upper right Dini derivative D+ui of ui(t) along the trajectories of Eq. (10), weget

DþuiðtÞ ¼ dijyiðtÞjr�1DþjyiðtÞj 6 dijyiðtÞj

r�1 �ciðtÞjyiðtÞj þXn

j¼1

jaijðtÞjLjjyjðtÞj þXn

j¼1

jbijðtÞjLjjyjðt � sjðtÞÞj( )

¼ �diciðtÞjyiðtÞjr þXn

j¼1

jaijðtÞjLjdijyiðtÞjr�1jyjðtÞj þ

Xn

j¼1

jbijðtÞjLjdijyiðtÞjr�1jyjðt � sjðtÞÞj

6 �rciðtÞdi

rjyiðtÞj

r þXn

j¼1

rjaijðtÞjLjdi

dj

� �1r di

rjyiðtÞj

r� �r�1

r dj

rjyjðtÞj

r� �1

r

þXn

j¼1

rjbijðtÞjLjdi

dj

� �1r di

rjyiðtÞj

r� �r�1

r dj

rj�yjðtÞjr

� �1r

¼ �rciðtÞuiðtÞ þXn

j¼1

rjaijðtÞjLjdi

dj

� �1r

ðuiðtÞÞðr�1Þ=rðujðtÞÞ1=r þXn

j¼1

rjbijðtÞjLjdi

dj

� �1r

ðuiðtÞÞðr�1Þ=rð�ujðtÞÞ1=r:

Let p ¼ r�1r , q ¼ 1

r and c(t) = e. It follows from Lemma 2 that the inequality

e� rciðtÞ þXn

j¼1

rjaijðtÞjLjdi

dj

� �1r

þXn

j¼1

ðr � 1ÞjbijðtÞjLjdi

dj

� �1r

þXn

j¼1

jbijðtÞjLjdi

dj

� �1r

ees < 0;

implies that system (1) is globally exponentially stable. The proof is completed. h

By applying Theorem 1, we can easily obtain the following corollaries:

Corollary 1. Eq. (1) is globally exponentially stable if there exist real constants r P 1, e > 0 such that

e� rciðtÞ þXn

j¼1

rjaijðtÞjLj þXn

j¼1

ðr � 1ÞjbijðtÞjLj þXn

j¼1

jbijðtÞjLjees < 0: ð13Þ

Corollary 2. Eq. (1) is globally exponentially stable if there exist real constants r P 1, e > 0, di > 0 (i = 1,2, . . . , n) such

that

Q. Zhang et al. / Chaos, Solitons and Fractals 36 (2008) 985–990 989

e� ciðtÞ þXn

j¼1

jaijðtÞjLj þXn

j¼1

jbijðtÞjLjees < 0: ð14Þ

Remark. In [11], by constructing Lyapunov functional Jiang et.al obtained several sufficient conditions ensuring globalexponential stability for non-autonomous cellular neural networks with delays. The conditions require that the timedelay be continuously differentiable. However, our conditions given here do not impose this restriction on delay. Hence,our conditions are less restrictive than those in [11].

4. An illustrative example

In this section, we will give an example showing the effectiveness of the conditions given here.

Example. Consider the following non-autonomous cellular neural networks with delay:

x01ðtÞ ¼ �c1ðtÞx1ðtÞ þ a11ðtÞf ðx1ðtÞÞ þ a12ðtÞf ðx2ðtÞÞþ b11ðtÞf ðx1ðt � s1ðtÞÞÞ þ b12ðtÞf ðx2ðt � s2ðtÞÞÞ;

x02ðtÞ ¼ �c2ðtÞx2ðtÞ þ a21ðtÞf ðx1ðtÞÞ þ a22ðtÞf ðx2ðtÞÞþ b21ðtÞf ðx1ðt � s1ðtÞÞÞ þ b22ðtÞf ðx2ðt � s2ðtÞÞÞ; ð15Þ

where the activation function is described by a PWL function fi(x) = 0.5(jx + 1j � jx � 1j). Clearly, fi(x) satisfy hypoth-esis (H) above, with L1 = L2 = 1. For model (15), taking

c1ðtÞ ¼ 6� sin t; c2ðtÞ ¼ 6þ cos t;

a11ðtÞ ¼ 1� sin t; a12ðtÞ ¼ sin t;

a21ðtÞ ¼ cos t; a22ðtÞ ¼ 1þ cos t;

b11ðtÞ ¼ sin t; b12ðtÞ ¼ sin t;

