ARTICLE IN PRESS

Neurocomputing 72 (2009) 3331–3336

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

$ Thi

(Grant

for Che

605210

China (G

Univers� Corr

116028,

E-m

laibing3

journal homepage: www.elsevier.com/locate/neucom

New LMT-based delay-dependent criterion for global asymptotic stability ofcellular neural networks$

Cheng-De Zheng a,b,�, Lai-Bing Lu a, Zhan-Shan Wang b

a School of Science, Dalian Jiaotong University, Dalian 116028, PR Chinab School of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China

a r t i c l e i n f o

Article history:

Received 23 July 2008

Received in revised form

3 December 2008

Accepted 13 January 2009

Communicated by T. Heskesinequality are considered. Three illustrative examples are given to demonstrate the effectiveness of the

Available online 5 March 2009

Keywords:

Global asymptotic stability (GAS)

Linear matrix inequality (LMI)

Cellular neural networks

Time-varying delay

Jensen integral inequality

12/$ - see front matter & 2009 Elsevier B.V. A

016/j.neucom.2009.01.013

s work was supported by the National Natura

nos. 60534010, 60572070, 60728307, 607740

ung Kong Scholars and Innovative Research

03), the National High Technology Research a

rant no. 2006AA04Z183) and the Postdoctora

ity (Grant no. 20080314).

esponding author at: School of Science, Dalia

PR China.

ail addresses: zhengchengde@dl.cn, cdzheng

01@sina.com (L.-B. Lu), zhanshan_wang@163

a b s t r a c t

The problem of global asymptotic stability analysis is studied for a class of cellular neural networks with

time-varying delay. By defining a Lyapunov–Krasovskii functional, a new delay-dependent stability

condition is derived in terms of linear matrix inequalities. The obtained criterion is less conser-

vative than some previous literature because free-weighting matrix method and the Jensen integral

proposed results.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

As neural networks have been successfully applied to variousfields such as image processing, pattern recognition and partialdifferential equations solving, the stability analysis of neuralnetworks has been extensively investigated. However, time delaysare frequently encountered in neural networks due to the finiteswitching speed of amplifiers and the inherent communication ofneurons. It is well known that the existence of time delay maycause divergence, oscillation, and even instability. Therefore,considerable efforts have been devoted to stability analysis ofneural networks with time delays. Among them, delay-dependentstability criteria [1,3–6,9,23] have attracted much attentionrecently because delay-dependent criteria make use of informa-tion on the length of delays, and are less conservative than delay-independent ones [7,8,10–22].

ll rights reserved.

l Science Foundation of China

48, 60774093), the Program

Groups of China (Grant no.

nd Development Program of

l Foundation of Northeastern

n Jiaotong University, Dalian

@djtu.edu.cn (C.-D. Zheng),

.com (Z.-S. Wang).

In order to get a less conservative result, we constructa new Lyapunov–Krasovskii functional and derive newsufficient condition for the globally asymptotic stability of theconcerned delayed neural network, which is delay-dependentand computationally efficient. Since the free-weighting matrixapproach [3,4,6] and the Jensen integral inequality [2] areinvolved, the relationship between the time-varying delay andits lower and upper bounds is considered, the result is lessconservative than some existing ones. Three illustrativeexamples are given to demonstrate the effectiveness of theproposed results.

2. Problem description

Considering the following cellular neural networks withinterval time-varying delays:

_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt � tðtÞÞÞ þ J, (1)

where xðtÞ ¼ ðx1ðtÞ; x2ðtÞ; . . . ; xnðtÞÞT2Rn is the neural state

vector, C ¼ diagfc1; c2; . . . ; cng is a positive diagonal matrix, A ¼

ðaijÞn�n; B ¼ ðbijÞn�n are known constant matrices, 0pt1ptðtÞpt2

is the time-varying delay, where t1; t2 are constants. J isthe constant external input vector, and f ðxðtÞÞ ¼ ðf 1ðx1ðtÞÞ;

f 2ðx2ðtÞÞ; . . . ; f nðxnðtÞÞÞT2 Rn denotes the neural activation func-

tion. It is assumed that f iðxiðtÞÞ ði ¼ 1;2; . . . ;nÞ are bounded and

ARTICLE IN PRESS

C.-D. Zheng et al. / Neurocomputing 72 (2009) 3331–33363332

there exist constants gj;sj such that (see [9])

gjpf jðs1Þ � f jðs2Þ

s1 � s2psj; j ¼ 1;2; . . . ;n (2)

for any s1; s2 2 R,s1as2:

From the well-known Brouwer’s fixed point theorem, system(1) always has an equilibrium point x�.

