a new method for complete stability analysis of cellular neural networks with time delay

1126 IEEE TRANSACTIONS ON NEURAL NETWORKS VOL. 21, NO. 7, JULY 2010

A New Method for Complete Stability Analysis ofCellular Neural Networks With Time Delay

Wu-Hua Chen and Wei Xing Zheng, Senior Member, IEEE

Abstract—This paper presents new complete stability resultsfor delayed cellular neural networks (DCNNs). A novel method isproposed for complete stability analysis of DCNNs. By applyingthe M-matrix theory and introducing some new estimationtechniques on the solutions of DCNNs, a simple and improvedcomplete stability criterion is derived. The new criterion unifiesthe delay-dependent and delay-independent complete stabilityconditions for DCNNs. Moreover, the obtained delay-dependentcriterion can give a larger upper bound of the time delay thanthe existing ones such that the complete stability can still beretained. Numerical examples are presented which show that thenew complete stability results for DCNNs are compared favorablywith the existing results.

Index Terms—Cellular neural networks (CNNs), completestability, delay-independent/delay-dependent criterion, M-matrix,time-delay.

I. Introduction

CELLULAR neural networks (CNNs) introduced by Chuaand Yang [8] have attracted a great deal of interest due to

their important applications in such fields as image processing,pattern recognition, and biology. In these practical applica-tions, CNNs are usually required to be stable. When a CNN isused for solving optimization problems, the CNN must possessa unique and globally asymptotically stable (GAS) equilibriumpoint for every input vector. On the other hand, when a CNNis applied to image processing where the main function ofthe CNN is to transform an input image into a correspondingoutput image, normally represented by an equilibrium point,it is very important that the CNN must be completely stablein the sense that every trajectory converges to an equilibriumpoint, possibly within a set of many equilibrium points. Notethat GAS implies complete stability, but not vice versa. Somesufficient conditions for complete stability of CNNs have beenobtained (see [6]–[8], [10], [12], [20], [22]).

Manuscript received July 11, 2009; accepted April 15, 2010. Date ofpublication June 14, 2010; date of current version July 8, 2010. This workwas supported in part by the National Natural Science Foundation of Chinaunder Grant 60864002, in part by the Australian Research Council, undera research grant, and by the University of Western Sydney, Sydney, NSW,Australia, under a research grant.

W.-H. Chen was with the School of Computing and Mathematics, Universityof Western Sydney, Sydney, NSW 1797, Australia. He is now with the Collegeof Mathematics and Information Science, Guangxi University, Nanning,Guangxi 530004, China (e-mail: wuhua [email protected]).

W. X. Zheng is with the School of Computing and Mathematics,University of Western Sydney, Sydney, NSW 1797, Australia (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNN.2010.2048925

In electronic implementation of neural networks, time de-lays are inevitable due to axonal conduction times and finiteswitching speeds of amplifiers. The existence of time delaysfrequently causes oscillation or even instability of neuralnetworks. In the past two decades, the stability analysis ofdelayed neural networks has attracted a large amount ofresearch interest. By applying Lyapunov-Krasovskii functionalmethods or Lyapunov-Razumkhin function methods, a vari-ety of competing results for global asymptotic stability ofdelayed neural networks have been accumulated (see [2]–[5], [11], [15], [16], and [25], and the references therein).However, the usual methods for global stability analysis ofdelayed neural networks with a unique equilibrium point arenot adequately applicable to complete stability analysis ofdelayed neural networks with multiple equilibrium points.For instance, the standard Lyapunov-Krasovskii functionalmethods or Lyapunov-Razumkhin function methods for globalasymptotic stability analysis of delayed neural networks are nolonger effective for complete stability analysis. So far, only afew complete stability criteria have been obtained for delayedCNNs (DCNNs) (see [9], [13], [17]–[19], [21], and [26]). In[18], it was proved that positive cell linking templates arecompletely stable almost everywhere. By means of a Lyapunovfunctional, it was shown in [9], [13] that if a DCNN satisfiessome symmetry conditions and the upper bound of the productbetween the delay and the norm of the feedback matrix isless than 2

3 , then the DCNN is completely stable. In [26],the above-mentioned upper bound was further improved tobe 1. In deriving the complete stability criteria in [9], [13],and [26], the symmetric conditions are essential. For DCNNswith nonsymmetric templates, delay-independent and delay-dependent complete stability criteria were derived in [17],[19], and [21]. Recently, by using the M-matrix technique,some new delay-dependent and delay-independent completestability conditions were obtained in [1] and [24] for DCNNswith time-varying delays.

In this paper, we develop a new method to establish thecomplete stability criteria for DCNNs. This method is builtupon the classification of the solutions of DCNNs accordingto the oscillation of their derivatives. For different types ofsolutions, by exploiting the special structure of DCNNs, somenew analysis techniques are utilized to explore the effect of thediagonal delays on their convergence behaviors. Under certainconditions, we prove that for each solution of a given DCNN,either it converges or the output of its maximal componentis eventually constant. Then by applying the mathematical

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CHEN AND ZHENG: A NEW METHOD FOR COMPLETE STABILITY ANALYSIS OF CELLULAR NN 1127

induction method on the system dimension of the DCNN, wearrive at a conclusion that the DCNN is completely stable. Theattractive feature of our method is that there is no need of thesymmetric condition and more information on the bounds ofthe diagonal delays of DCNNs are taken into account. Thecomplete stability criteria obtained in this paper include thedelay-independent complete stability results given in [1], [19],[21], and [24] as a special case. More importantly, our newdelay-dependent criterion improves the allowable upper boundfor complete stability of DCNNs over those criteria given in[1], [17], and [24].

The remainder of this paper is arranged as follows. InSection II, the problem under investigation is formulated,together with some definitions and lemmas. In Section III, thenew complete stability criteria for DCNNs are presented. Theproof of the main theoretical results is presented in SectionIV. Numerical illustration in comparison with the existingresults is made in Section V. Concluding remarks are givenin Section VI. Finally, the proofs of two lemmas are providedin the Appendix.

II. Problem Formulation

Consider DCNN described by the following delay differen-tial equations:

xi(t) = −xi(t) +n∑

j=1

aijyj(t) +n∑

j=1

bijyj(t − τij) + ui

i = 1, 2, . . . , n (1)

where n is the number of cells, xi(t) denotes the state of the ithcell at time t, A =

(aij

)is the instantaneous feedback matrix,

B =(bij

)is the delayed feedback matrix, yi(t) is the output

of the ith cell defined by

yi(t) =1

2(|xi(t) + 1| − |xi(t) − 1|) (2)

τij ≥ 0 are the time delays, and u = [u1 u2 . . . un]T is anexternal input vector.

DCNN (1) has the following initial condition:

x(t) = φ(t) t ∈ [−τ, 0]

where x(t) = [x1(t) x2(t) . . . xn(t)]T , τ = max1≤i,j≤n

{τij

}, and

φ(t) is a continuous function on [−τ, 0].Note that DCNN (1) reduces to the standard CNN when

B = 0.Some useful definitions and lemmas are now introduced.Definition 1: DCNN (1) is said to be completely stable if

for any continuous function φ(t), the solution x(t, φ) of (1)satisfies

limt→∞ x(t, φ) = const.

Definition 2: An n × n matrix P with nonpositive off-diagonal elements is called a nonsingular M-matrix if all itsprincipal minors are positive.

Lemma 1 (See [14]): P is a nonsingular M-matrix if andonly if there exists a positive diagonal matrix D such that PD

is a diagonally dominant matrix.

Lemma 2 (See [14]): Let D0 be an n×n positive diagonalmatrix and P be an n × n matrix with P =

(pij

). If D0 − |P |

is a nonsingular M-matrix with |P | �(|pij|

), then D0 + P is

nonsingular.

III. Main Results and Comparisons

In this section, the complete stability of DCNN (1) isinvestigated.

First of all, a nomenclature list is provided regarding thenotations to be used in the sequel.

