global stability of cellular neural networks with constant and variable delays

Nonlinear Analysis 53 (2003) 319–333www.elsevier.com/locate/na

Global stability of cellular neural networks withconstant and variable delays�

Xue Mei Lia ;1, Li Hong Huanga ; ∗, Huiyan ZhubaCollege of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082,

People’s Republic of ChinabDepartment of Mathematics, Nanhua University, Hengyang, Hunan 421001,

People’s Republic of China

Received 17 December 2001; accepted 1 March 2002

Abstract

This paper gives new conditions ensuring global asymptotic stability and global exponentialstability for cellular neural networks with constant delay and variable delay, respectively. Theseconditions are derived by using the essence of piecewise linearity of the output function ofcellular neural networks and by constructing Lyapunov functions and functionals. Furthermore,these conditions are signi2cantly weaker than those given in existing literature.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: DCNNs; Global asymptotic stability; Global exponential stability; Variable delay

1. Introduction

Since cellular neural networks (CNNs) were introduced by Chua and Yang in 1988[8], they have been widely studied both in theory and applications [9,5,20]. They havebeen successfully applied to signal processing, pattern recognition, optimization andassociative memories, especially in image processing and solving nonlinear algebraicequations. They are inherently local in nature and are easily to implement in very largescale integration (VLSI). Although electronic circuits of CNNs can be fabricated into

� This work was supported by the Foundation for University Excellent Teacher by the Ministry ofEducation of China and by the NNSF (10071016) of China.

∗ Corresponding author.E-mail address: [email protected] (L.H. Huang).

1 Current address: Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081,People’s Republic of China.

0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(02)00176 -1

320 X.M. Li et al. / Nonlinear Analysis 53 (2003) 319–333

chips by VLSI technology, the 2nite switching speed of ampli2ers and communicationtime will introduce the time delays in the interaction among the cells. Moreover, toprocess moving images, one must introduce the time delays in the signals transmit-ted among the cells. These lead to the model of CNNs with delay (DCNNs) [19].They have found applications in diGerent areas such as classi2cation of patterns andreconstruction of moving images.In CNN applications, convergence plays an important role. For example, in the ap-

plication of image processing where the main function of a CNN is to transform aninput image into a corresponding output image, normally represented by an equilib-rium point, it is necessary that the CNN must be completely stable in the sense thatevery trajectory converges to an equilibrium point; and in solving optimization prob-lems, the network must possess a unique and globally asymptotically stable (GAS)equilibrium point for every input vector. So far, some suIcient conditions ensuringcomplete stability for CNNs [6,7,12,23,24] and for DCNNs [10,12,21,22] have beenobtained, and some GAS results for CNNs can be found in [1,11,18]. In [2–4,16,17],a few results for GAS of DCNNs are presented and these results are independent ofthe delay parameter.The purpose of this paper is to establish some criteria for GAS of DCNNs. Section

2 introduces the model of DCNNs and notation, and gives some preliminary lemmason the output function. In Section 3, by using the essence of piecewise linearity of theoutput function of CNNs, we present a few new suIcient conditions ensuring GAS,among these conditions, some are independent of the delay parameter and some aredependent on the delay parameter. Moreover, these conditions are less restrictive thanthose given in the literature. In Section 4, conditions for global exponential stability ofDCNNs with variable delay are obtained by constructing Lyapunov functions. Examplesand discussion are given in Section 5.

