494 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 4, APRIL 2001

so that c := kerHc is the controllable part of . The following factsare equivalent:

1) there exists an autonomous behaviora = kerHa such that= c � a;

2) the pair(H�; Hc) is zero skew-prime.If any of the above conditions holds, thenVL(Ha) = VL(H�).

REFERENCES

[1] E. Fornasini, P. Rocha, and S. Zampieri, “State realization of 2-D finitedimensional autonomous systems,”SIAM J. Contr. Optim., vol. 31, no.6, pp. 1502–1517, 1993.

[2] E. Fornasini and M. E. Valcher, “nD polynomial matrices with appli-cations to multidimensional signal analysis,”Multidim. Syst. SignalProcess., vol. 8, no. 4, pp. 387–407, 1997.

[3] P. P. Khargonekar, T. T. Georgiou, and A. B. Öüzgler, “Skew-primepolynomial matrices: The polynomial-model approach,”Linear AlgebraAppl., vol. 50, pp. 403–435, 1983.

[4] V. K ucera and M. Sebek, “On dead-beat controllers,”IEEE Trans. Au-tomat. Contr., vol. AC-29, pp. 719–722, Aug. 1984.

[5] U. Oberst, “Multidimensional constant linear systems,”Acta Applic.Math., vol. 20, pp. 1–175, 1990.

[6] J. W. Polderman and J. C. Willems,Introduction to MathematicalSystems Theory: A Behavioral Approach. New York: Springer-Verlag,1997.

[7] P. Rocha, “Structure and representation of 2-D systems,” Ph.D. disserta-tion, Dept. of Math., Univ. of Groningen, Groningen, The Netherlands,1990.

[8] P. Rocha and J. C. Willems, “Controllability of 2-D systems,”IEEETrans. Automat. Contr., vol. 36, pp. 413–423, Apr. 1991.

[9] M. Sebek, “Two-sided equations and skew primeness,”Syst. ControlLett., vol. 12, no. 4, pp. 331–337, 1989.

[10] M. E. Valcher, “On the decomposition of two-dimensional behaviors,”Multidim. Syst. Signal. Process., vol. 11, no. 1/2, pp. 49–65, 2000.

[11] , “Characteristic cones and stability properties of two-dimensionalautonomous behaviors,”IEEE Trans. Circuits Syst. I, vol. 47, pp.290–302, Mar. 2000.

[12] W. A. Wolovich, “Skew prime polynomial matrices,”IEEE Trans. Au-tomat. Contr., vol. AC-23, pp. 880–887, Oct. 1978.

[13] J. Wood, E. Rogers, and D. Owens, “Controllable and autonomousnDlinear systems,”Multidim. Syst. Signal Process., vol. 10, pp. 33–70,1999.

[14] L. Xu, O. Saito, and K. Abe, “Bilateral polynomial matrix equations intwo indeterminates,”Multidim. Syst. Signal Process., vol. 1, no. 4, pp.363–379, 1990.

A Set of Stability Criteria for Delayed Cellular NeuralNetworks

Jinde Cao

Abstract—This brief presents a set of criteria on the global asymptoticstability of delayed cellular neural networks (DCNN) by constructing suit-able Lyapunov functionals, introducing ingeniously real parameters0, , , , , , , , with + = 1,

+ = 1, + = 1, + = 1 ( = 1 2 . . . ) andcombining with elementary inequality technique2 + . Thesecriteria are of theoretical and applicable important significance in signalprocessing, especially in speed detection of moving objects, processing ofmoving images and the design of networks since they possess infinitely ad-justable real parameters. This result is also discussed from the point of viewof its relationship to earlier results.

Index Terms—Cellular neural networks, delay, global asymptoticstability, inequality, Lyapunov functional, parameter, stable equilibriumpoints.