For any scalar d, define d+ = max{d, 0}, d− = max{−d, 0}.Index sets

N = {1, 2, . . . , n}N0 = {i ∈ N ; the input ui is known}N1 = {i ∈ N ; aii − 1 + bii > 0}N2 = {i ∈ N ; 1 − aii − bii > 0}N1 = {i ∈ N ; aii − 1 + bii > 0, bii > 0}N2 = {i ∈ N ; 1 − aii − bii > 0, bii < 0}

N =

⎧⎨⎩i ∈ N0; |ui| > |aii − 1| +

n∑j=1,j �=i

|aij| +n∑

j=1

|bij|⎫⎬⎭

Ni = Ni − N i = 1, 2

N1 =

⎧⎨⎩i; ui > |aii − 1| +

n∑j=1,j �=i

|aij| +n∑

j=1

|bij|⎫⎬⎭

N2 =

⎧⎨⎩i; ui < −|aii − 1| −

n∑j=1,j �=i

|aij| −n∑

j=1

|bij|⎫⎬⎭ .

Note that N = N1 ∪ N2.Notations associated with the entries of A and B

η(1)i = aii − 1 − |bii|

η(2)i = aii − 1 − |bii| +

2b+ii√

1 + 2(b+ii − 1 + aii)τii

η(3)i = aii−1+ bii− 2

(aii−1+ b+

ii

)[1− 1√

1+2b+iiτii

]

µi =e(1−aii)τii − 1

1 − aii

�i =

{12

(b−

ii τii

)2if b−

ii τii < 1

b−ii τii − 1

2 if b−ii τii ≥ 1

ξ(1)i = 1 − aii − |bii|

ξ(2)i = 1 − aii − |bii| +

2b−ii

1 + b−ii µi

ξ(3)i =

1 − aii − bii − �i(aii − 1 + |bii|)1 + �i

ω(1)i =

aii + bii − 1 − biiτii(|1 − aii| + bii)

1 + biiτii

ω(2)i =

b−ii − aii + 1 − b−

ii τii(|1 − aii| + b−ii )

1 + b−ii τii

.

Three matrices associated with A and B

Using the above notations, define the three matrices

W = (wij)n×n W =(wij

)n×n

W =(wij

)n×n


where wij , wij , and wij are given by

wij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

η(2)i i=j i∈N1 and aii <1

max{

η(1)i , η

(3)i

}i=j i∈N1 and aii ≥1

ξ(2)i i=j i∈N2 and aii <1

ξ(3)i i=j i∈N2 and aii ≥1

1 i=j i∈N−(|aij| + |bij|) i �= j i �∈N

0 i �=j i∈N

wij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

ω(1)i i = j i ∈ N1 i �∈ N

ω(2)i i = j i ∈ N2 i �∈ N

1 i = j i ∈ N

−(|aij| + |bij|) i �= j i �∈ N

0 i �= j i ∈ N

wij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

η(1)i i = j i ∈ N1

ξ(1)i i = j i ∈ N2

1 i = j i ∈ N

−(|aij| + |bij|) i �= j i �∈ N

0 i �= j i ∈ N .

Now our main result for complete stability of DCNN (1) isstated as follows.

Theorem 1: Assume that N1∪N2∪N = N . If W = (wij)n×n

is a nonsingular M-matrix, then DCNN (1) is completelystable.

The proof of Theorem 1 will be given in Section IV soas to maintain a smooth presentation flow. The followingdelay-dependent complete stability criterion on DCNN (1) wasobtained in [24].

Zeng-Wang Theorem [24]: Assume that N1 ∪ N2 ∪ N = N .If W = (wij)n×n is a nonsingular M-matrix, then DCNN (1) iscompletely stable.

Remark 1: In Theorem 1 in [24], ω(1)i and ω

(2)i were defined

as follows:

ω(1)i =

aii + bii − 1 − biiτii(|aii| + bii)

1 + biiτii

ω(2)i =

b−ii − aii + 1 − b−

ii τii(|aii| + b−ii )

1 + b−ii τii

.

But from the proof given in [24] it has been found that the term|aii| in ω

(1)i and ω

(2)i as previously defined in [24, Th. 1] should

be |1−aii|.1 Note that when N2 = N = ∅, the above-mentionedcriterion given in [24] was also independently obtained in [1].

It is easy to check that wij = wij when i �= j, but wii ≥ wii

when i = j. In particular, wii > wii when i ∈ N1 ∪ N2 andτii > 0. Therefore, our new Theorem 1 achieves an improve-ment over the delay-dependent complete stability conditionsobtained in [1] and [24].

1Z. Zeng and W.-H. Chen discussed this issue in [23].

From Theorem 1, it is straightforward to derive the follow-ing delay-independent complete stability criterion.

Corollary 1: Assume that N1 ∪ N2 ∪ N = N . If W =(wij)n×n is a nonsingular M-matrix, then DCNN (1) is com-pletely stable for any delay τij .

Proof: It is easy to see that wii ≥ wii, and wij = wij

when i �= j. So the condition that W is a nonsingular M-matrix implies that W is a nonsingular M-matrix. Thus, byTheorem 1, DCNN (1) is completely stable.

Remark 2: Corollary 1 is identical with the delay-independent complete stability theorem given in [24]. More-over, the delay-independent complete stability criteria obtainedin [1], [19], and [21] are simply a special case of Corollary1. So Theorem 1 extends and improves the complete stabilityresults given in [1], [19], [21], and [24].

Remark 3: It should be pointed out that the delay-dependent complete stability criterion and delay-independentcomplete stability criterion obtained in [24] are separated inthe sense that neither of them implies the other. However, inthis paper the delay-independent complete stability criteriongiven in Corollary 1 is just an immediate outcome of our newcomplete stability criterion given in Theorem 1. Thus, ournew complete stability criterion unifies the delay-dependentand the delay-independent complete stability criteria in asimple and efficient format, which indicates that it has broaderapplications in practice.

IV. Proof of Theorem 1

In this section, the detailed proof of Theorem 1 is presented.We start with some auxiliary lemmas that are helpful to theproof of Theorem 1.

A. Some Preliminaries

Lemma 3 (See [17]): For i = 1, 2, . . . , n, let yi(t) be de-fined by (2). Then

D+yi(t)=

⎧⎪⎪⎨⎪⎪⎩

0 if |xi(t)| > 1or |xi(t)| = 1 and D+|xi(t)| ≥ 0

xi(t) if |xi(t)| < 1or |xi(t)| = 1 and D+|xi(t)| < 0

where D+yi(t) denotes the right derivative of yi(t). Moreover

D+yi(t)

= N(xi(t))

⎡⎣−yi(t) +

n∑j=1

aijyj(t) +n∑

j=1

bijyj(t−τij) + ui

⎤⎦ (3)

where for i = 1, 2, . . . , n, N(xi(t)) is defined by

N(xi(t)) =

{1 if D+yi(t) = xi(t)0 if D+yi(t) = 0.

Lemma 4: Assume that the conditions of Theorem 1 hold.Then there exists a vector q =

[q1 q2 . . . qn

]Twith qi = 0 for

i ∈ N such that

D+zi(t)

= N(xi(t))

⎡⎣−zi(t) +

n∑j=1

aijzj(t)+n∑

j=1

bijzj(t−τij)+ ui

⎤⎦ (4)


where zi(t) = yi(t) − qi for i ∈ N ; ui = 0 for i ∈ N1 ∪ N2,

and ui =n∑

j=1, j∈N1∪N2

(aij + bij)qj + ui for i ∈ N . Moreover, if

limt→+∞ z(t) = const, where z(t) = [z1(t) z2(t) . . . zn(t)]T , thenlim

t→+∞ x(t) = const.

Proof: We first prove that there exists a vector q withqi = 0 for i ∈ N such that (4) holds. It follows from (3) that

D+zi(t)= D+yi(t)

= N(xi(t))

⎡⎣−zi(t)+

n∑j=1

aijzj(t)+n∑

j=1

bijzj(t−τij)

−qi +n∑

j=1

(aij + bij

)qj + ui

⎤⎦ i = 1, 2, . . . , n.

So there exists a vector q with qi = 0 for i ∈ N such that(4) holds if and only if the following linear algebraic equationadmits a solution:⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

−qi +n∑

j=1

(aij + bij

)qj = −ui i ∈ N1

qi −n∑

j=1

(aij + bij

)qj = ui i ∈ N2

qi = 0 i ∈ N .

(5)

Let aij = aij , bij = bij , ui = −ui, for i ∈ N1; aij = −aij , bij =−bij , ui = ui, for i ∈ N2; and aij = bij = ui = 0, for i ∈ N .Then define three new matrices W0 = diag(w11, w22, . . . , wnn),D0 = diag(d01, d02, . . . , d0n), P = (pij)n×n, where

d0i =

⎧⎨⎩

aii − 1 + bii i ∈ N1

1 − aii − bii i ∈ N2

1 i ∈ Npij =

{0 i = j

aij + bij i �= j.