2. Preliminaries

The dynamics of a continuous time DCNN can be described by the followingtime-delayed functional diGerential system

x(t) =−x(t) + Ay(x(t)) + A�y(x(t − �)) + u; (1)

where x(·) = [x1(·); : : : ; xn(·)]T is the state vector, y(x(·)) = [y1(x1(·)); : : : ; yn(xn(·))]Tis the output vector, u= [u1; : : : ; un]T is a constant input vector, A= [aij]n×n and A� =[a�ij]n×n are the feedback matrix and delayed feedback matrix, respectively, � is thedelay parameter, and the output function is de2ned as

yi(xi(·)) = 0:5(|xi(·) + 1| − |xi(·)− 1|); i = 1; : : : ; n: (2)

If A� = 0, then (1) becomes the standard CNN without delay.Assume that system (1) has an equilibrium point x∗ = [x∗1 ; : : : ; x

∗n]

T for a given u.Let y∗ =y(x∗)= [y∗1 ; : : : ; y

∗n ]

T. Then system (1) can be transformed into the followingform:

x(t) =−(x(t)− x∗) + A(y(x(t))− y∗) + A�(y(x(t − �))− y∗): (3)

X.M. Li et al. / Nonlinear Analysis 53 (2003) 319–333 321

The initial conditions x0( ) = �( ) associated with (1) are of the form

xi( ) = �i( ); ∈ [− �; 0];where �i( )∈C([− �; 0]; R); i = 1; : : : ; n.In this paper, we denote: a vector norm by ‖x‖= (x21 + · · ·+ x2n)1=2, a matrix norm

by ‖A‖ = (max{�; � is an eigenvalue of ATA})1=2, i.e. the spectral norm of matrix Aand

‖�− x∗‖0 = max−�6 60

‖�( )− x∗‖:

We will use D+ to denote the Dini derivative. For any continuous function V :R→ R,the Dini derivative of V (t) is de2ned as

D+V (t) = lim suph→0+

V (t + h)− V (t)h

:

If x(t) is a solution of (1), yi(t) ≡ yi(xi(t)) is piecewise linear and possibly is notderivable with respect to t (such as |xi(t0)| = 1), but its right derivative is existent,which is denoted by y i(t). By using |yi(t0+h)−yi(t0)|6 |xi(t0+h)−xi(t0)|, it followsthat

y i(t0) = limh→0+

yi(t0 + h)− yi(t0)h

=

0 if |xi(t0)|¿ 1;

xi(t0) if |xi(t0)|¡ 1;

0 if xi(t0) = 1 and xi(t0)¿ 0;

xi(t0) if xi(t0) = 1 and xi(t0)¡ 0;

0 if xi(t0) =−1 and xi(t0)6 0;

xi(t0) if xi(t0) =−1 and xi(t0)¿ 0:

Let ni(xi(t0)) = 1 iG y i(t0) = xi(t0), and ni(xi(t0)) = 0 iG y i(t0) = 0. Set N (x) =diag[n1(x1); : : : ; nn(xn)]. We have

Ny(t) = Nx(t); N y(t) = y(t); y(t) = Nx(t): (4)

To study the convergence of system (1), we need to study the output function, whoseproperties are useful for analyzing the convergence of system (1).

Lemma 1. We have

(i) (yi − y∗i )26 (yi − y∗i )(xi − x∗i )6 (xi − x∗i )2; and

(ii) (yi − y∗i )26 (yi − y∗i )(xi − y∗i )6 (yi − y∗i )(xi − x∗i ) (5)

for i = 1; : : : ; n.

Proof.

(i) Since yi(xi)) is piecewise linear, the result follows directly.


(ii) We 2rst prove the left inequality. If xi ∈ [− 1; 1], we have yi = xi, and the resultfollows easily. If xi ∈ (−∞;−1), we have yi=−1; yi−y∗i 6 0 and xi−y∗i ¡yi−y∗i ,then the left inequality holds. If xi ∈ (1;∞), we have yi = 1; yi − y∗i ¿ 0 andxi − y∗i ¿yi − y∗i implies that the left inequality also holds. By discussing x∗iinstead of xi, it is similarly to prove that the right inequality holds.

Lemma 2. Suppose that for the trajectory x(t) of (1), limt→∞ y(t) = y∗. Thenlimt→∞ x(t) = x∗.

