delay-dependent global asymptotic stability for delayed cellular neural networks

Commun Nonlinear Sci Numer Simulat 14 (2009) 1057–1063

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

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

Short communication

Delay-dependent global asymptotic stability for delayed cellularneural networks

Yanjun Shen *, Hui Yu, Jigui JianInstitute of Nonlinear Complex Systems, College of Science, China Three Gorges University, College Road 8, Yichang, Hubei 443002, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 October 2007Received in revised form 30 March 2008Accepted 20 April 2008Available online 30 April 2008

PACS:02.03.Yy

Keywords:Delay-dependent criteriaGlobal asymptotic stabilityLinear matrix inequalityTime-delay

1007-5704/$ - see front matter � 2008 Elsevier B.Vdoi:10.1016/j.cnsns.2008.04.012

* Corresponding author. Tel.: +86 0717 6390282.E-mail addresses: [email protected] (Y. Shen),

In this note, we address the problem of the existence of a unique equilibrium point andpresent delay-dependent global asymptotical stability for cellular neural networks withtime-delay. The LMI-based criteria are checkable easily. An example illustrates that theproposed conditions provide useful and less conservative results for the problem.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

Cellular neural networks are complex and large-scale nonlinear dynamical systems, while the dynamics of the delayedcellular neural network (DCNNs) are even richer and more complicated. They have been found applications in many areassuch as signal processing, pattern recognition, and static image processing [9–11]. Some of these applications require thatthe equilibrium points of the designed networks be stable. So, it is important to study the stability of DCNNs.

Recently, fundamental results have been established on uniqueness, global stability and global exponential stability of theequilibrium point for DCNNs [1–8]. For example, when the activation functions fiðxÞ ¼ 1

2 ðjxþ 1j � jx� 1jÞ ði ¼ 1; . . . ;nÞ, Arikand Tavsanoglu [5] and Cao [7] studied the global asymptotical stability for DCNNs via different approaches and obtainedseveral sufficient conditions, respectively. Cao and Zhou [8] and Zhang et al. [14] extended these results to more general classof DCNNs, which the activation functions fiðxÞ are nondecreasing, bounded and globally Lipschitz. Liao et al. [12] studied thestability of DCNNs with strictly increasing and slope-bounded activation functions by introducing the linear matrix inequal-ity (LMI) approach. By employing a more general Lyapunov functional, Arik [13] showed that the results of [12] were alsoapplicable to neural networks with increasing (not necessarily strictly) and slope-bounded activation functions. He et al.[3] presented a new Lyapunov–Krasovskii functional containing an integral term of state for DCNNs. The S-procedure wasemployed to handle the nonlinearities, and a less conservative global asymptotic stability criterion was derived. It is worthpointing out that all the above mentioned asymptotic stability criteria are delay-independent. When the size of the delay issmall [15,16] it is known that delay-dependent stability conditions are generally less conservative than delay-independentones. Based on this, Civalleri et al. proposed a delay-dependent asymptotical stability condition for symmetric DCNNs [17].

. All rights reserved.

[email protected] (H. Yu), [email protected] (J. Jian).

mailto:[email protected]



http://www.sciencedirect.com/science/journal/10075704

http://www.elsevier.com/locate/cnsns

1058 Y. Shen et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 1057–1063

Very recently, Xu et al. [1] provided improved criteria for the existence of a unique equilibrium point and its global asymp-totic stability of DCNNs. Both delay-dependent and delay-independent stability conditions have been proposed in terms ofLMIs, which can be checked easily.

In this paper, we consider the problem of global asymptotical stability for DCNNs with a new Lyapunov–Krasovskii func-tional containing an integral term of state for DCNNs. Then, by using the methods of Xu et al. [1] and He et al. [4], new LMI-based conditions for the existence of a unique equilibrium point and its global asymptotic stability are derived. A key featureof our approach is the introduction of some auxiliary matrix terms in the LMI-based conditions. This introduces flexibility inapplying the criteria in the analysis. The employment of the new Lyapunov–Krasovskii functional containing an integral termenable us to prove with ease the global asymptotic stability under relaxed conditions. Our main results are delay-dependentglobal asymptotic stability criteria for DCNNs. Note that it lends itself to a easy delay-independent criterion. Numerical exam-ples illustrate the proposed condition provide useful and less conservative results for the problem.