I. INTRODUCTION

IN a dynamical system which is globally asymptotically stable, thereexists a unique equilibrium state to which every solution converges.Cellular neural networks (CNNs) were introduced by L. O. Chua andL. Yang [1], [2] in 1988. They have found important applications insignal processing, especially in static image treatment. During the pastfew years, the problems of stability of cellular neural networks has beenone of the most active areas of research and has attracted the attentionof many researchers, we refer to [1]–[7], [9]–[20], [22]. One knowsthat the CNNs is formed by many units called cells, the structure ofthe CNNs is similar to that found in cellular automata, namely, anycell in a cellular neural network is connected only to its neighbor cells.A cell contains linear and nonlinear circuit elements, which typicallyare linear capacitors, linear resistors, linear and nonlinear controlledsources, and independent sources. The circuit diagram and connectionpattern implementing for the CNN can be found in [1], [2]. Delayed cel-lular neural networks (DCNNs) were first introduced in [3] and usedin various types of motion-related applications such as speed detectionof moving objects, processing of moving images and in pattern clas-sification. In order to achieve these tasks, a delay parameter was in-troduced into the CNNs system equation. The stability analysis in thiscase is much more difficult than for CNNs. There exist some resultsof stability for CNNs and DCNNs, we refer to [1], [2], [4], [13], [14],[20] and [5]–[7], [9]–[19], [22], respectively, and the references citedtherein. In this paper, we derived a set of criteria ensuring the globalasymptotic stability of DCNNs with more general output functions byconstructing suitable Lyapunov functionals [7]–[11], introducing in-geniously real parameterswi > 0, ��ij , ��ij , �

�

ij , ��

ij , �ij , �ij , �ij ,�ij 2 Rwith��ij+�

�

ij = 1,�ij+�ij = 1,��ij+��

ij = 1,�ij+�ij = 1(i; j = 1; 2; . . . ; n) and combining with elementary inequality tech-nique2ab � a2 + b

2. The results related in [1]–[7], [9]–[20], [22] andthe references cited therein are extended and improved. Moreover these

Manuscript received October 23, 2000; revised December 4, 2000. This workis supported in part by the Natural Science Foundation of Yunnan Province,China under Grant 1999F0017M and Grant 97A012G, and in part by the Sci-ence Foundation of Southeast University under Grant 9207011100 and Grant9207012049. This paper was recommended by Associate Editor P. Szolgay.

The author is with the Department of Applied Mathematics, Southeast Uni-versity, Nanjing 210096, R.O.C, and also with the Adult Education College,Yunnan University, Kunming 650091, R.O.C.

Publisher Item Identifier S 1057-7122(01)02951-8.

1057–7122/01$10.00 © 2001 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 4, APRIL 2001 495

criteria are independent of delays and possess infinitely adjustable realparameters. These are of prime importance and great interest in manyapplication fields and the design of networks, and are easy to check andapply in practice.

In the following, we investigated further a class of DCNNs, whichcan be described by delayed differential equations (namely, functionaldifferential equations)

x0i(t) = �cixi(t) +

n

j=1

aijfj(xj(t)) +

n

j=1

bijfj(xj(t� �j))

+ Ii; ci > 0; i = 1; 2; . . . ; n (1)

in whichn numbers of units in a neural network;xi(t) state of theith unit at timet;fj(xj(t)) output of thejth unit at timet;aij , bij , Ii, ci constant;aij strength of thejth unit on theith unit at timet;bij strength of thejth unit on theith unit at timet��j ;Ii external bias on theith unit;�j transmission delay along the axon of thejth unit

and is not negative constant,;ci rate with which theith unit will reset its potential

to the resting state in isolation when disconnectedfrom the network and external inputs.

Let A = faijg be the feedback matrix,B = fbijg be the delayedfeedback matrix. In the following, we assume that each of the relationbetween the output of the cellfi(i = 1; 2; . . . ; n) and the state of thecell possess following properties.

(H1) fi(i = 1; 2; . . . ; n) is bounded onR;(H2) There is a number�i > 0 such thatjfi(u) � fi(v)j �

�iju � vj for anyu; v 2 R.It is easy to find from(H2) thatfi is a continuous function onR.

In particular, if the relation between the output of the cell and the stateof the cell is described by a piecewise-linear functionfi(x) =

1

2(jx+

1j � jx � 1j), then it is easy to see that the functionfi clearly satisfythe hypotheses(H1) and(H2) above, and�i � 1(i = 1; 2; . . . ; n).The circuit implementing of (1) can be referred to [1], [3].