Thus (5) can be rewritten as

(D0 + P) q = u (6)

where u = [u1 u2 . . . un]T . It is easy to check that d0i ≥ wii

for all i ∈ N , and W = W0 − |P |. So, from the fact that W

is a nonsingular M-matrix, it follows that D0 − |P | is also anonsingular M-matrix. By Lemma 2, D0 + P is nonsingular,which yields that the linear algebraic equation (6) admits aunique solution. Therefore, there exists a vector q with qi = 0for i ∈ N such that (4) holds.

Next we prove that limt→+∞ z(t) = const implies lim

t→+∞ x(t) =const. If lim

t→+∞ z(t) = const, then limt→+∞ y(t) = const. For any

i ∈ {1, 2, . . . , n}, by the constant-variation-formula, we have

xi(t) = xi(0)e−t + e−t

∫ t

0esfi(y(s))ds

where fi(y(t)) =n∑

j=1aijyj(t) +

n∑j=1

bijyj(t − τij) + ui. We assume

that limt→+∞ fi(y(t)) = ci. If |ci| > 0, then there exists Ti > 0

such that |fi(y(t))| > |ci|/2 for t ≥ Ti. It follows thatlim

t→+∞∫ t

Tiesfi(y(s))ds = ∞. Thus, by l’Hopital’s rule

limt→+∞ xi(t) = lim

t→+∞ xi(0)e−t + limt→+∞e−t

∫ Ti

0esfi(y(s))ds

+ limt→+∞

∫ t

Tiesfi(y(s))ds

et

= limt→+∞ fi(y(t))

= ci.

If ci = 0, then for any ε > 0, there exists Ti1 > 0 such that|fi(y(t))| < ε/2 for t ≥ Ti1. We may assume that |fi(y(t))| ≤Mi for t ≥ 0, where Mi is some positive scalar. Choose Ti2 > 0such that

[|xi(0)| + Mi

(eTi1 − 1

)]e−Ti2 ≤ ε/2. Then for t >

max{Ti1, Ti2}

|xi(t)| ≤ e−t

[|xi(0)| + Mi

∫ Ti1

0esds +

ε

2

∫ t

Ti1

esds

]≤ [|xi(0)| + Mi

(eTi1 − 1

)]e−Ti2 + ε/2

< ε

which implies limt→+∞ xi(t) = 0. The proof of Lemma 4 is thus

complete.Remark 4: By (1) and the proof of Lemma 4, x(t) and z(t)

have the following relation:

xi(t) = −xi(t) + qi

n∑j=1

aijzj(t) +n∑

j=1

bijzj(t − τij)+ ui

i = 1, 2, . . . , n. (7)

Remark 5: Since limt→+∞ y(t) = const is equivalent to

limt→+∞ z(t) = const, we can conclude from Lemma 4 that iflim

t→+∞ y(t) = const, then limt→+∞ x(t) = const.

B. Idea of Proof of Theorem 1

Under the assumption that W is a nonsingular M-matrix,it follows from Lemma 1 that there exist positive constantsd1, d2, . . . , dn such that

wiidi −n∑

j=1,j �=i

|wij|dj > 0 i = 1, 2, . . . , n. (8)

Setαi =

n∑j=1,j �=i

|wij|dj/di (9)

c1 = min1≤i≤n

{wii − αi} (10)

c2 = max1≤i≤n

{|aii − 1| + |bii|} (11)

c3 = max1≤i≤n

⎧⎨⎩

n∑j=1,j �=i

|wij|⎫⎬⎭ + c1/2. (12)

Then by (8), we have c1 > 0 and c3 > 0. Moreover, from thedefinition of wii, one can verify that

aii − 1 + bii ≥ wii > αi i ∈ N1 (13)

1 − aii − bii ≥ wii > αi i ∈ N2. (14)For any given solution x(t) of DCNN (1), by the bounded-

ness of x(t), we have lim supt→+∞

|zi(t)| = βi < +∞ for all i ∈ N .

We assume thatβl/dl = max

1≤i≤n{βi/di} (15)

for some l ∈ N .


To prove Theorem 1, we will distinguish the followingtwo cases to discuss the convergence of x(t) according to theoscillation of D+zl(t).

1) Case A: D+zl(t) is eventually sign-definite, i.e., thereexists T0 > 0 such that D+zl(t) > 0 for all t ≥ T0 orD+zl(t) < 0 for all t ≥ T0.

2) Case B: D+zl(t) is oscillatory, i.e., there exists an in-creasing sequence {tm}, tm → +∞, such that D+zl(tm) =0.

Based on the above classification, we will prove that underthe conditions of Theorem 1, either x(t) converges or theoutput yl(t) of xl(t) is eventually constant. Then by applyingthe mathematical induction method on the system dimensionn of DCNN (1), we will finally prove that x(t) converges.

C. Behavior of x(t) for Case A

Proposition 1: Assume that the conditions of Theorem 1hold. If zl(t) belongs to Case A, then lim

t→+∞ x(t) = const.

Proof: Since zl(t) is eventually monotonic, we havelim

t→+∞ |zl(t)| = βl. We will prove βl = 0. By the contrary, βl > 0.Without loss of generality, we assume that lim

t→+∞ zl(t) = βl

and there exists T0 > 0 such that D+zl(t) > 0 for t ≥ T0.Then for any given σ ∈ (0, min{βl, c1βl/(c2 +c3)}), there existsT1 ≥ T0 + τ such that

βl−σ <zl(t)<βl+ σ; |zj(t)|<βj + σ; t≥T1−τ; j∈N . (16)

We first assert that there exists T2 ≥ T1 such that

|xl(t)| < 1 for t ≥ T2. (17)

Otherwise, there exists a sequence {tk}, tk → +∞, such that|xl(tk)| ≥ 1. If there exists an infinite subsequence {tkl

} of {tk}such that xl(tkl

) ≥ 1, then we have zl(tkl) = 1 − ql, which

contradicts the fact that D+zl(t) > 0 for t ≥ T0. Similarly, ifthere exists a subsequence {tkm

} of {tk} such that xl(tkm) ≤ −1,

then we also get a contradiction. So (17) holds.Then we assert that l �∈ N . By the contrary, without loss of

generality, we may assume that l ∈ N1. Using (3) and noticingthat |yl(t)| ≤ 1 and (17), we have

D+yl(t)≥ul−|all−1|−n∑

j=1,j �=l

|alj|−n∑

j=1

|blj|>0 for t≥T2.

It follows that yl(t) → +∞, which contradicts the fact that|yl(t)| ≤ 1.

In the following, we will divide two cases to lead to acontradiction.

1) Case A-1: l ∈ N1.By Lemma 4, and using (16), (17), and (9)–(13), wehave that for t ≥ T2

D+zl(t) ≥ ((all − 1)+ − (all − 1)−)zl(t) +

(b+

ll − b−ll

)×zl(t − τll) −

n∑j=1,j �=l

(|alj| + |blj|)(

dj

dl

βl + σ

)

>((all − 1)+ + b+

ll

)(βl − σ)

−((all−1)−+b−ll

)(βl+σ)−(αlβl+c3σ)

= (all−1+bll−αl)βl−σ(|all−1|+|bll|+c3)

≥ c1βl − σ(c2 + c3)

> 0

which implies that zl(t) → +∞. This is a contradic-tion.

2) Case A-2: l ∈ N2.By Lemma 4, and using (16), (17), (9)–(12), and (14),we have that for t ≥ T2

D+zl(t) ≤ ((all − 1)+ − (all − 1)−)zl(t)

+(b+

ll − b−ll

)zl(t − τll) + (αlβl + σc3)

≤ −(1 − all − bll − αl)βl + σ(|all − 1| + |bll|+c3)

≤ −(c1βl − σ(c2 + c3))

< 0

which implies that zl(t) → −∞. This is also a contra-diction.

Therefore, βl = 0, which implies limt→+∞ zi(t) = 0 for all i ∈

N . It follows from Lemma 4 that limt→+∞ x(t) = const. So the

proof of Proposition 1 is complete.

D. Behavior of x(t) for Case B

Proposition 2: Assume that the conditions of Theorem 1hold. If zl(t) belongs to Case B, then βl = 0 or

there exists T > 0 such that |yl(t)| = 1 for t ≥ T. (18)

We first give a lemma which plays an important role in theproof of Proposition 2.