The organization of this paper is as follows. The problem statement is given in Section 2. Section 3 presents the main re-sults. Numerical example is given in Section 4. Finally, the paper is concluded in Section 5.

2. Problem statement

Consider the following DCNNs, which is described by a nonlinear delayed differential equation of the form:
_uðtÞ ¼ �AuðtÞ þW0gðuðtÞÞ þW1gðuðt � sÞÞ þI; ð1Þ
where

uðtÞ ¼ ½u1ðtÞ; u2ðtÞ; . . . ;unðtÞ�T

is the state vector.

gðuðtÞÞ ¼ ½g1ðu1ðtÞÞ; g2ðu2ðtÞÞ; . . . ; gnðunðtÞÞ�T

is the neuron activation function and giðuiÞ satisfy

0 6giðn1Þ � giðn2Þ

n1 � n26 ri; gið0Þ ¼ 0; i ¼ 1; . . . ;n: ð2Þ

A ¼ diagfa1; a2; . . . ; ang > 0, W0, W1 are the interconnection matrices representing the weighting coefficients of the neurons,I ¼ ½I1; . . . ; In�T is a constant vector representing the bias. The scalar s > 0 is a constant delay of the system. System (1) is ageneral form of DCNNs, which has been studied by many researchers [1–4,8,12–14].

By [8], it can be seen that there exists an equilibrium u� ¼ ½u�1; . . . ;u�n�T for (1). Now, we shift the equilibrium point u�

of the system (1) to the origin by introducing a new state xðtÞ ¼ uðtÞ � u�, which transforms the system into thefollowing:

_xðtÞ ¼ �AxðtÞ þW0f ðxðtÞÞ þW1f ðxðt � sÞÞ; ð3Þ
where xðtÞ ¼ ½x1ðtÞ; . . . ; xnðtÞ�T is the state vector of the transformed system, and
f ðxðtÞÞ ¼ ½f1ðx1ðtÞÞ; . . . ; fnðxnðtÞÞ�T;

with

fiðxiðtÞÞ ¼ giðxiðtÞ þ u�i Þ � giðu�i Þ; i ¼ 1; . . . ;n;

and fið0Þ ¼ 0, for i ¼ 1; . . . ;n. Note that the functions fið�Þ satisfies the following conditions:

0 6fiðn1Þ � fiðn2Þ

n1 � n26 ri; i ¼ 1;2; . . . ;n:

Except for these conditions, no specific forms of the functions fi’s are required here and therefore in the following con-clusions. Due to this, information of fi’s is provided only through ri’s.

The following lemma is useful to derive the main results.

Lemma 1 [20]. Let D, S and P be real matrices of appropriate dimensions with P > 0. Then, for any vectors x, y with appropriatedimensions, the following inequality holds:

2xT DSy 6 xT DPDT xþ yT ST P�1Sy: ð4Þ

3. Main results

Now, we present a new delay-dependent asymptotic stability condition for system (3) in the following theorem.

Theorem 1. The origin of the DCNNs in (3) is the unique equilibrium point and it is globally asymptotically stable for any delay0 < s 6 �s if there exist symmetric matrices P > 0, Q > 0, R > 0, W > 0 and diagonal matrices T > 0, S > 0, D > 0 and matrices Yi

ði ¼ 1;2;3;4Þ, U such that the following LMIs hold:

Y. Shen et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 1057–1063 1059

�PA�APþ Rþ�sATWAþ Y1 þ YT1 �Y1 þ YT

2 PW0 �ADþRT ��sATWW0 þ YT3 þU PW1 ��sATWW1 þ YT

4 ��sY1

�YT1 þ Y2 �R� Y2 � YT

2 �YT3 RS� YT

4 �U ��sY2

WT0P� �sWT

0WAþ Y3 þUT �DAþ TR �Y3 DW0 þWT0DþQ �2T þ �sWT

0WW0 DW1 þ�sWT0WW1 ��sY3

WT1P��sWT

1WAþ Y4 SR� Y4 �UT WT1Dþ�sWT

1WW0 �Q � 2Sþ�sWT1WW1 ��sY4

��sYT1 ��sYT

2 ��sYT3 ��sYT

4 ��sW

266666664

377777775< 0;

ð5Þ

R UUT Q

� �P 0; ð6Þ

where R ¼ diagfr1; r2; . . . ; rng.