Throughout this paper, let kyk denote the Euclidean normof a vector y 2 Rn; WT ;W�1 denote the transpose, the inverse,respectively. Let W40 ðX0Þ denote a positive (nonnegative)definite symmetric matrix, I denote an identity matrix withcompatible dimension.

In order to prove the global asymptotic stability (GAS) of theequilibrium point x� of system (1), we will first simplify system (1)as follows. Let uð�Þ ¼ xð�Þ � x�; then we have,

_uðtÞ ¼ �CuðtÞ þ AgðuðtÞÞ þ Bgðuðt � tðtÞÞÞ, (3)

where uðtÞ ¼ ðu1ðtÞ;u2ðtÞ; . . . ;unðtÞÞT ; gjðujðtÞÞ ¼ f jðujðtÞ þ x�j Þ � f jðx

�j Þ

with gjð0Þ ¼ 0; j ¼ 1;2; . . . ;n: By assumption (2), we can see that

gjpgjðujðtÞÞ

ujðtÞpsj. (4)

Clearly, the equilibrium point of system (1) is GAS if and only ifthe zero solution of system (3) is GAS.

Through this paper, the Jensen integral inequality will be used,so it is listed as the following lemma.

Lemma 1 (Gu [2]). For any positive symmetric constant matrix

M 2 Rn�n, scalars r1or2 and vector function o : ½r1; r2� !Rn such

that the integrations concerned are well defined, then

Z r2

r1

oðsÞds

� �T

M

Z r2

r1

oðsÞds

� �pðr2 � r1Þ

Z r2

r1

oT ðsÞMoðsÞds.

3. GAS results of delayed cellular neural networks

First, we will present the GAS results for system (3) with_tðtÞpZo1:

Theorem 1. Under assumption (2) and 0pt1ptðtÞpt2; _tðtÞpZo1;suppose that there exist positive definite symmetric matrices Pi ði ¼

1; . . . ;7Þ;Q ¼ ½Qij�2�2;R ¼ ½Rij�2�2; S ¼ ½Sij�2�2; positive diagonal ma-

trices T1; T2;D ¼ diagfd1; d2; . . . ;dng;D ¼ diagfd1; d2; . . . ; dng and

real matrices XT¼ ½XT

1 XT2�;Y

T¼ ½YT

1 YT2�; Z

T¼ ½ZT

1 ZT2� such that the

following linear matrix inequality (LMIs) hold:

X1 ¼R X

XT P3

" #X0, (5)

X2 ¼S Y

YT P5

" #X0, (6)

X3 ¼Rþ S Z

ZT P3 þ P5

" #X0, (7)

O ¼

O11 O12 O13 O14 Y1 �Z1

� O22 0 O24 O25 O26

� � O33 O34 0 0

� � � O44 0 0

� � 0 0 O55 0

� � 0 0 0 O66

26666666664

37777777775o0, (8)