This paper is organized as follows. In Section II a set of new suffi-cient conditions are derived for the global asymptotic stability of theDCNNs on the parametersci, aij , bij , �j , wi > 0, ��ij , �

�ij , �

�ij , �

�ij ,

�ij , �ij , �ij , �ij 2 R (i; j = 1; 2; . . . ; n) by using Lyapunov func-tional method and combining with the techniques of inequality anal-ysis. Moreover, some examples to show the theory are given. In Sec-tion III we give some concluding remarks of the results.

II. STABILITY PROPERTIES IN THEDCNNS

To clarify our main results, we provide again following two lemmasand their proofs under more general conditions.

Lemma 1: For the DCNN (1), suppose that the output of the cellfi(i = 1; 2; . . . ; n) satisfy the hypotheses(H1) and(H2) above. Thenall solutions of the DCNN (1) remain bounded for[0;+1).

Proof: It is easy to observe that all solutions of the DCNN (1)satisfy differential inequalities of the form

�cixi(t)� i � x0i(t) � �cixi(t) + i (2)

where

i =

n

j=1

(jaij j+ jbij j) sups2R

jfj(s)j+ jIij

Using (2), one can easily prove that solutions of the DCNN (1) re-main bounded on[0;+1). This completes the proof.

Lemma 2: Let f(t) be a nonnegative function defined on[0;1)such thatf(t) is integrable on[0;1) (i.e., 1

0f(t)dt < +1) and

uniformly continuous on[0;1). Then,

limt!1

f(t) = 0:

Proof: Supposef(t) does not approach zero ast!1. This willmean that there exist a positive number� and a sequenceftkg ! 1such thatf(tk) > � > 0 for anyk � 1. The uniform continuity off(t) assures the existence of a positive� with the property thatf(t) >�=2 for jt� tkj � �, k � 1. We can without loss of generality assumethat the intervals(tk � �; tk + �) do not overlap. Therefore,

1

0

f(t)dt �

N

k=1

t +�

t ��

f(t)dt � N��

for any positive integerN and this contradicts the integrability off(t)on [0;1). Hence, the lemma follows.

Remark 1: Proof of the lemma can also refer to [21, pp. 4–5].Theorem 1: For the DCNN (1), suppose that the outputs of the cell

fi(i = 1; 2; . . . ; n) satisfy the hypotheses(H1) and(H2) above andthere exist costantswi > 0, ��ij , �

�ij , �

�ij , �

�ij , �ij , �ij , �ij , �ij 2

R(i; j = 1; 2; . . . ; n) such that

n

j=1

jaij j2� �

2�

j +wj

wi

jajij2� �

2�

i + jbij j2� �

2�

j

+wj

wi

jbjij2� �

2�

i < 2ci; i = 1; 2; . . . ; n

in which��ij ,��ij ,�

�ij , �

�ij ,�ij ,�ij ,�ij , �ij (i; j = 1; 2; . . . ; n) are any

real constant numbers with��ij +��ij = 1,�ij +�ij = 1, ��ij + ��ij =1, �ij + �ij = 1, �j(j = 1; 2; . . . ; n) is constant numbers of thehypotheses(H2). Then, the equilibriumx� of the DCNN (1) is globallyasymptotically stable independent of delays.

Proof: If x� = (x�1; x�2; . . . ; x

�n)

T is an equilibrium ofthe DCNN (1), one can derive from (1) that the deviationsyi(t) = xi(t)� x�i (i = 1; 2; . . . ; n) satisfy

y0i(t) = �ciyi(t) +

n

j=1

aij(fj(x�

j + yj(t))� fj(x�

j ))

+

n

j=1

bij(fj(x�

j + yj(t� �j))� fj(x�

j )) (3)

obviously,(0; 0; . . . ; 0)T is an equilibrium of (3). To prove the globalasymptotic stability ofx� of the DCNN (1), it is sufficient to provethe global asymptotic stability of the trivial solution of (3). The localexistence of solutions of (1) and (3) follows by the method of steps, theexistence on[0;+1) will be a consequence of our analysis below.

We consider the Lyapunov functional defined by

V (t) = V (y)(t) = 1

2

n

i=1

wi y2i (t) +

n

j=1

� jbij j2� �

2�

j

t

t��

y2j (s)ds (4)

wherewi > 0, �ij , �ij(i; j = 1; 2; . . . ; n) are any real constantnumbers.