Proof. By the variations of constants formula, (3) implies that

‖x(t)− x∗‖

6e−t[‖x(0)−x∗‖+

∫ t

0es(‖A‖‖y(s)−y∗‖+ ‖A�‖‖y(s− �)−y∗‖) ds

]: (6)

As limt→∞‖y(t)−y∗‖=0; (6) implies that limt→∞‖x(t)−x∗‖=0; i.e. limt→∞ x(t)=x∗.

Lemma 3. Suppose that there exists a positive constant � such that ‖y(t) − y∗‖6 �‖�−x∗‖0 for the trajectory x(t) of (1) with the initial condition x0( )=�( ) andt¿ 0. Then there exists a positive constant � such that ‖x(t)− x∗‖6 �‖�− x∗‖0 fort¿ 0.

Proof. By the variations of constants formula; the proof is similar to that of Lemma2; and it is omitted.The following Lyapunov function plays an important role in the proofs of stability

in Sections 3 and 4.

Lemma 4. The Lyapunov function

V1(x) = (y − y∗)TD(x − y∗)− 12 (y − y∗)TD(y − y∗)

satis9es

(i) 12 (y − y∗)TD(y − y∗)6V1(x)6 (y − y∗)TD(x − x∗); and

(ii) D+V1 = (y − y∗)TDx;

where D is a positive diagonal matrix and D+V1 is the right derivative of V1 withrespect to t along the trajectories of system (1).

Proof.

(i) By (5); the result follows easily.(ii) By (4); we have

D+V1 = y TD(x − y∗) + (y − y∗)TDx − y TD(y − y∗)= y TND(x − y∗) + (y − y∗)TDx − y TND(y − y∗)


= y TDN (y − y∗) + (y − y∗)TDx − y TDN (y − y∗)= (y − y∗)TDx:

In the results in Sections 3 and 4, we will require the Lyapunov diagonally stablematrix.

De�nition (Hershkowitz [14]): A real square matrix A is said to be Lyapunov diago-nally stable if there exists a positive diagonal matrix D such that DA+ATD is positivede2nite.

3. Global stability results

In this section, we study the global asymptotic stability of (1) with constant delay.Let H denote A+ A� and I denote the identity matrix.

Lemma 5. If I − H is Lyapunov diagonally stable; then for every input vector u;system (1) has a unique equilibrium point.

Proof. The equilibrium points of (1) satis2es the algebraic equation

−x + (A+ A�)y(x) + u= 0:

From Theorem 3 in [11]; it follows that the above algebraic equation has a uniquesolution.

Theorem 1. If �‖A�‖¡ 1 and there exists a positive diagonal matrix D such that

D(H − I) + (H − I)TD + 2�‖A�‖(1− �‖A�‖)−1‖D‖‖H − I‖Iis negative de9nite; then for every input vector u satisfying ‖u‖6 ‖(H − I)−1‖−1;system (1) has a unique equilibrium point which is GAS.

Proof. The hypothesis implies that I−H is Lyapunov diagonally stable. Hence; system(1) has a unique equilibrium point x∗ by Lemma 5; and because of ‖u‖6 ‖(H −I)−1‖−1; we have ‖x∗‖6 1 and y∗ = x∗. In order to prove its GAS; we consider thefollowing Lyapunov functional

V (xt) = (y(t)− y∗)TD(x(t)− y∗)− 12(y(t)− y∗)TD(y(t)− y∗)

+∫ t

t−�[A�(y(s)− y(t))]Tf(s− t)[A�(y(s)− y(t))] ds;

where f( ) is a positive function with its continuous derivative on [−�; 0]; which willbe determined in the following proof.