where ‘‘*’’ are entries readily inferred by symmetry, and

O11 ¼ � P1C � CP1 þ CPC þ X1 þ XT1 þ Q11 þ P6 þ P7 þ t2R11

þ ðt2 � t1ÞS11 � 2GT1S� 2ðDS� DGÞC �1

t2P2,

O12 ¼ �X1 þ XT2 þ Z1 � Y1 þ t2R12 þ ðt2 � t1ÞS12 þ

1

t2P2,

O13 ¼ P1A� CðD� DÞ þ Q12 � CPAþ ðDS� DGÞAþ ðSþGÞT1,

O14 ¼ P1B� CPBþ ðDS� DGÞB,

O22 ¼ � ð1� ZÞQ11 � X2 � XT2 þ Z2 þ ZT

2 � Y2 � YT2 þ t2R22

þ ðt2 � t1ÞS22 � 2GT2S�1

t2P2 �

1

t2 � t1ðP2 þ 2P4Þ,

O24 ¼ �ð1� ZÞQ12 þ ðSþGÞT2; O25 ¼ Y2 þ1

t2 � t1P4,

O26 ¼1

t2 � t1ðP2 þ P4Þ � Z2,

O33 ¼ Q22 þ ðD�DÞAþ ATðD� DÞ þ AT PA� 2T1,

O34 ¼ ðD�DÞBþ AT PB,

O44 ¼ �ð1� ZÞQ22 þ BT PB� 2T2; O55 ¼ �1

t2 � t1P4 � P6,

O66 ¼ �1

t2 � t1ðP2 þ P4Þ � P7,

P ¼ t2ðP2 þ P3Þ þ ðt2 � t1ÞðP4 þ P5Þ; G ¼ diagfg1; g2; . . . ; gng,

S ¼ diagfs1;s2; . . . ;sng,

then the equilibrium point of system (3) is unique and GAS.

Proof. Firstly, we show the uniqueness of the equilibrium pointby contradiction. To end this, let u be the equilibrium point of thedelayed recurrent neural network (3), then we have

�Cuþ ðAþ BÞgðuÞ ¼ 0.

Now suppose ua0: It is easy to see that

2fuTðP1 þDS� DGÞ þ gT ðuÞðD� DÞgf�Cuþ ðAþ BÞgðuÞg ¼ 0. (9)

By inequality (4), we get

� 2uTSðT1 þ T2ÞGuþ 2u

TðT1 þ T2ÞðGþ SÞgðuÞ � 2gT ðuÞ

�ðT1 þ T2ÞgðuÞX0.

This together with Eq. (9) gives

uTE11uþ 2u

TE12gðuÞ þ gT ðuÞE22gðuÞX0,

i.e.

½uT

gT ðuÞ�E11 E12

� E22

" #u

gðuÞ

" #X0, (10)

where

E11 ¼ �ðP1 þDS� DGÞC � CðP1 þDS� DGÞ � 2GSðT1 þ T2Þ,

E12 ¼ ðP1 þDS� DGÞðAþ BÞ � CðD� DÞ þ ðT1 þ T2ÞðGþ SÞ,

E22 ¼ ðD� DÞðAþ BÞ þ ðAþ BÞT ðD�DÞ � 2ðT1 þ T2Þ.

On the other hand, let

P ¼I I 0 0 I I

0 0 I I 0 0

� �,

ARTICLE IN PRESS

C.-D. Zheng et al. / Neurocomputing 72 (2009) 3331–3336 3333

multiplying (8) by P and PT on its left and right side, respectively,we have

~E11 E12 þ ZQ12 � CPðAþ BÞ

� E22 þ ZQ22 þ ðAþ BÞT PðAþ BÞ

" #o0,

where ~E11 ¼ E11 þ CPC þ ZQ11 þ t2ðR11 þ R12 þ RT12 þ R22Þ þ ðt2 �

t1ÞðS11 þ S12 þ ST12 þ S22Þ:

That is

E11 E12

� E22

" #þ

I I

0 0

" #ðt2Rþ ðt2 � t1ÞSÞ

I 0

I 0

" #þ ZQ

þ�C

ðAþ BÞT

" #P½�C Aþ B�o0.

Note that P40;QX0;RX0; SX0; we obtain

E11 E12

� E22

" #o0.

Obviously, this contradicts with (10). The contradiction implies

that u ¼ 0: That is, the origin of the delayed recurrent neural

networks (3) is the unique equilibrium point.

Next, we show the unique equilibrium point of (3) is GAS.

Consider the following Lyapunov–Krasovskii functional:

VðuðtÞÞ ¼ uT ðtÞP1uðtÞ þ 2Xn

i¼1

di

Z uiðtÞ

0fgiðsÞ � gisgds

�

þdi

Z uiðtÞ

0fsis� giðsÞgds

�

þ

Z t

t�t2

Z t

y_uTðsÞðP2 þ P3Þ _uðsÞds dy

þ

Z t�t1

t�t2

Z t

y_uTðsÞðP4 þ P5Þ _uðsÞds dy

þX2

i¼1

Z t

t�ti

uT ðsÞPiþ5uðsÞdsþ

Z t

t�tðtÞxTðuðsÞÞQxðuðsÞÞds,

(11)

where xTðuðsÞÞ ¼ ðuT ðsÞ; gT ðuðsÞÞÞ:

For convenience, we denote ut ¼ uðt � tðtÞÞ: The time derivative

of functional (11) along the trajectories of system (3) is obtained

as follows:

_VðuðtÞÞ ¼ 2uT ðtÞP1 _uðtÞ þ 2fgT ðuðtÞÞ � uT ðtÞGgD _uðtÞ þ 2fuT ðtÞS

� gT ðuðtÞÞgD _uðtÞ þ t2 _uTðtÞðP2 þ P3Þ _uðtÞ

�

Z t

t�t2

_uTðsÞðP2 þ P3Þ _uðsÞdsþ ðt2 � t1Þ _u

TðtÞ

�ðP4 þ P5Þ _uðtÞ �

Z t�t1

t�t2

_uTðsÞðP4 þ P5Þ _uðsÞdsþ uT ðtÞ

�ðP6 þ P7ÞuðtÞ �X2

i¼1

uT ðt � tiÞPiþ5uðt � tiÞ

þ xTðuðtÞÞQxðuðtÞÞ � ð1� _tðtÞÞxT

ðutÞQxTðutÞ. (12)

It is clear that the following equations are true:

Z t

t�t2

_uTðsÞP2 _uðsÞds ¼

Z t

t�tðtÞ_uTðsÞP2 _uðsÞdsþ

Z t�tðtÞ

t�t2

_uTðsÞP2 _uðsÞds,

(13)

Z t�t1

t�t2

_uTðsÞP4 _uðsÞds ¼

Z t�t1

t�tðtÞ_uTðsÞP4 _uðsÞdsþ

Z t�tðtÞ

t�t2

_uTðsÞP4 _uðsÞds.

(14)

By using the Jensen integral inequality (Lemma 1), we obtain

�

Z t

t�tðtÞ_uTðsÞP2 _uðsÞdsp�

1

tðtÞ

Z t

t�tðtÞ_uðsÞds

� �T

P2

Z t

t�tðtÞ_uðsÞds

p�1

t2½uðtÞ � ut�

T P2½uðtÞ � ut�, (15)

�

Z t�t1

t�tðtÞ_uTðsÞP4 _uðsÞdsp�

1

tðtÞ � t1

Z t�t1

t�tðtÞ_uðsÞds

� �T

P4

Z t�t1

t�tðtÞ_uðsÞds

p�1

t2 � t1½uðt � t1Þ � ut�

T P4½uðt � t1Þ � ut�,

(16)

�

Z t�tðtÞ

t�t2

_uTðsÞðP2 þ P4Þ _uðsÞds

p�1

t2 � tðtÞ

Z t�tðtÞ

t�t2

_uðsÞds

� �T

ðP2 þ P4Þ

Z t�tðtÞ

t�t2

_uðsÞds

p�1

t2 � t1½ut � uðt � t2Þ�

T ðP2 þ P4Þ½ut � uðt � t2Þ�. (17)

On the other hand, based on Leibniz–Newton formula, for

any real matrix Xi;Yi; Zi ði ¼ 1;2Þ with compatible dimensions,

we get

0 ¼ 2fuT ðtÞX1 þ uTtX2g uðtÞ � ut �

Z t

t�tðtÞ_uðsÞds

� �, (18)

0 ¼ 2fuT ðtÞY1 þ uTtY2g uðt � t1Þ � ut �

Z t�t1

t�tðtÞ_uðsÞds

� �, (19)

0 ¼ 2fuT ðtÞZ1 þ uTtZ2g ut � uðt � t2Þ �

Z t�tðtÞ

t�t2

_uðsÞds

� �. (20)

Furthermore, from inequality (4), the following matrix inequal-

ities hold for any positive diagonal matrices T1; T2 with compa-

tible dimensions

0p� 2uT ðtÞST1GuðtÞ þ 2uT ðtÞT1ðGþSÞgðuðtÞÞ � 2gT ðuðtÞÞT1gðuðtÞÞ,

(21)

0p� 2uTtST2Gut þ 2uT

tT2ðGþSÞgðutÞ � 2gT ðutÞT2gðutÞ. (22)

Moreover, for any real symmetric matrices R; S with compatible

dimensions, we have

0 ¼ t2kT ðtÞRkðtÞ �Z t

t�tðtÞkT ðtÞRkðtÞds�

Z t�tðtÞ

t�t2

kT ðtÞRkðtÞds, (23)

0 ¼ ðt2 � t1ÞkT ðtÞSkðtÞ �Z t�t1

t�tðtÞkT ðtÞSkðtÞds�

Z t�tðtÞ

t�t2

kT ðtÞSkðtÞds,

(24)

where kT ðtÞ ¼ ðuT ðtÞ;uTt Þ:

Therefore it can be seen from (12) to (24) that

_VðuðtÞÞpzTðtÞOzðtÞ �

Z t

t�tðtÞBT ðt; sÞX1Bðt; sÞds

�

Z t�t1

t�tðtÞBT ðt; sÞX2Bðt; sÞds�

Z t�tðtÞ

t�t2

BT ðt; sÞX3Bðt; sÞds,

where BT ðt; sÞ ¼ ½uT ðtÞ;uTt ; _u

TðsÞ� and

zTðtÞ ¼ ½uT ðtÞ uT

t gT ðuðtÞÞ gT ðutÞ uT ðt � t1Þ uT ðt � t2Þ�.

Therefore, if Oo0 and XiX0; i ¼ 1;2;3; then _VðuðtÞÞp� �kuðtÞk2

holds for a sufficiently small �40: This implies the GAS of system

(3) with assumption (2), which completes the proof. &

ARTICLE IN PRESS

C.-D. Zheng et al. / Neurocomputing 72 (2009) 3331–33363334

Remark 1. Theorem 1 provides an LMI-based sufficientcondition for the GAS of the delayed neural network in (3). Oneadvantage of the LMI approach is that the LMI condition can bechecked numerically more efficiently than those in [8,21,22]which are not LMIs with respect to the parameters to bedetermined.

Remark 2. Unlike the Lyapunov–Krasovskii functionaland the proof technique in [4], this paper introduced theterms of G;D; P2; P4;Q12; used Jensen integral inequality. Theadvantage is that Theorem 1 can lead to less conservativenessthan [4]. In fact, let G;D;P2; P4;Q12 all be 0; Theorem 1 of thispaper is equal to Theorem 1 of [4]. Furthermore, this paperproved the uniqueness of equilibrium point of above network,while [4] did not.

When tðtÞ is not differentiable or _tðtÞ is unknown, Q

will no longer be helpful to improve the stability condition since�ð1� ZÞQ may be nonnegative definite. Therefore, by settingQ ¼ 0; an easy delay-dependent criterion is derived as follows forunknown Z:

0 5 10 15 20 25 30−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

u1

u2

u3

u4

example 1

Time t

u(t)

Fig. 1. The dynamical behavior of the system (3) in Example 1.

Table 1Calculated maximal upper bounds of t2 for various t1 and Z of Example 1.

t1 Methods Z ¼ 0 Z ¼ 0:1 Z ¼ 0:5 Z ¼ 0:9 Z ¼ 0:95 Unknown Z

0 [6] 3.5841 3.2775 2.1502 1.3164 1.2708 1.2598

[3] 3.5841 3.2793 2.2245 1.5847 1.5514 1.5444

[4] 3.5841 3.3039 2.5376 2.0853 2.0485 2.0389

Corollary 1. Under assumption (2) and 0pt1ptðtÞpt2; _tðtÞpZo1;suppose that there exist positive definite symmetric matrices

Pi ði ¼ 1; . . . ;7Þ;Q ;R; S; positive diagonal matrices T1; T2;D;D and

real matrices X;Y ; Z such that Xl ðl ¼ 1;2;3ÞX0 and the following

LMI holds:

O11 O12 O13 O14 Y1 �Z1

� O22 0 ðSþ GÞT2 O25 O26

� � O33 O34 0 0

� � � O44 0 0

� � 0 0 O55 0

� � 0 0 0 O66

26666666664

37777777775o0,

where

O11 ¼ � P1C � CP1 þ CPC þ X1 þ XT1 þ P6 þ P7 þ t2R11

þ ðt2 � t1ÞS11 � 2ðDS� DGÞC � 2GT1S�1

t2P2,

O13 ¼ P1A� CðD� DÞ � CPAþ ðDS� DGÞAþ ðSþ GÞT1,

O22 ¼ � X2 � XT2 þ Z2 þ ZT

2 � Y2 � YT2 þ t2R22 þ ðt2 � t1ÞS22

� 2GT2S�1

t2P2 �

1

t2 � t1ðP2 þ 2P4Þ,

O33 ¼ ðD� DÞAþ ATðD� DÞ þ AT PA� 2T1,

O44 ¼ BT PB� 2T2,

and the other parameters are determined in Theorem 1, then the

equilibrium point of system (3) is unique and GAS.