496 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 4, APRIL 2001

Calculating the derivative ofV along the solution of (3), we get

dV

dt=

n

i=1

wi yi(t)(�ciyi(t) +

n

j=1

� aij(fj(x�

j + yj(t))� fj(x�

j ))

+

n

j=1

bij(fj(x�

j + yj(t� �j))� fj(x�

j ))

+1

2

n

j=1

jbij j2�

�2�

j (y2j (t)� y2

j (t� �j))

�

n

i=1

wi �ciy2

i (t) +

n

j=1

jaij j�j jyi(t)kyj(t)j

+

n

j=1

jbij j�j jyi(t)kyj(t� �j)j

+1

2

n

j=1

jbij j2�

�2�

j (y2j (t)� y2

j (t� �j))

=

n

i=1

wi �ciy2

i (t) +

n

j=1

� (jaij j�

��

j jyi(t)j)(jaij j�

��

j jyj(t)j)

+

n

j=1

(jbij j�

��

j jyi(t)j)(jbij j�

��

j jyj(t� �j)j)

+1

2

n

j=1

jbij j2�

�2�

j (y2j (t)� y2

j (t� �j)) (5)

where��ij , ��

ij , ��

ij , ��

ij , �ij , �ij , �ij , �ij (i; j = 1; 2; . . . ; n) are anyreal constant numbers with��ij +��ij = 1,�ij +�ij = 1, ��ij + ��ij =1, �ij + �ij = 1. Now estimating the right side of (5) by using theelementary inequality2ab � a2 + b2, we have

dV

dt�

n

i=1

wi �ciy2

i (t) +

n

j=1

1

2

� ((jaij j�

��

j jyi(t)j)2 + (jaij j

���

j jyj(t)j)2)

+

n

j=1

1

2((jbijj

���

j jyi(t)j)2

+ (jbij j�

��

j jyj(t� �j)j)2)

+1

2

n

j=1

jbij j2�

�2�

j (y2j (t)� y2

j (t� �j))

=

n

i=1

wi �ciy2

i (t) +

n

j=1

1

2

� (jaij j2�

�2�

j jyi(t)j2 + jaij j

2��2�

j jyj(t)j2)

+

n

j=1

1

2(jbij j

2��2�

j jyi(t)j2

+ jbij j2�

�2�

j jyj(t� �j)j2)

+1

2

n

j=1

jbij j2�

�2�

j (y2j (t)� y2

j (t� �j))

=

n

i=1

wi �ci +1

2

n

j=1

� (jaij j2�

�2�

j +wj

wi

jajij2�

�2�

i

+ jbij j2�

�2�

j +wj

wi

jbjij2�

�2�

i ) y2

i (t)

� �r

n

i=1

y2

i (t) (6)

where

r = min1�i�n

wi ci �1

2

n

j=1

jaij j2�

�2�

j +wj

wi

� jajij2�

�2�

i + jbij j2�

�2�

j

+wj

wi

jbjij2�

�2�

i > 0:

A consequence of (6) is that

V (y)(t) + rt

0

n

i=1

y2

i (s)ds � V (y)(0): (7)

It follows from (7) that

+1

0

n

i=1

y2

i (t)dt < +1: (8)

According to Lemma 1,xi(t) is bounded on(0;+1). This impliesboundedness ofyi(t), y0i(t) and on(0;+1); henceyi(t), y2i (t) is uni-formly continuous on(0;+1). By Lemma 2, it follows that

limt!+1

n

i=1

y2

i (t) = 0: (9)

It follows from (9) that zero solution of (3) is globally asymptoticallystable for any delays, thus the equilibrium of the DCNN (1) is alsoglobally asymptotically stable independent of delays. This completesthe proof.

Applying Theorem 1 above, we easily prove the following Corol-laries:

Corollary 1: For the DCNN (1), suppose that the outputs of the cellfi(i = 1; 2; . . . ; n) satisfy the hypotheses(H1) and(H2) above andthere exist constants��ij , �

�ij , �

�ij , �

�ij , �ij , �ij , �ij , �ij 2 R(i; j =

1; 2; . . . ; n) such that

n

j=1

(jaij j2�

�2�

j + jajij2�

�2�

i + jbij j2�

�2�

j

+ jbjij2�

�2�

i ) < 2ci; i = 1; 2; . . . ; n

in which��ij ,��ij ,��ij , �

�ij ,�ij ,�ij ,�ij , �ij (i; j = 1; 2; . . . ; n) are any

real constant numbers with��ij +��ij = 1,�ij +�ij = 1, ��ij + ��ij =1, �ij + �ij = 1, �j(j = 1; 2; . . . ; n) is constant numbers of thehypotheses(H2) above. Then the equilibriumx� of the DCNN (1) isalso globally asymptotically stable independent of delays.