b21ðtÞ ¼ cos t; b22ðtÞ ¼ cos t;

s1ðtÞ ¼ s2ðtÞ ¼ cos2 t:

then we have 0 6 s(t) 6 s = 1. In Corollary 2, choose e = 0.5, one can easily check that

e� c1ðtÞ þ ja11ðtÞj þ ja12ðtÞj þ e0:5ðjb11ðtÞj þ jb12ðtÞjÞ ¼ �4:5þ 4:2974j sin tj < 0;

e� c2ðtÞ þ ja21ðtÞj þ ja22ðtÞj þ e0:5ðjb21ðtÞj þ jb22ðtÞjÞ ¼ �4:5þ 4:2974j cos tj < 0:

Hence, it follows from Corollary 2 that the system (15) is globally exponentially stable.

References

[1] Arik S. An improved global stability result for delayed cellular neural networks. IEEE Trans Circuits Syst I 2002;49:1211–4.[2] Arik S. On the global asymptotic stability of delayed cellular neural networks. IEEE Trans Circuits Syst I 2000;47:571–4.[3] Arik S. An analysis of global asymptotic stability of delayed cellular neural networks. IEEE Trans Neural Networks

2002;13:1239–42.[4] Cao J. Global stability conditions for delayed CNNs. IEEE Trans Circuits Syst I 2001;48:1330–3.[5] Cao J, Wang J. Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans

Circuits Syst I 2003;50:34–44.[6] Chen A, Cao J, Huang L. An estimation of upperbound of delays for global asymptotic stability of delayed Hopfiled neural

networks. IEEE Trans Circuits Syst I 2002;49:1028–32.[7] Chua LO, Yang L. Cellular neural networks:theory and applications. IEEE Trans Circuits Syst I 1988;35:1257–90.[8] Feng CH, Plamondon R. On the stability analysis of delayed neural networks systems. Neural Networks 2001;14:1181–8.[9] Hardy GH, Littlewood JE, Polya G. Inequalities. 2nd ed. Cambridge University Press; 1952.

[10] Huang H, Cao J. On global asymptotic stability of recurrent neural networks with time-varying delays. Appl Math Comput2003;142:143–54.

[11] Jiang H, Li Z, Teng Z. Boundedness and stability for nonautonomous cellular neural networks with delay. Phys Lett A2003;306:313–25.

990 Q. Zhang et al. / Chaos, Solitons and Fractals 36 (2008) 985–990

[12] Liao X, Chen G, Sanchez EN. LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE TransCircuits Syst I 2002;49:1033–9.

[13] Liao XX, Wang J. Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays. IEEETrans Circuits Syst I 2003;50:268–74.

[14] Mohamad S, Gopalsamy K. Exponential stability of continuous-time and discrete-time cellular neural networks with delays. ApplMath Comput 2003;135:17–38.

[15] Mukherjea A. Real and Functional Analysis. New York: Mcgraw-Hill; 1978.[16] Roska T, Wu CW, Balsi M, Chua LO. Stability and dynamics of delay-type general neural networks. IEEE Trans Circuits Syst

1992;39:487–90.[17] Roska T, Wu CW, Chua LO. Stability of cellular neural network with dominant nonlinear and delay-type templates. IEEE Trans

Circuits Syst 1993;40:270–2.[18] Zeng Z, Wang J, Liao X. Global exponential stability of a general class of recurrent neural networks with time-varying delays.

IEEE Trans Circuits Syst I 2003;50:1353–8.[19] Zhang J. Globally exponential stability of neural networks with variable delays. IEEE Trans Circuits Syst I 2003;50:288–90.[20] Zhang Q, Wei X, Xu J. Stability analysis for cellular neural networks with variable delays. Chaos, Solitons & Fractals

2006;28:331–6.[21] Zhang Q, Wei X, Xu J. Delay-dependent exponential stability of cellular neural networks with time-varying delays. Chaos,

Solitons & Fractals 2005;23:1363–9.[22] Zhang Q, Wei X, Xu J. New stability conditions for neural networks with constant and variable delays. Chaos, Solitons & Fractals

2005;26:1391–8.[23] Zhang Q, Wei X, Xu J. On global exponential stability of nonautonomous delayed neural networks. Chaos, Solitons & Fractals

2005;26:965–70.[24] Zhou D, Cao J. Globally exponential stability conditions for cellular neural networks with time-varying delays. Appl Math

Comput 2002;131:487–96.