Calculating the right derivative D+V of V with respect to t along the trajectories ofsystem (1), and using (3), Lemma 4(ii), and Lemma 1(i), we have

D+V 6−(y(t)− y∗)TD(y(t)− y∗) + (y(t)− y∗)TD(A+ A�)(y(t)− y∗)+ (y(t)− y∗)TDA�(y(t − �)− y(t))− [A�(y(t − �)− y(t))]Tf(−�)[A�(y(t − �)− y(t))]

−∫ t

t−�[A�(y(s)− y(t))]Tf′(s− t)[A�(y(s)− y(t))] ds

− 2∫ t

t−�[A�(y(s)− y(t))]Tf(s− t)A�N [− (y(t)− y∗)

+ (A+ A�)(y(t)− y∗) + A�(y(t − �)− y(t))] ds

=∫ 0

−�

{(y(t)− y∗)T D(H − I) + (H − I)TD

2�(y(t)− y∗)

+ (y(t)− y∗)T D2�A�(y(t − �)− y(t))

+ [A�(y(t − �)− y(t))]T D2�

(y(t)− y∗)

−[A�(y(t − �)− y(t))]T f(−�)�

[A�(y(t − �)− y(t))]

− [A�(y(t + )− y(t))]Tf′( )[A�(y(t + )− y(t))]

− [A�(y(t + )− y(t))]Tf( )A�N (H − I)(y(t)− y∗)

− (y(t)− y∗)T[f( )A�N (H − I)]TA�(y(t + )− y(t))

− [A�(y(t + )− y(t))]Tf( )A�NA�(y(t − �)− y(t))

− [A�(y(t − �)− y(t))]T(f( )A�N )TA�(y(t + )− y(t))}

d

≡∫ 0

−��(t; )TQN ( )�(t; ) d ; (7)

where

QN ( ) =

12�

[D(H − I) + (H − I)TD] D=(2�) −[f( )A�N (H − I)]T

D=(2�) −f(−�)I=� −(f( )A�N )T

−f( )A�N (H − I) −f( )A�N −f′( )I


and

�(t; )T = ((y(t)− y∗)T; [A�(y(t − �)− y(t))]T; [A�(y(t + )− y(t))]T)≡ (�T1 ; �

T2 ; �

T3 ):

Let �0 denote the minimum eigenvalue of − 12 [D(H−I)+(H−I)TD]. By the hypothesis,

it implies �0¿ 0. Hence,

�(t; )TQN ( )�(t; )6−�0�‖�1‖2 + 1

�‖D‖‖�1‖‖�2‖ − f(−�)

�‖�2‖2 − f′( )‖�3‖2

+ 2f( )‖H − I‖‖A�‖‖�1‖‖�3‖+ 2f( )‖A�‖‖�2‖‖�3‖: (8)In order to derive that the quadratic form (8) is negative de2nite, it is necessary that

f(−�)¿ ‖D‖24�0

(9)

and

(4�0f(−�)− ‖D‖2)f′( )

− 4�(‖D‖‖H − I‖+ f(−�)‖H − I‖2 + �0)‖A�‖2f2( )¿ 0: (10)

If there exists f( )∈C1[− �; 0] such that (9) and (10) hold, then we derive that thequadratic form (8) is negative de2nite. Set

f( ) =l

k(c − ) ;

where 0¡l¡ 1; c¿ 0 to ensure the continuity of function f( ) on [− �; 0] and

k =4�(‖D‖‖H − I‖+ f(−�)‖H − I‖2 + �0)‖A�‖2

4�0f(−�)− ‖D‖2 :

The constraint c¿ 0 has to be satis2ed, it implies that f(−�)=l=k(c+�)¡ 1=k(c+�),i.e. 1=kf(−�)− �¿c. We can 2nd l∈ (0; 1) and c¿ 0, and the derivative of functionf( ) shows that inequality (10) is always veri2ed if

1kf(−�) − �¿ 0:

By replacing the expression of k, we obtain

�2‖A�‖2‖H − I‖2f2(−�)

− [�0 − �2‖A�‖2(‖D‖‖H − I‖+ �0)]f(−�) + ‖D‖24

¡ 0: (11)