Proof. First, we prove the uniqueness of the equilibrium point by contradiction. Assume that �x is the equilibrium point of thedelayed DCNNs in (3). Then, we have

�A�xþ ðW0 þW1Þf ð�xÞ ¼ 0: ð7Þ

Suppose f ð�xÞ–0. By (7), we can obtain

2�xTP½�A�xþ ðW0 þW1Þf ð�xÞ� ¼ 0; ð8Þ

and

2f Tð�xÞD½�A�xþ ðW0 þW1Þf ð�xÞ� ¼ 0: ð9Þ

Note that

2f Tð�xÞD½�A�xþ ðW0 þW1Þf ð�xÞ� ¼ 2f Tð�xÞ½DðW0 þW1Þ � S� T�f ð�xÞ þ 2f Tð�xÞðSþ TÞf ð�xÞ � 2f Tð�xÞDA�x

6 2f Tð�xÞ½DðW0 þW1Þ � S� T�f ð�xÞ þ 2f Tð�xÞðSRþ TR� DAÞ�x: ð10Þ

Then, from (8)–(10) we can obtain that

2�xTP½�A�xþ ðW0 þW1Þf ð�xÞ� þ 2f Tð�xÞðSRþ TR� DAÞ�xþ 2f Tð�xÞ½DðW0 þW1Þ � S� T�f ð�xÞP 0;

i.e.

��xTðPAþ APÞ�xþ f Tð�xÞJf ð�xÞ þ 2f Tð�xÞKT�x P 0; ð11Þ

where J ¼ DðW0 þW1Þ þ ðW0 þW1ÞTD� 2S� 2T , K ¼ PðW0 þW1Þ þ RSþ RT � AD. Then, by Lemma 1, we have

f Tð�xÞ½J þ KT½AP þ PA��1K�f ð�xÞP 0: ð12Þ

On the other hand, (5) can be rewritten as follows:

�PA� AP þ Rþ Y1 þ YT1 �Y1 þ YT

2 PW0 � ADþ RT þ YT3 þ U PW1 þ YT

4 ��sY1

�YT1 þ Y2 �R� Y2 � YT

2 �YT3 RS� YT

4 � U ��sY2

WT0P þ Y3 þ UT � DAþ TR �Y3 DW0 þWT

0Dþ Q � 2T DW1 ��sY3

WT1P þ Y4 SR� Y4 � UT WT

1D �Q � 2S ��sY4



4 ��sW

2666666664

3777777775

þ �s

�AT

0

WT0

WT1

0

266666664

377777775W �A 0 W0 W1 0½ � < 0;

ð13Þ

which implies that


2 PW0 þ YT3 þ U � ADþ RT PW1 þ YT

4 ��sY1

�YT1 þ Y2 �R� Y2 � YT

2 �YT3 RS� YT

4 � U ��sY2


0Dþ Q � 2T DW1 ��sY3


1D �Q � 2S ��sY4



4 ��sW

266666664

377777775< 0: ð14Þ


Pre- and post-multiplying (14) by

I I 0 0 00 0 I I 0

� �

and its transpose, respectively, we have

�AP � PA KKT J

� �< 0; ð15Þ

which, by the Schur complement formula, results in

J þ KTðAP þ PAÞ�1K < 0:

This contradicts with (12). Therefore, we can conclude that the equilibrium point f ð�xÞ, which, by (7), implies �x ¼ 0. That isthat the origin of the DCNNs in (3) is the unique solution to (7), thus, (3) has a unique equilibrium point.