Lemma 5: Assume that the conditions of Theorem 1 hold.If zl(t) belongs to Case B, then βl = 0, or

{βl = 1 − ql > 0 and there exists an increasingsequence{tk}, tk → +∞ such that xl(tk) ≥ 1

(19)

or

{βl = |1 + ql| > 0 and there exists an increasingsequence{tk}, tk → +∞ such that xl(tk) ≤ −1.

(20)

In order to prove Lemma 5, it suffices to prove that (19) or(20) holds when βl > 0. Without loss of generality, we mayassume that lim sup

t→+∞zl(t) = βl > 0. Since D+zl(t) is oscillatory,

it follows that there exists a sequence {tm}, tm → +∞, suchthat D+zl(tm) = 0 and lim

m→+∞ zl(tm) = βl. For the sequence{zl(tm)}, there are two possible cases: the first case is that

xl(tm) > −1 for sufficient large tm (21)

whereas the second case is that there exists a subsequence {tmk}

of {tm}, tmk→ +∞, such that xl(tmk

) ≤ −1. If the second caseholds, then

zl(tmk) = yl(tmk

) − ql = −1 − ql

which yields βl = −1 − ql. It follows from βl > 0 that (20)holds.


Hence, to complete the proof of Lemma 5, we only need toprove (19) under condition (21). In fact, under condition (21),if we can prove the following assertion:{

there exists an increasing sequence {tk}tk → +∞ such that xl(tk) ≥ 1

(22)

then zl(tk) = yl(tk)−ql = 1−ql. Noticing that zl(t) = yl(t)−ql ≤1 − ql for all t ≥ 0, we obtain βl = 1 − ql. It follows that (19)holds.

According to the analysis made above, we conclude thatthe proof of Lemma 5 comes down to the proof of (22)under condition (21). In what follows, we will distinguish threecases: l ∈ N1, l ∈ N2, and l ∈ N , to prove that (22) holdsunder condition (21).

Lemma 6: Assume that the conditions of Theorem 1 hold,zl(t) belongs to Case B, βl > 0, l ∈ N1, and {tm} is aninfinite increasing sequence such that tm → +∞, D+zl(tm) = 0and lim

m→+∞ zl(tm) = βl. If condition (21) is satisfied, then (22)holds.

Proof: See the Appendix.Lemma 7: Assume that the conditions of Theorem 1 hold,

zl(t) belongs to Case B, βl > 0, l ∈ N2, and {tm} is aninfinite increasing sequence such that tm → +∞, D+zl(tm) = 0,and lim


Proof: See the Appendix.Lemma 8: Assume that the conditions of Theorem 1 hold,

zl(t) belongs to Case B, βl > 0, l ∈ N , and {tm} is aninfinite increasing sequence such that tm → +∞, D+zl(tm) = 0,and lim


Proof: Without loss of generality, we assume that l ∈N1. By the contrary, if there is no increasing sequence {tk},{tk} → +∞, such that xl(tk) ≥ 1, then for sufficiently large t,we have xl(t) < 1. From condition (21), there exists T1 > 0such that |xl(tm)| < 1 for tm ≥ T1. So we have N(xl(tm)) = 1for tm ≥ T1. Using (3) and noticing that |yl(t)| ≤ 1, we havethat for tm ≥ T1

D+yl(tm) ≥ ul−|all−1| −n∑

j=1,j �=l

|alj| −n∑

j=1

|blj| > 0

which implies D+zl(tm) > 0. This is a contradiction. Hence,the proof of Lemma 8 is complete.

With Lemmas 6–8 and the analysis given before Lemma 6,we can conclude that Lemma 5 holds.

Using Lemma 5, we are now able to wrap up the proof ofProposition 2.

Proof of Proposition 2: If βl = 0, then Proposition 2holds. In the sequel, we will consider only the case of βl > 0.That is, under the assumption of βl > 0, we will prove (18).

By Lemma 5, we have obtained that if zl(t) belongs toCase B and βl > 0, then there are two possible cases.

1) Case B1: βl = 1 − ql and there exists an increasingsequence {tk}, {tk} → +∞, such that xl(tk) ≥ 1.

2) Case B2: βl = |1 + ql| and there exists an increasingsequence {tk}, {tk} → +∞, such that xl(tk) ≤ −1.

We only consider Case B1 since the proof for Case B2 issimilar. If (18) is not true, then for any T ≥ 0, there existtk ≥ T and t ∈ (tk, tk+1) such that xl(t) < 1. As done in theproofs of Lemmas 6–8, we will divide three cases to deducea contradiction.

1) Case B1-1: l ∈ N1.For tk ≥ 2τ, let t∗1 = sup{t ∈ [tk, t); xl(t) ≥ 1}. Thenxl(t∗1 ) = 1, xl(t) < 1, t ∈ (t∗1 , t ], and xl(t∗1 ) ≤ 0. It followsthat zl(t∗1 ) = 1 − ql = βl. By (7)

xl(t∗1 ) = −xl(t

∗1 ) + ql +

n∑j=1

aljzj(t∗1 ) +n∑

j=1

bljzj(t∗1 − τlj)

= (all − 1)(1 − ql) + bllzl(t∗1 − τll)

+n∑

j=1,j �=l

[aljzj(t∗1 ) + bljzj(t∗1 − τlj)]

= (all − 1)βl + bllzl(t∗1 − τll)

+n∑

j=1,j �=l

[aljzj(t∗1 ) + bljzj(t∗1 − τlj)].

Then applying the analogous argument to that in theproof of Lemma 6, we can conclude that xl(t∗1 ) > 0,which is a contradiction.

2) Case B1-2: l ∈ N2.For tk ≥ 2τ, let t∗2 = inf{t ∈ [t, tk+1); xl(t) ≥ 1}. Thenxl(t∗2 ) = 1, xl(t) < 1, t ∈ [t, t∗2 ), and xl(t∗2 ) ≥ 0. It followsthat zl(t∗2 ) = 1 − ql = βl. By (7)

xl(t∗2 ) = −(1 − all)βl + bllzl(t

∗1 − τll)

+n∑

j=1,j �=l

[aljzj(t∗1 ) + bljzj(t∗1 − τlj)].

Then using the analogous argument to that in the proofof Lemma 7, we can conclude that xl(t∗2 ) < 0, which isa contradiction.

3) Case B1-3: l ∈ N .We may assume that l ∈ N1 since the case of l ∈ N2 issimilar. Let t∗1 be as defined in Case B1-1. Then xl(t∗1 ) =yl(t∗1 ) = 1 and xl(t∗1 ) ≤ 0. Using (1) and noticing that|yl(t)| ≤ 1, we have

xl(t∗1 ) = −1 + all +

n∑j=1,j �=l

aljyl(t∗1 )

+n∑

j=1

bljyl(t∗1 − τll) + ul

≥ ul − |all − 1| −n∑

j=1,j �=l

|alj| −n∑

j=1

|blj|

> 0

which contradicts the fact that xl(t∗1 ) ≤ 0.

Therefore, (18) holds. Finally, the proof of Proposition 2 iscomplete.

E. Proof of Theorem 1

Using Propositions 1 and 2, we will first show that one-cellDCNN is completely stable.

Proposition 3: Consider the one-cell DCNN

x(t) = −x(t) + ay(t) + by(t − τ) + u (23)


where y(t) = |x(t)+1|−|x(t)−1|2 . If one of the following conditions

holds:

1) a < 1 and a − 1 − |b| + 2b+√1+2(b+−1+a)τ

> 0;

2) a ≥ 1 and max {a − 1 − |b|,a − 1 + b − 2(a − 1 + b+)

{1 − 1√

1+2b+τ

}}> 0;

3) a < 1 and 1 − a − |b| + 2b−1+b−µ

> 0;

4) a ≥ 1 and 1−a−b−�(|b|+a−1)1+�

> 0;

5) |u| > |a − 1| + |b|;where

µ =e(1−a)τ − 1

1 − aand � =

⎧⎪⎪⎨⎪⎪⎩

1

2

(b−τ

)2if b−τ < 1

b−τ − 1

2if b−τ ≥ 1

then the one-cell DCNN (23) is completely stable.Proof: According to the preceding analysis, for any

solution x(t) of DCNN (23), x(t) must belong to one ofthe two cases: Case A or Case B. Under one of condi-tions 1)–5), if x(t) belongs to Case A, then by Proposition1, lim

t→+∞ x(t) = const. If x(t) belongs to Case B, then byProposition 2, lim

t→+∞ x(t) = const, or there exists T > 0such that y(t) = 1 or y(t) = −1 for t ≥ T . Conse-quently, by Remark 5, we always have lim

t→+∞ x(t) = const.Hence, the one-cell DCNN (23) is completely stable underone of conditions 1)–5). The proof of Proposition 3 is thuscomplete.