This paper 3.8363 3.5209 2.7176 2.2151 2.1339 2.0771

1 [4] 3.5841 3.3068 2.5802 2.2736 2.2510 2.2393

This paper 3.8363 3.5234 2.7280 2.3510 2.3084 2.2739

2 [4] 3.5841 3.3125 2.7500 2.6468 2.6361 2.6299

This paper 3.8363 3.5289 2.8392 2.7064 2.6884 2.6735

4. Illustrative examples

In this section, we provide three numerical examples todemonstrate the effectiveness and less conservativeness of ourdelay-dependent stability criteria.

Example 1. Consider system (3) with

C ¼ diagf1:2769;0:6231;0:9230;0:4480g;

A ¼

�0:0373 0:4852 �0:3351 0:2336

�1:6033 0:5988 �0:3224 1:2352

0:3394 �0:0860 �0:3824 �0:5785

�0:1311 0:3253 �0:9534 �0:5015

26664

37775;

B ¼

0:8674 �1:2405 �0:5325 0:0220

0:0474 �0:9164 0:0360 0:9816

1:8495 2:6117 �0:3788 0:8428

�2:0413 0:5179 1:1734 �0:2775

26664

37775;

f 1ðsÞ ¼ 0:05685ðjsþ 1j � js� 1jÞ; f 2ðsÞ ¼ 0:06395ðjsþ 1j � js� 1jÞ;

f 3ðsÞ ¼ 0:3997ðjsþ 1j � js� 1jÞ; f 4ðsÞ ¼ 0:1184ðjsþ 1j � js� 1jÞ:

Obviously, the activation functions are bounded and satisfyassumption (2) with

s1 ¼ 0:1137; s2 ¼ 0:1279; s3 ¼ 0:7994; s4 ¼ 0:2368,

g1 ¼ g2 ¼ g3 ¼ g4 ¼ 0.

For this model, from Theorem 1 we can verify that ð0;0;0;0ÞT isthe unique equilibrium point and is GAS. The GAS with the initialstate ð�0:2;0:5;�0:4;0:2ÞT is shown in Fig. 1.

The same model has been studied in [3,4,6]. The comparison

among Theorem 1 and Corollary 1 in this paper and those in

[3,4,6] are listed in Table 1, where ‘‘unknown Z’’ means that Z can

be arbitrary value or tðtÞ can be not differentiable.

However, for any t1 and Z; all of the criteria given in

[5,8,9,13,14,21,22] fail to conclude whether the model is GAS or

ARTICLE IN PRESS

Table 3Calculated maximal upper bounds of t2 for various t1 and Z of Example 3.

t1 Methods Z ¼ 0:4 Z ¼ 0:8 Z ¼ 0:9 Z ¼ 0:95 Z ¼ 0:99 Unknown Z

0 [5] 0.4067 0.3928 0.3920 0.3920 0.3920 0.3920

This paper 1.7654 1.4270 1.3888 1.3660 1.3465 1.3418

1 [5] 1.3901 1.3794 1.3782 1.3777 1.3774 1.3774

This paper 2.6190 2.3514 2.3461 2.3435 2.3420 2.3418

2 [5] 2.3900 2.3794 2.3782 2.3777 2.3773 2.3773

This paper 3.6172 3.3506 3.3456 3.3432 3.3420 3.3418

100 [5] 100.3898 100.3794 100.3782 100.3777 100.3773 100.3772

This paper 101.6141 101.3506 101.3456 101.3432 101.3420 101.3418

Table 2Calculated maximal upper bounds of t2 for various t1 and Z of Example 2.