Next, we prove that the unique equilibrium point of (3) is globally asymptotically stable. Construct the followingLyapunov–Krasovskii functional:

VðxðtÞÞ ¼ xTðtÞPxðtÞ þ 2Xn

j¼1dj

Z xj

0fjðsÞdsþ

Z t

t�s

xðsÞf ðxðsÞÞ

� �T R U

UT Q

� �xðsÞ

f ðxðsÞÞ

� �dsþ

Z 0

�s

Z t

tþb

_xTðaÞW _xðaÞdadb:

Calculating the derivative of VðxðtÞÞ along the solution of system (3) yields

_VðxðtÞÞ ¼ 2xTðtÞP _xðtÞ þ 2Xn

j¼1

djfjðxjðtÞÞ _xjðtÞ þxðtÞ

f ðxðtÞÞ

� �T R UUT Q

� �xðtÞ

f ðxðtÞÞ

� �� xðt � sÞ

f ðxðt � sÞÞ

� �T R UUT Q

� �xðt � sÞ

f ðxðt � sÞÞ

� �

þ 1s

Z t

t�s½s _xTðtÞW _xðtÞ � s _xTðbÞW _xðbÞ � 2sxTðtÞY1 _xðbÞ þ 2xTðtÞY1xðtÞ � 2xTðtÞY1xðt � sÞ � 2sxTðt � sÞY2 _xðbÞ

þ 2xTðt � sÞY2xðtÞ � 2xTðt � sÞY2xðt � sÞ � 2sf TðxðtÞÞY3 _xðbÞ þ 2f TðxðtÞÞY3xðtÞ � 2f TðxðtÞÞY3xðt � sÞ� 2sf Tðxðt � sÞÞY4 _xðbÞ þ 2f Tðxðt � sÞÞY4xðtÞ � 2f Tðxðt � sÞÞY4xðt � sÞ�db:

ð16Þ
Noting that T > 0 and S > 0 are diagonal matrices, we have
0 6 2f TðxðtÞÞTRxðtÞ � 2f TðxðtÞÞTf ðxðtÞÞ; ð17Þ0 6 f Tðxðt � sÞÞSRxðt � sÞ � 2f Tðxðt � sÞÞSf ðxðt � sÞÞ; ð18Þ

Then, it follows form (16)–(18) that

_VðxðtÞÞ 6 1s

Z t

t�s½2xTðtÞP _xðtÞ þ 2

Xn

j¼1

djfjðxjðtÞÞ _xjðtÞ þxðtÞ

f ðxðtÞÞ

� �T R U

UT Q

� �xðtÞ

f ðxðtÞÞ

� ��

xðt � sÞf ðxðt � sÞÞ

� �T R U

UT Q

� �xðt � sÞ

f ðxðt � sÞÞ

� �

þ �s _xTðtÞW _xðtÞ � s _xTðbÞW _xðbÞ � 2sxTðtÞY1 _xðbÞ þ 2xTðtÞY1xðtÞ � 2xTðtÞY1xðt � sÞ � 2sxTðt � sÞY2 _xðbÞþ 2xTðt � sÞY2xðtÞ � 2xTðt � sÞY2xðt � sÞ � 2sf TðxðtÞÞY3 _xðbÞ þ 2f TðxðtÞÞY3xðtÞ � 2f TðxðtÞÞY3xðtÞ� 2sf Tðxðt � sÞÞY4 _xðbÞ þ 2f Tðxðt � sÞÞY4xðtÞ � 2f Tðxðt � sÞÞY4xðt � sÞ � 2f TðxðtÞÞTf ðxðtÞÞ þ 2f TðxðtÞÞTRxðtÞ

� 2f Tðxðt � sÞÞSf ðxðt � sÞÞ þ 2f Tðxðt � sÞÞSRxðt � sÞ�db ¼ 1s

Z t

t�snTðt; bÞXnðt; bÞdb;