Corollary 2: If the relation between the output of the cell and thestate of the cell is described by a piecewise-linear functionfi(x) =1

2(jx + 1j � jx � 1j) and there exist costants��ij , �

�ij , d�ij , �ij 2

R(i; j = 1; 2; . . . ; n) such that

n

j=1

(jaij j2� + jajij

2� + jbij j2� + jbjij

2� ) < 2ci;

i = 1; 2; . . . ; n

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 4, APRIL 2001 497

in which ��ij , ��

ij , �ij , �ij (i; j = 1; 2; . . . ; n) are any real constantnumbers with��ij + ��ij = 1, �ij + �ij = 1. Then the equilibriumx�

of the DCNN (1) is also globally asymptotically stable independent ofdelays.

Remark 2: In Corollary 1 above,

(1) If we take

(i) ��ij = ��i , ��ij = ��i , ��ij = ��i , ��ij = ��i , �ij = �i,�ij = �i, �ij = �i, �ij = �i;

(ii) ��ij = ��j , ��ij = ��j , ��ij = ��i , ��ij = ��i , �ij = �i,�ij = �i, �ij = �i, �ij = �i;

(iii) ��ij = ��i , ��ij = ��i , ��ij = ��i , ��ij = ��i , �ij = �i,�ij = �i, �ij = �j , �ij = �j ;

(iv) ��ij = ��j , ��ij = ��j , ��ij = ��i , ��ij = ��i , �ij = �i,�ij = �i, �ij = �j , �ij = �j ;

(v) ��ij = ��i , d��ij = ��i , ��ij = ��i , ��ij = ��i , �ij = �j ,�ij = �j , �ij = �j , �ij = �j ;

(vi) ��ij = ��j , ��ij = ��j , ��ij = ��i , ��ij = ��i , �ij = �j ,�ij = �j , �ij = �j , �ij = �j ;

(vii) ��ij = ��i , ��ij = ��i , ��ij = ��i , ��ij = ��i , �ij = �j ,�ij = �j , �ij = �i, �ij = �i;

(viii) ��ij = ��j , ��ij = ��j , ��ij = ��i , ��ij = ��i , �ij = �j ,�ij = �j , �ij = �i, �ij = �i

(i; j = 1; 2; . . . ; n), respectively, then we have derived [19,Theorem], i.e., the results in [19] is actually a special case ofCorollary 1 above;

(2) If we take

(i) ��ij = ��ij = �ij = �ij = 1

2, ��ij = ��ij = 1

2, �ij = �ij = 1

2;

(ii) ��ij = ��ij = �ij = �ij = 1

2, ��ij = 1, ��ij = 0, �ij = �ij

= 1

2;

(iii) ��ij = ��ij = �ij = �ij = 1

2, ��ij = 0, ��ij = 1, �ij = �ij

= 1

2;

(iv) ��ij = ��ij = �ij = �ij = 1

2, ��ij = ��ij = 1

2, �ij = 1,

�ij = 0

(i; j = 1; 2; . . . ; n), respectively, then we can derive main re-sults in [9], that is, [9] is also actually a special case of Corol-lary 1 above;

(3) If we take��ij = ��ij = �ij = �ij = 1

2, ��ij = ��j , ��ij = ��j , �ij =

�j , �ij = �j , we can also derive from Corollary 1 above that[7, Theorem 1] holds.

In the following, we will give some examples showing the conditionsgiven in the brief hold for different classes of feedback matrices thanthose given in [15], [12] and [22]. First, we restate the results of [12],[15] and [22].

Theorem 2 [12]: if i) A has nonnegative off-diagonal elements, ii)B has nonnegative elements and iii)�(A+B) is row sum dominant,then the system defined by (1) has a globally asymptotically stableequilibrium point for every constant input when the relation betweenthe output of the cell and the state of the cell is described by a piece-wise-linear functionfi(x) = 1

2(jx + 1j � jx � 1j) andci � 1.