Using the vertex coordinates of the parabola, we take

f(−�) = �0 − �2‖A�‖2(‖D‖‖H − I‖+ �0)2�2‖A�‖2‖H − I‖2 ; (12)


then (11) holds if

�2‖A�‖2‖D‖2‖H − I‖2 − [�0 − �2‖A�‖2(‖D‖‖H − I‖+ �0)]2¡ 0

which is equivalent to

�‖A�‖‖D‖‖H − I‖¡�0 − �2‖A�‖2(‖D‖‖H − I‖+ �0): (13)

By the assumption of the theorem, (13) is equivalent to

�0(1− �‖A�‖)¿�‖A�‖‖D‖‖H − I‖: (14)

The assumption of the theorem can imply (14). So far, we have proven that theassumption of the theorem implies inequality (10). Next, we must verify that (12)satis2es (9). In fact, the assumption of the theorem can imply that �‖A�‖‖D‖‖H −I‖¡�0. Using (13), it follows that (12) satis2es (9). Thus, we have proven that thequadratic form (8) is negative de2nite. By (7) and (8), there exists a constant r ¿ 0such that

D+V 6− r‖�1‖2 =−r‖y(t)− y∗‖2: (15)

By Proposition 2.2 in [21], the solution of DCNN (1) is bounded. Hence, V (t) ≡ V (xt)is bounded and y(t) is uniformly continuous on [0;∞). It follows from (15) that

limt→∞y(t) = y

∗:

By Lemma 2, we have limt→∞ x(t)=x∗, i.e. x∗ is globally attractive. In the following,we will prove that x∗ is stable. In fact, D+V 6 0 implies that V (t)6V (0). By Lemma1(i), Lemma 4(i) and the expression of V (t), we obtain

12 (y(t)− y∗)TD(y(t)− y∗)6V1(t)6V (t)

and

V (t)6 (x(0)− x∗)TD(x(0)− x∗) +∫ 0

−�f(s)‖A�‖2‖x(s)− x(0)‖2 ds

6 (‖D‖+ 4�f(0)‖A�‖2)‖�− x∗‖20which derive that ‖y(t) − y∗‖6 �‖� − x∗‖0 with some constant �. By Lemma 3, x∗

is stable. The proof of Theorem 1 is complete.

Corollary 1. Suppose that H +HT is negative de9nite; and �‖A�‖6 1=(1+ ‖H − I‖).Then for every input vector u satisfying ‖u‖6 ‖(H − I)−1‖−1; system (1) has aunique equilibrium point which is GAS.


Proof. The assumptions of the corollary implies that the condition of Theorem 1 holdsby taking D = I ; and hence the conclusion of Corollary 1 follows immediately fromTheorem 1.

Remark 1. In most of the dynamic neural network applications; either (i) the externalinputs u(t) are 2xed at some constant values; usually zero; and the input data are fedinto the network via initial conditions rather than the external inputs; or (ii) the inputdata are fed via the external inputs; with resetting of the initial conditions to zero beforethe network is run (see [13]). Thus; though we constrain the external input vector uin Theorem 1; Theorem 1 can be used in most of the CNN applications.

Theorem 2. If there exists a positive diagonal matrix D such that

D(A− I) + (A− I)TD + 2‖DA�‖I

is negative de9nite; then for every input vector u; system (1) has a unique equilibriumpoint which is GAS.