ð19Þ
where 2 3
X ¼



4 �sY1

�YT1 þ Y2 �R� Y2 � YT

2 �YT3 RS� YT

4 � U �sY2


0Dþ Q � 2T DW1 �sY3


1D �Q � 2S �sY4

�sYT1 �sYT

2 �sYT3 �sYT

4 �sW

66666647777775þ �s

�AT

0WT

0

WT1

0

2666664

3777775W

�AT

0WT

0

WT1

0

2666664

3777775

T

;

where nðtÞ ¼ ½xTðtÞ; xTðt � sÞ; f TðxðtÞÞ; f Tðxðt � sÞÞ; _xTðbÞ�T. Applying the Schur complement formula to (5), we can obtain thatfor all 0 6 s 6 �s



4

�YT1 þ Y2 �R� Y2 � YT

2 �YT3 RS� YT

4 � U


0Dþ Q � 2T DW1


1D �Q � 2S

266664

377775þ s

Y1

Y2

Y3

Y4

26664

37775W�1

Y1

Y2

Y3

Y4

26664

37775

T

þ �s

�AT

0W0

W1

26664

37775W

�AT

0WT

0

WT1

26664

37775

T

< 0:

ð20Þ


By the Schur complement formula, (20) implies that X < 0. Then,

_VðxðtÞÞ 6 �akxðtÞk2; ð21Þ

where a ¼ kminð�XÞ > 0. Thus, the system (3) is asymptotically stable for any delay 0 < s 6 �s. This completes the proof. h

Remark 1. In order to prove the sufficient condition of Theorem 2 [1], Xu used Lemma 1 to obtain the bound of some crossproduct terms. In this paper, a new Lyapunov-Krasovkii functional is proposed in Theorem 1. The advantage of this is that thebounding techniques on some cross product terms are not involved.

Remark 2. The free-weighting matrices Y1, Y2, Y3 and Y4 in the sufficient conditions (5) introduce some flexibility in check-ing the asymptotical stability for DCNNs. Thus, Theorem 1 will be less conservative than the condition derived in [1]. Such atechnique has been used recently by some researchers [4,18,19,22,23], to reduce the conservatism of the conventional LMIapproaches.

Remark 3. Unlike the Lyapunov–Krasovskii functionals in [22,23], the one given above contains not only an integral of termof state,

R tt�s xTðsÞRxðsÞds, but also an integral of a cross product term,

R tt�s xTðsÞUf ðxðsÞÞds. The advantage of this is that The-

orem 1 can lead to better stability results for DCNNs.

Next, we provide delay-independent asymptotical stability condition in the following corollary.

Corollary 1. The origin of the DCNNs in (3) is the unique equilibrium point and it is globally asymptotically stable for all delays > 0 if there exist symmetric P > 0, Q > 0, R > 0 and three diagonal matrices D > 0, S > 0, T > 0 and matrix U such that thefollowing LMIs hold:

�PA� AP þ R 0 PW0 � ADþ TRþ U PW1

0 �R 0 RS� U

WT0P � DAþ RT þ UT 0 K DW1

WT1P SR� UT WT

1D �Q � 2S

266666664

377777775< 0; ð22Þ

R U

UT Q

" #P 0; ð23Þ

where R ¼ diagfr1; r; . . . ; rng, K ¼ DW0 þWT0Dþ Q � 2T.

Proof. The uniqueness of the equilibrium point can be established by following a similar line as in the proof of Theorem 1, soit is omitted. Construct the following Lyapunov–Krasovskii functional:

VðxðtÞÞ ¼ xTðtÞPxðtÞ þ 2Xn

j¼1

dj

Z xj

0fjðsÞdsþ

Z t

t�s

xðsÞf ðxðsÞÞ

� �T R U

UT Q

� �xðsÞ

f ðxðsÞÞ

� �:

We can prove that the unique point of (3) is globally asymptotically stable. This completes the proof. h

Remark 4. In this case, if we set A ¼ I and U ¼ 0, then Corollary 1 in this note yields Theorem 1 in [3] with constant delay.

4. Numerical example

This section provides a numerical example borrowed form [1] to demonstrate the effectiveness of the criteria presented inthis paper.