Theorem 3 [15]: Let

sij =1� aii � jbiij; i = j,�(jaij j+ jbij j); i 6= j

with nj=1 jbjij 6= 0 for everyi. If S = fsijg is a nonsingularM

matrix (i.e., the real part of every eigenvalue ofS is positive), then thesystem defined by (1) has a globally asymptotically stable equilibriumpoint when the relation between the output of the cell and the state ofthe cell is described by a piecewise-linear functionfi(x) = 1

2(jx +

1j � jx � 1j) andci � 1.Theorem 4 [22]: If i) �(A+AT ) is positive definite; ii)kBk2 � 1

wherekBk2 = (�max(BTB))1=2, then the unique equilibrium point

is globally asymptotically stable when the relation between the outputof the cell and the state of the cell is described by a piecewise-linearfunctionfi(x) = 1

2(jx + 1j � jx � 1j) andci � 1.

Example 1: Consider the cellular neural networks with delays

x0

1(t) = �c1x1(t) + a11f(x1(t)) + a12f(x2(t))

+b11f(x1(t� �1)) + b12f(x2(t� �2)) + I1;

x0

2(t) = �c1x2(t) + a21f(x1(t)) + a22f(x2(t))

+b21f(x1(t� �1)) + b22f(x2(t� �2)) + I2;

(10)

where the relation between the output of the cell and the state of thecell is described by a piecewise-linear functionfi(x) � f(x) = 1

2(jx

+ 1j � jx � 1j), �1 > 0, �2 > 0. It is easy to prove the Example1 has unique equilibrium. In Corollary 2, taking��ij = ��ij = �ij = �ij= 1

2(i; j = 1; 2), c1 = c2 = 1, a11 = a22 = 1

8, a12 = a21 = 1

4,

b11 = b22 = �1=4, b12 = b21 = 1

4i.e.,

A =1

8

1

4

1

4

1

8

; B =� 1

4

1

4

1

4� 1

4

we can easily check that

c1 > ja11j +1

2(ja21j + ja12j)

+ jb11j +1

2(jb21j+ jb12j)

c2 > 1

2(ja21j + ja12j)

+ ja22j +1

2(jb21j+ jb12j) + jb22j

so the unique equilibrium is globally asymptotically stable.The matrices�(A+ AT ) and�(A+ B) are obtained as

�(A+AT ) =� 1

4� 1

2

� 1

2� 1

4

�(A+B) =1

8� 1

2

� 1

2

1

8

We can easily check that�(A+AT ) is not positive definite. Therefore,the condition in [22] does not hold. Also,�(A+B) is not diagonallyrow dominant. Thus the condition given in [12] does not also hold.

Example 2: Consider the cellular neural networks with delays

x0

1(t) = �c1x1(t) + a11f(x1(t)) + a12f(x2(t))

+b11f(x1(t� �1)) + b12f(x2(t� �2)) + I1;

x0

2(t) = �c1x2(t) + a21f(x1(t)) + a22f(x2(t))

+b21f(x1(t� �1)) + b22f(x2(t� �2)) + I2;

(11)

where the relation between the output of the cell and the state of the cellis described by a piecewise-linear functionfi(x) � f(x) = 1

2(jx +

1j � jx � 1j), �1 > 0, �2 > 0. It is easy to prove the Example 2has unique equilibrium. In Corollary 2, taking��ij = ��ij = �ij = �ij =1

2(i; j = 1; 2), c1 = c2 = 2, a11 = a22 = 1

4, a12 = a21 = 0,

b11 = b22 = 0, b12 = b21 = 1 i.e.,

A =1

40

0 1

4

B =0 1

1 0

obviously

c1 > ja11j +1

2(ja21j + ja12j)

+ jb11j +1

2(jb21j+ jb12j)

c2 > 1

2(ja21j+ ja12j) + ja22j

+ 1

2(jb21j+ jb12j) + jb22j

so the unique equilibrium is globally asymptotically stable.

498 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 4, APRIL 2001

The matrixS in Theorem 3 is obtained as

S =3

4�1

�1 3

4

Since the system equations discussed in [15] requireci = 1(i = 1; 2),the condition given in [15] is not applied. In addition, we can also seethatS is not a nonsingularM matrix, so the condition given in [15]does not also hold for the matrices given above.