Based on the above Propositions 1–3, we are now readyto apply the mathematical induction to finish the proof ofTheorem 1.

Proof of Theorem 1: By Proposition 3, Theorem 1 holdsfor any one-cell DCNN. Assume that Theorem 1 holds forany (n−1)-cell DCNN (n ≥ 2). We will show that Theorem 1holds for any n-cell DCNN.

To avoid the notation confusion, for a given k-cell DCNN,we use the notations W(k) to denote the correspondingmatrix W defined in Section III, and use the notationsN1(k), N2(k), N (k) to denote the corresponding index setsN1, N2, N defined in Section III, respectively. Now givenan n-cell DCNN (1) satisfying the conditions of Theorem1, let x(t) be any solution of the n-cell DCNN (1). Letlim supt→+∞

|zi(t)| = βi, then 0 ≤ βi < +∞, i ∈ N . Assume

that βl/dl = max1≤i≤n{βi/di}. Then zl(t) must belong to oneof the two cases: Case A or Case B. If zl(t) belongs to CaseA, then by Proposition 1, lim

t→+∞ x(t) = const. If zl(t) belongs toCase B, then by Proposition 2, there are two possible cases:lim

t→+∞ x(t) = const, or yl(t) = 1 or yl(t) = −1 for t ≥ T .If lim

t→+∞ x(t) = const, then Theorem 1 holds. If yl(t) = 1 oryl(t) = −1 for t ≥ T , then setting yl(t) = ν for t ≥ T ,where ν = 1 or −1, the n-cell DCNN (1) for t ≥ T + τ

becomes⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x1(t) = −x1(t) +n∑

j=1,j �=l

[a1jyj(t) + b1jyj(t − τ1j)

]+(a1l + b1l)ν + u1

. . . . . .

xl−1(t) = −xl−1(t)

+n∑

j=1,j �=l

[al−1,jyj(t) + bl−1,jyj(t − τl−1,j)

]+(al−1,l + bl−1,l)ν + ul−1

xl(t) = −xl(t) +n∑

j=1,j �=l

[aljyj(t)+bljyj(t − τlj)

]+(all + bll)ν + ul

xl+1(t) = −xl+1(t)

+n∑

j=1,j �=l

[al+1,jyj(t) + bl+1,jyj(t − τl+1,j)

]+(al+1,l + bl+1,l)ν + ul+1

. . . . . .

xn(t) = −xn(t) +n∑

j=1,j �=l

[anjyj(t) + bnjyj(t − τnj)

]+(anl + bnl)ν + un.

.

The above system shows that the behavior ofx1(t), . . . , xl−1(t), xl+1(t), . . . , xn(t) for t ≥ T +τ is independentof xl(t). Thus, for t ≥ T +τ, x1(t), . . . , xl−1(t), xl+1(t), . . . , xn(t)form an (n − 1)-cell DCNN. Let N ′

1(n − 1) = N1(n) − {l},N ′

2(n − 1) = N2(n) − {l}, N ′(n − 1) = N (n) − {l}. Onecan verify that if i �= l and i ∈ N (n), then i ∈ N (n − 1).So N1(n − 1) ⊆ N ′

1(n − 1), N2(n − 1) ⊆ N ′2(n − 1),

and N ′(n − 1) ⊆ N (n − 1). Let W ′(n − 1) be the(n − 1) × (n − 1) matrix obtained by eliminating thelth row and the lth column from W(n). Since anyprincipal submatrix of an M-matrix is also an M-matrix, it follows that W ′(n − 1) is an M-matrix. Set

N (n − 1) =2⋃

j=1

(N ′

j(n − 1) − Nj(n − 1))

then one can find that W(n−1) can be derived from W ′(n−1)by replacing wii with 1 and wij (i �= j) with 0 for alli ∈ N (n−1). Since W ′(n−1) is an M-matrix, it is easy to seethat W(n−1) is also an M-matrix. Thus, the (n−1)-cell DCNNformed by x1(t), . . . , xl−1(t), xl+1(t), . . . , xn(t) for t ≥ T + τ

satisfies the conditions of Theorem 1. By the assumption,lim

t→+∞ xi(t) = const, i = 1, . . . , l − 1, l + 1, . . . , n. Then itfollows that lim

t→+∞ y(t) = const. By Remark 5, limt→+∞ x(t) =

const. In other words, the n-cell DCNN (1) is completelystable. Therefore, the proof of Theorem 1 is finally complete.

Remark 6: In DCNN (1), we have assumed that the delaysare constant. From the proofs of Proposition 1 and Proposition2, it is easy to see that when the constant non-diagonal delaysτij and the constant diagonal delays τii with i ∈ N1∪N2∪N arereplaced by time-varying delays τij(t) and τii(t), respectively,where N1 = {i; aii − 1 − |bii| > 0}, our complete stabilitycriterion will still hold. However, it is observed from the proofof Lemma 6 (see the Appendix) that the estimation technique


is not applicable to the case when τii with i ∈ N1 − N1 aretime-varying. The problem of how to remove the restriction inTheorem 1 that the delays τii with i ∈ N1 − N1 are constantwill be investigated in our future work.

V. Illustrative Examples

In this section, we present three numerical examples whichshows that our delay-dependent complete stability results areless conservative than the existing ones.

Example 1: Consider the one-cell DCNN

x(t) = −x(t) + y(t) + y(t − τ) + u. (24)

When u = 0, it was shown in [19] that DCNN (24) is notcompletely stable with τ = 3π

2 . However, the result in [19]cannot be used to determine the complete stability of DCNN(24). On the other hand, applying the results of [1], [9], [13],[17], [24], and [26] to the one-cell DCNN (24) gives that theupper bounds of the delay τ such that (24) is completely stableare 1, 2

3 , 23 , 1

2 , 1, 1, respectively. Applying our Theorem 1 orProposition 3 to DCNN (24), the upper bound of the delay τ

for complete stability has been found to be 32 for any input u.

When |u| > 1, by Corollary 1 or Proposition 3, the DCNN(24) is completely stable for any delay τ.

Example 2: Consider the two-cell DCNN[x1(t)x2(t)

]= −

[x1(t)x2(t)

]+

[2 0.20.4 2

] [y1(t)y2(t)

]

+

[−0.5 0.20.2 1

][y1(t − τ1)y2(t − τ2)

]−[0.51

]. (25)

One can verify that

W =

[0.5 −0.4

−0.6 0

]

is not a nonsingular M-matrix, so Corollary 1 or the resultsin [19] and [21] cannot be used to determine the completestability of DCNN (25). Because A + B is not symmetricand there does not exist a positive diagonal matrix D suchthat matrices DA and DB are symmetric, the results in [9],[13], and [26] cannot be applied to DCNN (25) either. SinceN2 = N = ∅ and N1 = {2}, it follows that N1∪N2∪N �= {1, 2}.So the result in [24] is also not applicable to DCNN (25).Applying Theorem 1 in [17] to (25) with τ1 ≡ τ2 ≡ τ, it isfound that DCNN (25) is completely stable when τ < 0.1551.By Theorem 1 in [1], it is found that when τ2 < 0.6129,DCNN (25) is completely stable for any delay τ1. Nowconsider using our Theorem 1. It is noted that N1 = {1, 2}and

W =

[0.5 −0.4

−0.6 4/√

1 + 2τ2 − 2

].

It is easy to check that when τ2 < 0.8007, W is a nonsingularM-matrix. So when τ2 < 0.8007, the two-cell DCNN (25) iscompletely stable for any delay τ1.

The two-cell DCNN (25) has seven equilibriums: (1.4, 2.6),(1.08, 0.2), ( 15

19 , 519 ), (0.2, 2.12), (−1.6, 1.4), (−1.68, 0.8),

(−2.4, −4.6). To better illustrate Example 2, we have com-puted numerically the solutions of the two-cell DCNN (25)

Fig. 1. Behavior of x(t) in delayed cellular neural network (25).

Fig. 2. Behavior of x(t) in the DCNN described in Example 3.

with τ1 = 10 and τ2 = 0.8 corresponding to several constantinitial functions, as shown in Fig. 1. One can see that alltrajectories tend to one of the above mentioned equilibriumsexcept ( 15

19 , 519 ).