Example 1. Consider the DCNNs (3) with parameter

A ¼

1:2769 0 0 0

0 0:6231 0 0

0 0 0:923 0

0 0 0 0:448

266664

377775;

W0 ¼

�0:0373 0:4852 �0:3351 0:2336�1:6033 0:5988 �0:3224 1:23520:3394 �0:086 �0:3824 �0:5785�0:1311 0:3253 �0:9534 �0:5015

26664

37775;


W1 ¼

0:8674 �1:2405 �0:5325 0:0220:0474 �0:9164 0:036 0:98161:8495 2:6117 �0:3788 0:8428�2:0413 0:5179 1:1734 �0:2775

26664

37775;

R ¼ diagf0:1137; 0:1279; 0:7994; 0:2386g:

For this DCNNs, using Matlab and the LMI Control Toolbox [21], we can check the asymptotical stability for all delays sat-isfying 0 6 s 6 �s ¼ 1:4224 by Theorem 2 [1], 0 6 s 6 �s ¼ 3:5509 by Theorem 1 [22] and by Theorem 1 [23], respectively.However, Theorem 1 in this paper yields the largest allowed value of �s ¼ 3:6502. This shows that the condition given by The-orem 1 is much less conservative than the one in [1] in checking the asymptotic stability of a given DCNNs. We obtain a solu-tion as follows:

P ¼

1:7751 0:7678 �0:8836 �0:4192

0:7678 3:0117 �1:2203 �0:0786

�0:8836 �1:2203 3:6523 �0:2317

�0:4192 �0:0786 �0:2317 1:6600

266664

377775;

Q ¼

116:3045 6:6840 �16:7720 20:78756:6840 67:8381 �4:8381 �12:8679�16:7720 �4:8381 14:3657 0:937120:7875 �12:8679 0:9371 14:2889

26664

37775;

R ¼

2:3580 0:0188 0:0541 �0:01240:0188 1:2542 0:4085 �0:11550:0541 0:4085 0:9546 �0:4263�0:0124 �0:1155 �0:4263 0:3576

26664

37775;

W ¼

0:1079 0:2946 �0:2209 �0:03770:2946 0:8661 �0:7592 0:0141�0:2209 �0:7592 1:0719 �0:4806�0:0377 0:0141 �0:4806 0:5399

26664

37775;

D ¼

0:0013 0 0 0

0 0:0001 0 0

0 0 1:6306 0

0 0 0 0:0001

266664

377775;

S ¼

256:6749 0 0 00 124:3914 0 00 0 1:0046 00 0 0 2:7193

26664

37775;

T ¼

158:8460 0 0 0

0 59:3360 0 0

0 0 12:7373 0

0 0 0 21:0728

266664

377775;

Y1 ¼

�0:0251 �0:0675 0:0480 0:0116

�0:0753 �0:2206 0:1904 �0:0002

0:0529 0:1841 �0:2664 0:1241

0:0087 �0:0081 0:1339 �0:1462

266664

377775;

Y2 ¼

0:0275 0:0750 �0:0561 �0:0097

0:0773 0:2274 �0:1996 0:0040

�0:0584 �0:2002 0:2806 �0:1244

�0:0109 0:0010 �0:1253 0:1434

266664

377775;


Y3 ¼

�0:0041 �0:0059 �0:0137 0:0216�0:0096 �0:0271 0:0198 0:00410:0116 0:0371 �0:0438 0:01360:0010 0:0015 0:0041 �0:0062

26664

37775;

Y4 ¼

0:0215 0:0632 �0:0660 0:01340:0345 0:1000 �0:0828 �0:00400:0036 0:0075 0:0041 �0:0121�0:0013 �0:0021 �0:0050 0:0078

26664

37775;

U ¼

�6:9297 1:3009 �1:0147 �0:72680:9044 �2:3688 �1:3844 0:48865:3365 0:3354 �2:1041 0:4190�3:3765 0:2442 1:3461 �0:8093

26664

37775:

5. Conclusion

This paper has investigated the problems of the existence of a unique equilibrium and its global asymptotic stability forDCNNs. By construction of a new Lyapunov–Krasovskii functional, both delay-dependent and delay-independent sufficientLMI-based conditions have been developed. These conditions can be checked easily. Example has been provided to demon-strate the validity of the proposed results.