III. CONCLUSIONS

A set of criteria have been derived ensuring the global asymptoticstability of delayed cellular neural networks (DCNNs) with more gen-eral output functions by introducing ingeniously infinitely real param-eters, constructing suitable Lyapunov functionals and applying someanalysis techniques, and these criteria are independent of delays andpossess infinitely adjustable real parameterswi > 0, ��ij , ��ij , ��ij ,��ij , �ij , �ij , �ij , �ij are any real numbers with��ij + ��ij = 1,�ij +�ij = 1, ��ij + ��ij = 1, �ij + �j = 1 (i; j = 1; 2; . . . ; n). These areof prime importance and great interest in many application fields andthe design of networks. In addition, the methods of this paper may ex-tended to discuss more complicated systems such as Hopfield neuralnetworks(HNNs) and bi-directional associative memory (BAM) net-works.

ACKNOWLEDGMENT

The author would like to thank the referees and the associate editorfor suggesting improvements in presentation.

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[7] , “Global stability analysis in delayed cellular neural networks,”Phys. Rev. E, vol. 59, pp. 5940–5944, 1999.

[8] J. Cao and S. Wan, “The global asymptotic stability of Hopfield neuralnetwork with delays” (in in Chinese),J. Biomath, vol. 12, pp. 60–63,1997.

[9] J. Cao and D. Zhou, “Stability analysis of delayed cellular neural net-works,” Neural Netw., vol. 11, pp. 1601–1605, 1998.

[10] J. Cao, “Stability in cellular neural networks with delays,” inProc. 14thIFAC’99 World Congr., vol. E, 1999, pp. 71–74.

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[13] T. Roska, C. W. Wu, M. Balsi, and L. O. Chua, “Stability and dynamicsof delay-type general and cellular neural networks,”IEEE Trans. Cir-cuits Syst. I, vol. 39, pp. 487–490, 1992.

[14] M. Gilli, “Stability of cellular neural networks and delayed cellularneural networks with nonposive templates and nonomonotonic outputfunctions,”IEEE Trans. Circuits Syst. I, vol. 41, pp. 518–528, 1994.

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Optimal -Labeling of Strong Products of Cycles

Pranava K. Jha

Abstract—The (2 1)-labeling of a graph is an abstraction of as-signing integer frequencies to radio transmitters such that i) transmittersthat are one unit of distance apart receive frequencies that differ by atleast two, and ii) transmitters that are two units of distance apart receivefrequencies that differ by at least one. The least span of frequencies insuch a labeling is referred to as the -number of the graph. It is shownthat if 1 and . . . are each a multiple of3 + 2, then( . . . ) is equal to the theoretical minimum of3 + 1,

where denotes the cycle of length and denotes the strong productof graphs.

Index Terms—Cycle, frequency assignment, graph theory, (2 1)-la-beling, -number, strong product.

I. INTRODUCTION

CONSIDER the problem of assigning frequencies to radio transmit-ters at various nodes in a territory. The transmitters that are close mustreceive frequencies that are sufficiently apart, for otherwise they maybe at the risk of interfering with each other. The spectrum of frequenciesis an important resource on which there are increasing demands due tomodern communication needs, both civil and military. This calls for anefficient management of the spectrum. It is assumed that the transmit-ters are identical and the signal propagation is isotropic.

The foregoing problem, with the objective of minimizing the spanof frequencies, was first placed on a graph-theoretical footing in 1980by Hale [1] who established its equivalence to the generalized vertexcoloring problem, that is known to be computationally hard. (Verticescorrespond to transmitter locations and their labels to radio frequencies,while adjacencies are determined by geographical “closeness” of thetransmitters.)

Roberts [2] subsequently proposed a variation to the problem inwhich distinction is made between transmitters that are “close,” andthose that are “very close.” This enabled Griggs and Yeh [3] to for-

Manuscript receivedJanuary 25, 2000; revised October 30, 2000. This paperwas recommended by Associate Editor K. Thulasiraman.

The author is with the Department of Computer Science, St. Cloud StateUniversity, St. Cloud, MN 56301-4498 USA (e-mail: pkjha@eeyore.stcloud-state.edu).

Publisher Item Identifier S 1057-7122(01)02875-6.

02875–6/01$10.00 © 2001 IEEE