Proof. For any x∈Rn (the real n-space);

xT[D(H − I)+(H − I)TD]x = xT[D(A− I)+(A− I)TD]x+xT[DA�+(A�)TD]x

6 xT[D(A− I) + (A− I)TD]x + 2‖DA�‖‖x‖2

= xT[D(A− I) + (A− I)TD + 2‖DA�‖I ]x;

the hypothesis implies that I −H is Lyapunov diagonally stable; and by Lemma 5; forevery input vector u; system (1) has a unique equilibrium point x∗. To show that x∗

is GAS; consider the Lyapunov functional

V (xt) = (y(t)− y∗)TD(x(t)− y∗)− 12(y(t)− y∗)TD(y(t)− y∗)

+%02

∫ t

t−�(y(s)− y∗)T(y(s)− y∗) ds;

where %0 is the minimum eigenvalue of − 12 [D(A−I)+(A−I)TD]; and by the hypothesis;

%0¿ 0. By evaluating the time right derivative of V along the trajectories of system(1); and by using (3); Lemma 4(ii); and Lemma 1(i); we obtain

D+V 6−(y(t)− y∗)TD(y(t)− y∗)+ (y(t)− y∗)TD[A(y(t)− y∗) + A�(y(t − �)− y∗)]

+%02(y(t)− y∗)T(y(t)− y∗)− %0

2(y(t − �)− y∗)T(y(t − �)− y∗)

6 (y(t)− y∗)T 12[D(A− I) + (A− I)TD + %0](y(t)− y∗)


+ ‖DA�‖‖y(t)− y∗‖‖y(t − �)− y∗‖ − %02‖y(t − �)− y∗‖2

6−%02‖y(t)− y∗‖2 − %0

2‖y(t − �)− y∗‖2

+ ‖DA�‖‖y(t)− y∗‖‖y(t − �)− y∗‖: (16)

The hypothesis of the theorem implies that %0¿ ‖DA�‖. Thus; the quadratic form (16)is negative de2nite; and there exists a positive constant r such that

D+V 6− r‖y(t)− y∗‖2:Similar to the proof of Theorem 1; we can prove that x∗ is GAS. The proof of Theorem2 is thus complete.

Corollary 2. Suppose that A+AT is negative de9nite and ‖A�‖6 1. Then system (1)has a unique equilibrium point which is GAS for every input vector u.

Proof. The hypotheses of the corollary derive the condition of Theorem 2.

Corollary 3. If there is a constant &¿ 0 such that A + AT + &I is negative de9niteand ‖A�‖6√

1 + &; then system (1) has a unique equilibrium point which is GASfor every input vector u.

Proof. Since ‖A�‖6√1 + &6 1 + &=2; and

A+ AT − 2I + 2‖A�‖I6A+ AT + &I

is negative de2nite; by taking D = I ; the condition of Theorem 2 holds. By Theorem2; we obtain the conclusion of Corollary 3.From the proof of Corollary 3, it easily implies the following corollary.

Corollary 4. If there is a constant &¿ 0 such that A + AT + &I is negative de9niteand ‖A�‖6 1 + 1

2 &; then system (1) has a unique equilibrium point which is GASfor every input vector u.

Remark 2. Corollary 2 is the main result of [3] and Corollary 3 is the main theoremof [17]; i.e. main results in [3;17] are special cases of Theorem 2. Obviously; when& �=0; Corollary 4 is superior to Corollary 3.

Remark 3. To date; conditions proposed in the literature on global asymptotic stabilityof DCNNs are independent of the delay parameter �. However; the condition presentedin Theorem 1 relates to the delay parameter �. In order to ensure GAS for DCNNs; wemay use Theorem 1 for small delay � and Theorem 2 for large delay �; respectively.

Remark 4. To the best of the authors’ knowledge; the best one of existing results onGAS of CNNs without delay has been obtained under the assumption that I − A isLyapunov diagonally stable (i.e. Theorem 4 in [11] with D PG

−1= I for CNNs). When

A� = 0; Theorem 2 in this paper is just this result.


Remark 5. When the delay � is suIciently small; the condition of Theorem 1 statesthat Lyapunov diagonal stability of I − H implies GAS of (1) for small ‖u‖.