Example 3: Consider a three-cell DCNN (1), where τij = τ,i = 1, 2, τ3j = τ, u = (0.2, 0.15, 1.2)T , and

A =

⎡⎣ 1 0 0

0 1 00 0 1

⎤⎦ , B =

⎡⎣ 1 0.6 0.1

0.5 −1.5 0.10.2 0.6 0.3

⎤⎦ .

One can verify that N1 = N1 = {1}, N2 = N2 = {2}, andN = {3}. By Zeng-Wang Theorem [24], one has that

W =

⎡⎣ 1−τ

1+τ−0.6 −0.1

−0.5 1.5(1−1.5τ)1+1.5τ

−0.10 0 1

⎤⎦ .

It is straightforward to check that W is a nonsingular M-matrixif and only if τ < 0.3035. That is to say, by the result of [24],for any τ, the upper bound of the delay for complete stability


of this DCNN is 0.3035. Now apply our Theorem 1 to thisexample. It is easy to see that

W =

⎡⎣

21+2τ

−0.6 −0.1−0.5 1.5(1−�2)

1+�2τ−0.1

0 0 1

⎤⎦

with

�2 =

{12 (1.5τ)2 if 1.5τ < 11.5τ − 1

2 if 1.5τ ≥ 1.

When 1.5τ ≥ 1, W is a nonsingular M-matrix if and only ifτ < 0.5436. But it does not satisfy the assumption 1.5τ ≥ 1.Now we consider the case of 1.5τ < 1. In this case, W is anonsingular M-matrix if and only if τ < 0.5351. Thus, by ourcriterion, for any τ, the upper bound of the delay for completestability of this DCNN is 0.5351. The simulation results withτ = 0.53 and τ = 2 are depicted in Fig. 2, where all thetrajectories converge to one of the two equilibrium points at(2.6, 0.5, 3) and (− 9

5 , − 16 , 2.2).

From the above comparisons, it may be concluded that thenew complete stability criterion is less conservative than thosederived in the existing results.

VI. Conclusion

The problem of complete stability for delayed cellularneural networks has been investigated. The complete stabilitycriteria obtained in this paper depend upon the connectionmatrix, the time delay and the external inputs, and havebeen characterized in terms of M-matrix. The new completestability criterion unifies the delay-dependent and the delay-independent complete stability conditions and include theexisting M-matrix based criteria for complete stability as a spe-cial form. Moreover, our delay-dependent complete stabilitycondition has improved the allowable upper bound of the timedelay over the existing ones in the literature such that the com-plete stability for DCNNs can be retained. Through numericalexamples, such improvements of our complete stability criteriaover the existing results have been clearly observed.

APPENDIX

Proof of Lemma 6: By the contrary, if there is noincreasing sequence {tk}, {tk} → +∞, such that xl(tk) ≥ 1,then for sufficiently large t, we have xl(t) < 1. From condition(21), there exists T1 > 0 such that |xl(tm)| < 1, xl(t) < 1, fortm, t ≥ T1. So we have N(xl(tm)) = 1 for tm ≥ T1.

First, we consider the case that all ≥ 1 and η(1)l ≥ η

(3)l .

In this case, we have wll = η(1)l . For any given σ ∈

(0, βl min{1, c1/(c2 + c3)}), there exists T2 ≥ T1 + τ such that|zj(t)| ≤ βj + σ, j ∈ N , zl(tm) > βl − σ for t, tm ≥ T2 − τ.Thus, by (4), (9)–(13), for tm ≥ T2, we obtain

D+zl(tm) > (all − 1)βl − |bll|(βl + σ)

−(αlβl + σ(c3 + |all − 1|))≥ (η(1)

l − αl)βl − σ(c2 + c3)

≥ c1βl − σ(c2 + c3)

> 0

which contradicts D+zl(tm) = 0.

Next we consider the case of all < 1, and the case thatall ≥ 1 and η

(3)l > η

(1)l . In these two cases, we have wll = η

(2)l

if all < 1; and wll = η(3)l if all ≥ 1. Then by (13), we have

bll > 0. Here we need to introduce some notations. For givenσ ≥ 0, set

γ1(σ) = (1 − all + αl)βl + (|1 − all| + c3)σ (26)

γ0(σ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

all − 1

bll

γ1(σ) + (bll + αl)βl

+(bll + c3)σ if all < 1(all − 1 + bll + αl)βl

+(all − 1 + bll + c3)σ if all ≥ 1

(27)

γ2(σ) = (|all − 1| + bll + αl)βl

+(|all − 1| + bll + c3)σ (28)

γ3(σ) = bll(βl − σ) − γ1(σ) (29)

γ4(σ) =

⎧⎪⎨⎪⎩

all − 1

bll

γ1(σ) + αlβl + c3σ if all < 1

(all − 1 + αl)βl + (all − 1 + c3)σif all ≥ 1

(30)

h(σ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

2γ2(σ)[γ1(σ) + bll(γ4(σ)+bll(βl + σ))τll − bll(βl − σ)]−(bll(βl+σ)−γ1(σ))2 if all < 1

2γ2(σ)[γ1(σ) + bll(γ4(σ)+bll(βl + σ))τll − bll(βl − σ)]+γ2

3 (σ)+2γ3(σ)(γ1(σ)−bll(βl+σ))if all ≥ 1.

(31)

By the definitions (26)–(30) of γi(σ), i = 0, 1, . . . , 4, andusing (13) and bll > 0, one can verify that

γ0(σ) > 0, γ4(σ) + bll(βl + σ) > 0, γ4(σ) + γ1(σ) > 0. (32)

Moreover, by some computations, one can obtain that

h(0) =β2l×

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[1 + 2τll(bll − 1 + all)]α2l

+2(bll+1−all)[1+ 2τll(bll−1+all)]αl

−(bll − 1 + all)[3bll + 1 − all

−2τll(bll + 1 − all)2] if all < 1(1 + 2bllτll)α2

l

+2(bll + all − 1)(1 + 2bllτll)αl

−(bll + all − 1)2(3 − 2bllτll)if all ≥ 1.

Then for given all, bll and τll, h(0) is a quadratic function ofαl, i.e., h(0) = h(0)(αl). Set ρ(s) = h(0)(s). It is easy to checkthat ρ(s) < 0 if and only if

s <

{η

(2)l if all < 1

η(3)l if all ≥ 1.

Thus, by (13) and (14), we have h(0) = ρ(αl) < 0. Since h(σ)is continuous on σ, there exists σ1 > 0 such that

h(σ) < 0 σ ∈ [0, σ1). (33)

From the definition (26) of γ1(σ), one can verify that βl −σ >

γ1(σ)bll

is equivalent to (bll + all − 1 − αl)βl > (bll + |1 −all| + c3)σ. Set σ2 = bll+all−1−αl

bll+|1−all|+c3βl, then by (13), σ2 > 0 and

βl − σ >γ1(σ)

bll

σ ∈ [0, σ2). (34)


Let

s1(σ) =bll(βl + σ) − γ1(σ)

bllγ2(σ)and s2(σ) =

γ3(σ)

bllγ0(σ).

It is noted that γ2(σ) = γ0(σ) when all ≥ 1. So we have

s1(σ) > s2(σ) for any σ > 0 when all ≥ 1. (35)

Now we consider the case of all < 1. In this case, let g0(σ) =s1(σ) − s2(σ). Then we have

g0(0) = − (bll − 1 + all − αl)(1 − all)

bll(bll + 1 − all + αl)(bll − 1 + all).