Acknowledgements

The authors thank the anonymous reviewers for their useful comments. This work was supported by the National ScienceFoundation of China (No. 60773190), the Scientific Innovation Team Project of Hubei Provincial Department of Education(T200809), Natural Science Research Project of Hubei Provincial Department of Education under the Grant (D20081306).

References

[1] Xu S, Lam J, Ho DWC, Zou Y. Novel global asymptotic stability criteria for delayed cellular neural networks. IEEE Trans Circ Syst II 2005;52(6):349–53.[2] Xu S, Lam J. A new approach to exponential stability analysis of neural networks with time-varying delays. Neural Networks 2006;19:76–83.[3] He Y, Wu M, She JH. An improved global asymptotic stability criterion for delayed cellular neural networks. IEEE Trans Neural Networks

2006;17(1):250–2.[4] He Y, Wu M, She JH. Delay-dependent exponential stability of delayed neural networks with time-varying delay. IEEE Trans Circ Syst II

2006;53(7):553–7.[5] Arik S, Tavsanoglu V. Equilibrium analysis of delayed CNNs. IEEE Trans Circ Syst I 1998;45(2):168–71.[6] Arik S. An improved global stability result for delayed cellular neural networks. IEEE Trans Circ Syst I 2002;49(8):1211–4.[7] Cao J. Global stability conditions for delayed CNNs. IEEE Trans Circ Syst I 2001;48(12):1330–3.[8] Cao J, Zhou D. Stability analysis of delayed cellular neural networks. Neural Networks 1998;11:1601–5.[9] Chua LO, Yang L. Cellular neural networks: applications. IEEE Trans Circuits Syst 1988;35(12):1273–90.

[10] Kohonen T. Self-organizing maps. Berlin, Germany: Springer-Verlag; 2001.[11] Cichocki A, Unbehauen R. Neural networks for optimization and signal processing. Chichester, UK: Wiley; 1993.[12] Liao X, Chen G, Sanchez EN. LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans Circ Syst I

2002;49(7):1033–9.[13] Arik S. Global asymptotic stability of a larger class of neural networks with constant time delay. Phys Lett A 2003;311:504–11.[14] Zhang Q, Ma R, Xu J. Stability of cellular neural networks with delay. Electron Lett 2001;37:575–6.[15] Kolmanovskii VB, Myshkis AD. Introduction to the theory and applications of functional differential equations. Dordrecht, The Netherlands: Kluwer;

1999.[16] Xu S, Lam J, Yang C. Robust stability analysis and stabilization for uncertain linear neutral delay systems. Int J Syst Sci 2002;33:1195–206.[17] Civalleri PP, Gilli M, Pandolfi L. On stability of cellular neural networks with delay. IEEE Trans Circuits Syst I 1993;40(2):157–65.[18] Xu S, Lam J, Zhong M. New exponential estimates for time-delay systems. IEEE Trans Automa Control 2006;51(9):1501–5.[19] Lin C, Wang Q, Lee T. A less conservative robust stability test for linear uncertain time-delay systems. IEEE Trans Automa Control 2006;51(1):87–91.[20] Boyd S, Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelhhia: SIAM; 1994.[21] Gahinet P, Nemirovski A, Laub AJ, Chilali M. LMI control toolbox for use with Matlab. In: The Math Works Inc.: users guide. Natick, MANatick (MA): The

Mathworks, Inc.; 1995.[22] He Y, Liu GP, Rees D, Wu M. Stability analysis for neural networks with time-varying interval delay. IEEE Trans Neural Networks 2007;18(6):1850–4.[23] Hua C, Long C, Guan X. New results on stability analysis of neural networks with time-varying delays. Phys Lett A 2006;352:335–40.

delay-dependent global asymptotic stability for delayed cellular neural networks

Documents