4. Global exponential stability for variable delay

In this section, we consider the case where the delay in the system (1) is nonconstant.Speci2cally, we study the model of CNNs with variable delay as

x(t) =−x(t) + Ay(x(t)) + A�y(x(t − �(t))) + u; (17)

where �(t) is nonnegative and bounded, say 06 �(t)6 �.Joy [16] recently considered the absolute stability for generalized output function of

(17), where it is assumed that �(t) is diGerentiable.To obtain the main result in this section, we need the following two lemmas.

Lemma 6 (Hunhai Hou and Jixin Qian [15]). Assume that V : [ − �;∞) → R+ is acontinuous nonnegative function; 06 �(t)6 �; and there are two positive constants aand b with a¿b such that for t¿ 0; the following inequality holds:

D+V (t)6− aV (t) + bV (t − �(t)):Then there exists a positive constant r such that

V (t)6 PV (0) exp(−rt) for t¿− �;where PV (0) = sup−�6s60 V (s).

Lemma 7. Suppose that x∗ is the unique equilibrium point of (17) and there are twopositive constants � and % such that for the trajectory x(t) of (17) with the initialcondition x0( ) = �( ) and t¿− �;

‖y(t)− y∗‖6 �‖�− x∗‖0 exp(−%t): (18)

Then there exist two positive constants � and � such that for t¿ 0;

‖x(t)− x∗‖6 �‖�− x∗‖0 exp(−�t):

Proof. By using the variations of constants formula and (18); (17) implies that

‖x(t)− x∗‖6 e−t[‖x(0)− x∗‖

+∫ t

0es(‖A‖‖y(s)− y∗‖+ ‖A�‖‖y(s− �(s))− y∗‖) ds

]

6 e−t[‖x(0)− x∗‖+ �‖�− x∗‖0(‖A‖+ ‖A�‖e%�)

∫ t

0e(1−%)s ds

]

6 �‖�− x∗‖0e−�t ;where � = min{1; %} for % �=1; and 0¡�¡ 1 for % = 1; and � can be determinedcorrespondingly.


The main result in this section is as follows:

Theorem 3. If there exists a positive diagonal matrix D = diag[d1; : : : ; dn] such thatthe following conditions hold

(i) DA+ ATD − D + ‖DA�‖I is negative semide9nite;(ii) ‖D−1‖‖DA�‖¡ 1;

then for every input vector u; system (17) has a unique equilibrium point which isglobally exponentially stable.

Proof. The equilibrium points of (17) are the solutions of the following equation

−x + (A+ A�)y(x) + u= 0:

To show the uniqueness of the equilibrium point; by Lemma 5; it suIces to show thatI −H (where H = A+ A�) is Lyapunov diagonally stable. By condition (ii); we have‖DA�‖D−1D − D¡ 0. Noticing that for any x∈Rn;

xT[D(H − I) + (H − I)TD]x6 xT[DA+ ATD − D + ‖DA�‖I + ‖DA�‖D−1D − D]x;

by condition (i); it follows that I−H is Lyapunov diagonally stable. Let x∗ denote theunique equilibrium point. Next; we will show that x∗ is globally exponentially stable.To this end; let us consider the Lyapunov function

V1(x) = (y − y∗)TD(x − y∗)− 12 (y − y∗)TD(y − y∗):

By Lemma 4(ii); Lemma 1(ii); and the condition (i); the right derivative of V1 alongthe trajectories of (17) satis2es

D+V1 = −(y(t)− y∗)TD(x(t)− x∗)+ (y(t)− y∗)TD[A(y(t)− y∗) + A�(y(t − �(t))− y∗)]

6−V1(x(t))− 12(y(t)− y∗)TD(y(t)− y∗)

+ (y(t)− y∗)T DA+ ATD2

(y(t)− y∗)

+12‖DA�‖(‖y(t)− y∗‖2 + ‖y(t − �(t))− y∗‖2)

= −V1(x(t)) + 12(y(t)− y∗)T[DA+ ATD − D + ‖DA�‖I ](y(t)− y∗)