It follows from bll > 0 and (13) that g0(0) < 0. Thus thereexists σ3 > 0 such that g0(σ) < 0 for σ ∈ [0, σ3), that is

s1(σ) < s2(σ) for any σ ∈ [0, σ3) when all < 1. (36)

Set σ4 = min{βl, σ1, σ2, σ3}, where σ3 = σ3 if all < 1, andσ3 = +∞ if all ≥ 1. For any given σ ∈ (0, σ4), there existsT1 > τ such that |zj(t)| < βj + σ, j ∈ N , t ≥ T1 − τ, and

zl(tm) > βl − σ for tm ≥ T1 − τ. (37)

It follows from (15), (9), and (12) that for t ≥ T1

D+zl(tm) > (all − 1)+(βl − σ) − (all − 1)−(βl + σ)

+ bllzl(tm − τll)

−n∑

j=1,j �=l

(|alj| + |blj|)(βj + σ)

≥ (all − 1)βl − |all − 1|σ + bllzl(tm − τll)

−n∑

j=1,j �=l

(|alj| + |blj|)(dj

dl

βl + σ)

> − [(1 − all + αl)βl + (|1 − all| + c3)σ]

+ bllzl(tm − τll)

= −γ1(σ) + bllzl(tm − τll) (38)

where γ1(σ) is defined in (26). If we can prove that

zl(tm − τll) ≥ γ1(σ)

bll

(39)

then by (38), we will obtain D+zl(tm) > 0, which contradictsD+zl(tm) = 0. So the proof will be complete. In what follows,we will use the method of contradiction to prove (39). If (39)is not true, then we have

zl(tm − τll) <γ1(σ)

bll

. (40)

Under assumption (40), if we can further prove zl(tm) < βl−σ,then a contradiction follows from (37). To this end, it isobserved that combining (34), (37), and (40) together andusing the continuity of zl(t), there exists t1m ∈ (tm − τll, tm)such that

zl(t1m) =γ1(σ)

bll

and zl(t) >γ1(σ)

bll

for t ∈ (t1m, tm]. (41)

The above property of zl(t) makes it possible to make anestimation of zl(tm) by integrating the lth equation of (4) onboth sides from t1m to tm. In the following, we divide foursteps to prove zl(tm) < βl − σ under assumption (40).

1) Step 1: We assert that

|xl(t)| < 1 for t ∈ [t1m, tm]. (42)

Otherwise, by xl(t) < 1 for t ≥ T1, we will have t ∈[t1m, tm] such that xl(t) ≤ −1. Then zl(t) = −1−ql. It isnoted that zl(t) = yl(t) − ql ≥ −1 − ql for all t ≥ 0. Soγ1(σ)bll

= −1 − ql. This induces a contradiction from (40)that zl(tm − τll) <

γ1(σ)bll

= −1 − ql.2) Step 2: We make an estimation about the length of tm −

t1m. By (37), (41) and (42), we get

βl − σ − γ1(σ)

bll

< zl(tm) − zl(t1m)

=∫ tm

t1m

D+zl(t)dt

≤∫ tm

t1m

((all−1)zl(t)+bll(βl+σ)+αlβl+σc3)dt

≤

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

∫ tm

t1m

((all−1)γ1(σ)

bll

+bll(βl+σ)+αlβl+σc3)dt

if all < 1∫ tm

t1m

((all−1)(βl + σ)+bll(βl+σ)+αlβl+σc3)dt

if all ≥ 1= γ0(σ)(tm − t1m)

where γ0(σ) is defined in (27). By (32), γ0(σ) > 0. Itfollows that

tm−t1m >bll(βl − σ) − γ1(σ)

bllγ0(σ)=

γ3(σ)

bllγ0(σ)= s2(σ). (43)

By (34), s2(σ) > 0.3) Step 3: We make an estimation of zl(t) on [t1m−τll, t1m].

For t ∈ [t1m − τll, t1m], it is easy to obtain

|D+zl(t)| ≤ γ2(σ)

where γ2(σ) is defined in (28). So for t ∈ [t1m − τll, t1m]

zl(t) = zl(t) − zl(t1m) + zl(t1m)

= zl(t1m) −∫ t1m

t

D+zl(s)ds

≤ zl(t1m) + γ2(σ)(t1m − t)

= γ1(σ)/bll + γ2(σ)(t1m − t).

Thus, for t ∈ [t1m−τll, t1m], we get an estimate of zl(t) as

zl(t) ≤ min {βl + σ, γ1(σ)/bll + γ2(σ)(t1m − t)} . (44)

4) Step 4: Using (41)–(44), we prove zl(tm) < βl − σ. By(41), (42) and (4), we have

zl(tm) = zl(t1m) +∫ tm

t1m

D+zl(t)dt

≤ zl(t1m) +∫ tm

t1m

[(all − 1)zl(t)

+ bllzl(t − τll) + αlβl + c3σ]dt

≤ γ1(σ)

bll

+∫ tm

t1m

[bllzl(t − τll) + γ4(σ)]dt

where γ4(σ) is defined in (30).


It is noted that t1m ∈ (tm − τll, tm), so t − τll ∈ [t1m − τll, t1m]for t ∈ [t1m, tm]. By (44), zl(t − τll) has two differentestimations over [t1m, tm]. To reduce the conservatism, weinsert a point t2m in the integral interval [t1m, tm], where t2m

will be determined later. On the integral interval [t1m, t2m], weuse βl + σ to bound zl(t − τll), while on the integral interval[t2m, tm], we use γ1(σ)/bll+γ2(σ)(t1m−t+τll) to bound zl(t−τll).Then, we have

zl(tm) ≤ γ1(σ)

bll

+γ4(σ)(tm−t1m)+∫ t2m

t1m

bll(βl+σ)dt

+∫ tm

t2m

bll(γ1(σ)

bll

+ γ2(σ)(t1m − t + τll))dt

=γ1(σ)

bll

+[γ4(σ) + bll(βl + σ) − bllγ2(σ)ν

]×(t2m − t1m) + (γ1(σ) + γ4(σ))ν

+1

2bllγ2(σ)(2τll − ν)ν (45)

where ν = tm − t2m. The estimation (43) obtained in Step 2guarantees that for any s ∈ [0, s2(σ)], there exists t2m ∈[t1m, tm] such that ν = s. In order to enlarge the second term ofthe right side in inequality (45) as a function of ν, we requirethat t2m ∈ [t1m, tm] be chosen so as to make the coefficient oft2m − t1m in (45) nonnegative, that is

ν = tm − t2m ≤ γ4(σ) + bll(βl + σ)

bllγ2(σ)� s3(σ). (46)

By (32), we have s3(σ) > 0. Noticing that t2m − t1m = tm −t1m − (tm − t2m) ≤ τll − ν, it follows from (45) that

zl(tm) ≤ βl − σ + g(σ, ν) (47)

where

g(σ, ν) =1

2bllγ2(σ)ν2 + (γ1(σ) − bll(βl + σ))ν

+(γ4(σ)+bll(βl+σ))τll +γ1(σ)

bll

−(βl−σ).

Now we need to choose t2m ∈ [t1m, tm] satisfying the constraintν ≤ min{s2(σ), s3(σ)} such that g(σ, ν) < 0. For given σ > 0,it is easy to see that g(σ, ν) achieves its minimum when

ν = s1(σ) =bll(βl + σ) − γ1(σ)

bllγ2(σ).

From (32) and (36), we have s1(σ) < min{s2(σ), s3(σ)} whenall < 1. So we can choose some t2m ∈ [t1m, tm] such thatν = s1(σ) for the case of all < 1. However, when all ≥ 1,s1(σ) > s2(σ) for any σ > 0 by (35). So there may be not2m ∈ [t1m, tm] such that ν = s1(σ). Noticing that when all ≥ 1,by (32) and γ2(σ) = γ0(σ), we have s2(σ) ≤ s3(σ), so wechoose some t2m ∈ [t1m, tm] such that ν = s2(σ) for the caseof all ≥ 1. Thus, by setting

ν =

{s1(σ) if all < 1

s2(σ) if all ≥ 1

we get

g(σ, ν) =1

2bllγ2(σ)h(σ)

where h(σ) is defined in (31). By (33), h(σ) < 0. It followsfrom (47) that zl(tm) < βl −σ, which contradicts (37). Hence,(39) holds and the proof of Lemma 6 is finally complete.

Proof of Lemma 7: By the contrary, if there is noincreasing sequence {tk}, {tk} → +∞, such that xl(tk) ≥ 1,then for sufficiently large t, we have xl(t) < 1. From condition(21), there exists T1 > 0 such that |xl(tm)| < 1, xl(t) < 1, fortm, t ≥ T1. So we have N(xl(tm)) = 1 for tm ≥ T1.