t1 Methods Z ¼ 0:76 Z ¼ 0:8 Z ¼ 0:85 Z ¼ 0:9 Z ¼ 0:95 Unknown Z

0 [5] 11.4405 5.6225 4.5075 4.2448 4.2426 4.2426

[3] 14.5340 7.2083 5.7608 5.3274 5.2769 5.2769

[4] 20.5135 10.1217 8.0686 7.4622 7.3977 7.3977

This paper 25.8290 13.0158 10.4913 9.6516 9.3993 9.3711

1 [5] 12.4401 5.6226 5.5076 5.2449 5.2426 5.2426

[4] 21.3875 10.8601 8.7276 8.0760 7.9985 7.9985

This paper 26.5007 13.3606 10.6876 9.8375 9.6602 9.6602

2 [5] 13.4401 6.6226 6.5076 6.2449 6.2426 6.2426

[4] 22.3880 11.8602 9.7276 9.0760 8.9985 8.9985

This paper 27.5005 14.3606 11.6876 10.8275 10.6602 10.6602

100 [5] – – – – – –

[4] 120.3883 109.8602 107.7275 107.0760 106.9985 106.9985

This paper 125.4997 111.9289 109.6877 108.8274 108.6602 108.6602

C.-D. Zheng et al. / Neurocomputing 72 (2009) 3331–3336 3335

not. Therefore, we can say that for this system the results in

this paper are effective and less conservative than those in

[3–6,8,9,13,14,21,22].

Example 2. Consider system (1) with

C ¼1 0

0 1

" #; A ¼

0:3 0

0 0:2

" #; B ¼

0:1 �0:2

0 0:1

" #,

f 1ðsÞ ¼ ðjsþ 1j � js� 1jÞ; f 2ðsÞ ¼ ðjsþ 1j � js� 1jÞ.

Obviously, the activation functions are continuous and satisfyassumption (2) with S ¼ 2I;G ¼ 0:

For this model, from Theorem 1 we can verify that ð0;0ÞT is the

unique equilibrium point and is GAS.

The comparison among Theorem 1 and Corollary 1 in this paper

and those in [3–5] are listed in Table 2, where ‘‘–’’ denotes the

result failing to verify the GAS of the equilibrium point.

However, for any t1 and given Z; all the criteria given in

[8,9,21,22] fail to conclude whether the model is GAS or not.

Therefore, we can say that for this system the results in this

paper are much effective and less conservative than those in

[3–5,8,9,21,22].

Example 3. Consider system (3) with

C ¼ diagf1:34;1:08;1:46g,

A ¼

1:08 �0:52 0:05

�0:08 0:68 0:19

0:14 0:29 �0:58

264

375; B ¼

2:45 �0:64 1:2

0:45 0:88 0:47

0:07 �1:56 0:97

264

375,

s1 ¼ 0:31 s2 ¼ 0:40; s3 ¼ 0:23,

g1 ¼ �0:31; g2 ¼ �0:40; g3 ¼ �0:23.

Obviously, none of the results in [1,3,4,6,8,21,22] are applicableto ascertain the stability of this model. The comparison amongTheorem 1 and Corollary 1 in this letter and those in [5] are listedin Table 3.

Therefore, we can say that for this system the results in this

paper are more effective and less conservative than those in

[1,3–6,8,13,21,22].

5. Conclusion

Several sufficient conditions are derived to guarantee the GASof CNN with time-varying delays, which are different from theexisting ones and have wider application fields than some results

ARTICLE IN PRESS

C.-D. Zheng et al. / Neurocomputing 72 (2009) 3331–33363336

in literature. The obtained results can be expressed in the form ofLMI and are easy to be verified. Three illustrative examples arealso given to demonstrate the effectiveness of the proposedresults.

Acknowledgment

The authors would like to thank all of the referees and theAssociate Editor for their constructive comments, which led to asignificant improvement in the paper.

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Cheng-De Zheng was born in Shandong, China, in1966. He received the B.S. degree in mathematics fromJilin University of China in 1987, and the M.S. and Ph.D.degrees in computational mathematics from DalianUniversity of Technology of China in 1998 and 2004,respectively. He joined the Department of Mathe-matics, Dalian Jiaotong University, China, in 1987.Since 2004, he has been a Professor. His main researchinterests are stability of recurrent neural networks,numerical approximation, fuzzy set and its application.

Lai-Bing Lu was born in Shandong, China, in 1982. Hereceived the B.S. degree in mathematics from QufuNormal University, Shandong, China, in 2005. Now heis pursuing the M.S. degree in applied mathematics inDalian Jiaotong University.

Zhan-Shan Wang was born in Liaoning, China, in 1971.He received the Master Degree in control theory andcontrol engineering from Fushun Petroleum Institute,Fushun, China, in 2001. He received the Ph.D. degree incontrol theory and control engineering from North-eastern University, Shenyang, China, in 2006. He is nowan associate professor in Northeastern University. Hisresearch interests include stability analysis of recur-rent neural networks, fault diagnosis, fault tolerantcontrol and nonlinear control.