+12‖DA�‖‖y(t − �(t))− y∗‖2

6−V1(x(t)) + 12‖DA�‖‖y(t − �(t))− y∗‖2:


By Lemma 4(i); we have

‖y(t − �(t))− y∗‖26 2min16i6n{di} V1(x(t − �(t))) = 2‖D−1‖V1(x(t − �(t))):

Hence; for V1(t) ≡ V1(x(t)) we get

D+V16− V1(t) + ‖D−1‖‖DA�‖V1(t − �(t)):By Lemma 6 and condition (ii); there exists a constant r ¿ 0 such that

V1(t)6V1(0) exp(−rt) for t¿− �:Again by Lemma 4(i); we obtain

‖y(t)− y∗‖26 2‖D−1‖V1(t)6 2‖D−1‖ PV 1(0) exp(−rt)6 2‖D−1‖‖D‖‖�− x∗‖20 exp(−rt);

for t¿− �. That is‖y(t)− y∗‖6 (2‖D−1‖‖D‖) 12 ‖�− x∗‖0 exp

(− rt

2

)for t¿− �:

Thus; x∗ is globally exponentially stable by Lemma 7.

Corollary 5. Suppose that

(i) A+ AT is negative semide9nite; and(ii) ‖A�‖¡ 1.

Then system (17) has a unique equilibrium point which is globally exponentiallystable for every input vector u.

Proof. It is easy to check that the conditions of Theorem 3 are satis2ed by takingD = I ; and hence Theorem 3 implies Corollary 5.

Remark 6. The conditions of Corollary 5 are near the conditions of Theorem 1 in [3]for GAS of DCNNs with constant delay (i.e. Corollary 2 in this paper).

5. Examples and discussion

Example 1. Consider the following matrices

A=

[−1 0

2 −1

]and A� =

[−) )

0 −)

];

where )¿ 0. We have ‖A�‖=). It is easily veri2ed that A+AT is not negative de2nite;therefore; the results of [3;17] (i.e. Corollaries 2 and 3) cannot be applied. Furthermore;


the matrix

S =

[1− a11 − |a�11| −(|a12|+ |a�12|)−(|a21|+ |a�21|) 1− a22 − |a�22|

]=

[2− ) −)−2 2− )

]

is a nonsingular M -matrix for 0¡)¡ 3 −√5; and by Theorem 1 in [2]; system (1)

has a unique and globally asymptotically stable equilibrium point. But for )¿ 3−√5;

S is not a nonsingular M -matrix (see [14] for more details on nonsingular M -matrix);thus; the condition given in [2] does not hold.For 3−√

56 )¡ 1, take D = I , then

(A− I) + (A− I)T + 2‖A�‖I =[−4 + 2) 2

2 −4 + 2)

]

is negative de2nite, by Theorem 2 in this paper, system (1) has a unique and globallyasymptotically stable equilibrium point.For )¿ 1, we may discuss the GAS of system (1) by using Theorem 1. Since

‖H − I‖= 2 + )+√2); ‖A�‖= ), take D = I , then

H − I + (H − I)T + 2�‖A�‖(1− �‖A�‖)−1‖H − I‖I

=

[−4− 2)+ �)(2 + )+√2))(1− �))−1 2 + )

2 + ) −4− 2)+ �)(2 + )+√2))(1− �))−1

]

is negative de2nite if and only if

�¡2 + )

)(6 + 3)+ 2√2)): (19)

Hence, we also obtain the GAS of system (1) for )¿ 1 if � satis2es (19) and ‖u‖6 1.

Example 2. We continue to consider these matrices in Example 1; but the delay isvariable and bounded. It is easily seen that A+AT is negative semide2nite and ‖A�‖=).By Corollary 5; system (17) has a unique and globally exponentially stable equilibriumpoint if )¡ 1. The discussion in Example 1 represents that this result for variable delayis better.

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global stability of cellular neural networks with constant and variable delays

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