First, we consider the case that all < 1. Then wll = ξ(2)l . We

distinguish two subcases to deduce a contradiction. The firstsubcase is that ξ

(1)l = 1 − all − |bll| > αl. In this subcase, for

any given σ ∈ (0, βl min{1, (ξ(1)l − αl)/(c2 + c3)}), there exists

T2 ≥ T1 + τ such that |zj(t)| ≤ βj + σ, j ∈ N , zl(tm) > βl − σ,for t, tm ≥ T2 − τ. Thus, by (4) and (9)–(12), for tm ≥ T2, weobtain

D+zl(tm) < −(1 − all)βl + |bll|(βl + σ)

+ (αlβl + σ(c3 + |all − 1|))≤ −

((ξ(1)

l − αl)βl − σ(c2 + c3))

< 0

which contradicts D+zl(tm) = 0.The second subcase is that ξ

(1)l ≤ αl. In this subcase,

condition (8) implies bll < 0. Set

c4 =1

1−all

∣∣e(1−all)τll (1−all+b−ll +c3)+b−

ll +c3

∣∣ .For any given σ ∈ (0, βl min{1, c1/(c2+c3+b−

ll c4)}), there existsT2 ≥ T1 + τ such that |zj(t)| ≤ βj + σ for t ≥ T2 − τ, j ∈ N .For t ∈ [tm − τll, tm], it follows from (4) and (9)–(12) that

D+zl(t) ≤Nl(xl(t)) {(all − 1)zl(t)

+ ((b−ll + αl)βl + (b−

ll + c3)σ)}

which yields

D+

(e

∫ t

0(1−all)p(s)ds

zl(t)

)

≤p(t)e∫ t

0(1−all)p(s)ds((b−

ll +αl)βl+(b−ll +c3)σ) (48)

where p(t) = Nl(xl(t)). Integrating the above inequality fromtm − τll to tm gives

zl(tm − τll)

≥{e

∫ tm

tm−τll

(1−all)p(s)ds(

1− b−ll +αl

1−all

)+

b−ll +αl

1−all

}βl

−σ

{e

∫ tm

tm−τll

(1−all)p(s)ds(

1+b−

ll +c3

1−all

)+

b−ll +c3

1−all

}.

Since 1 − all ≤ b−ll + αl by ξ

(1)l ≤ αl, it follows from the above

inequality that

zl(tm − τll)

≥{

e(1−all)τll

(1− b−

ll +αl

1−all

)+

b−ll +αl

1−all

}βl−σc4.


Furthermore, by (4) and (8), we obtain

D+zl(tm) ≤ −(1 − all)βl + bllzl(tm − τll)

+ (αlβl + σ(c3 + |all − 1|))≤ −{[(1 − all − b−

ll )(1 + b−ll µl) + 2b−

ll

−αl(1 + b−ll µl)

]βl

− σ(|all − 1| + c3 + b−ll c4)

}= −(1 + b−

ll µl) {(wll − αl)βl − σ(|all − 1|+c3 + b−

ll c4)/(1 + b−ll µl)

}≤ −(1 + b−

ll µl){c1βl − σ(c2 + c3 + b−

ll c4)}

< 0

which contradicts D+zl(tm) = 0.Now, we consider the case of all ≥ 1. Then wll = ξ

(3)l .

Moreover, by (14), we have bll < 0. For any scalar σ > 0,define

γ1(σ) = (all − 1 + αl)βl + (all − 1 + c3)σ (49)

γ2(σ) = (b−ll + all − 1 + αl)βl

+ (b−ll + all − 1 + c3)σ. (50)

It follows from (14) that γi(σ) > 0 for any σ > 0, i = 1, 2.Set

g1(σ) = βl − σ − γ1(σ)

b−ll

g2(σ) = γ2(σ)�l + γ1(σ) − b−ll (βl − σ).

Using (14) and bll < 0, one can verify that

g1(0) = βl

b−ll + 1 − all

b−ll

> 0,

g2(0) = βl[(b−ll − 1 + all)�l − (b−

ll + 1 − all)

+ αl(�l + 1)] < 0.

Then there exists σ1 > 0 such that

g1(σ) > 0 and g2(σ) < 0 for σ ∈ [0, σ1]. (51)

For any given σ ∈ (0, min{βl, σ1}], there exists T1 > τ suchthat |zj(t)| < βj + σ, j ∈ N , for t ≥ T1 − τ, and

zl(tm) > βl − σ for tm ≥ T1 − τ. (52)

It is noted from (4), (9), (12), and (15) that

D+zl(tm) < −b−ll zl(tm − τll) + γ1(σ) (53)

where γ1(σ) is defined in (49). If we can prove that

zl(tm − τll) ≥ γ1(σ)

b−ll

(54)

then by (53), we will obtain D+zl(tm) < 0, which contradictsthe fact that D+zl(tm) = 0. So the proof will be complete.As done in the proof of Lemma 6, we will prove (54) bycontradiction. If (54) is not true, we will have

zl(tm − τll) <γ1(σ)

b−ll

. (55)

In what follows, we will prove zl(tm) < βl − σ underassumption (55). Then a contradiction follows from (52).

First, by (51), (52), (55), and using the continuity of zl(t),there exists t1m ∈ (tm − τll, tm) such that

zl(t1m) =γ1(σ)

b−ll

and zl(t) >γ1(σ)

b−ll

for t ∈ (t1m, tm]. (56)

In the following, we will divide three steps to complete theproof of zl(tm) < βl − σ.

1) Step 1: We prove

|xl(t)| < 1 t ∈ [t1m, tm]. (57)

The proof is similar to the proof of (42), so it is omitted.2) Step 2: We give an estimation of D+zl(t) on [t1m−τll, tm].

For t ∈ [t1m − τll, tm], by (4), (9), (12) and (15), it iseasy to prove that

D+zl(t) ≤ γ2(σ) for t ∈ [t1m − τll, tm] (58)

where γ2(σ) is defined in (50).It follows from (56)–(58) that for t ∈ [t1m, tm]

D+zl(t) ≤ −b−ll zl(t − τll) + γ1(σ)

= −b−ll zl(t − τll) + b−

ll zl(t1m)

= b−ll

∫ t1m

t−τll

D+zl(s)ds

≤ b−ll γ2(σ)(t1m − t + τll).

Combining the above inequality with (58) yields

D+zl(t) ≤ γ2(σ) min{1, b−ll (t1m − t + τll) for t∈ [t1m, tm].

(59)3) Step 3: Using (59) and applying the twice estimation

technique, we prove zl(tm) < βl −σ. Integrating inequal-ity (59) on both sides from t1m to tm gives

zl(tm) − zl(t1m) =∫ tm

t1m

D+zl(s)ds ≤ γ2(σ)I (60)

where I =∫ tm

t1mmin{1, b−

ll (t1m − t + τll)}dt.

The integrand in I involves the minimum of two differentfunctions. To reduce the conservatism, we insert a point t2m

in the integral interval [t1m, tm], where t2m will be determinedlater. On the interval [t1m, t2m], we use the constant function 1as the bound of the integrand, while on the interval [t2m, tm],we use the function b−

ll (t1m − t + τll) as the bound of theintegrand. Then

I ≤∫ t2m

t1m

1dt +∫ tm

t2m

b−ll (t1m − t + τll)dt

= t2m−t1m+1

2b−

ll (2t1m−tm−t2m+2τll)(tm − t2m)

=1

2b−

ll ν22 − (1 + b−

ll (ν1 − τll))ν2 + ν1

� H(ν1, ν2) (61)

where ν1 = tm − t1m, ν2 = tm − t2m. For given ν1, one can verifythat H2(s) = H(ν1, s) achieves its minimum when s = s1(ν1) =1b−

ll

+ ν1 − τll.


If b−ll τll ≥ 1 and ν1 ≥ τll − 1

b−ll

, then s1(ν1) ∈ [0, ν1]. Sowe can choose t2m ∈ [t1m, tm] such that ν2 = s1(ν1). It followsfrom (61) and ν1 ≤ τll that

I ≤ − 1

2b−ll

(1 + b−ll (ν1 − τll))

2 + ν1 ≤ − 1

2b−ll

+ τll. (62)

If b−ll τll ≥ 1 and ν1 < τll − 1

b−ll

, then we choose t2m = tm,i.e., ν2 = 0. It follows from (61) that

I ≤ ν1 < τll − 1

b−ll

. (63)

If b−ll τll < 1, then we choose t2m = t1m, i.e., ν2 = ν1. It

follows from (61) and ν1 ≤ τll that

I ≤ −1

2b−

ll ν21 + b−

ll τllν1 ≤ 1

2b−

ll τ2ll . (64)

Combining (62)–(64) together gives

I ≤ �l

b−ll

=

{− 1

2b−ll

+ τll if b−ll τll ≥ 1

12b−

ll τ2ll if b−

ll τll < 1.

Substituting the estimate of I into (60) and using (51) and(56) gives

zl(tm) ≤ βl−σ+1

b−ll

[γ2(σ)�l+γ1(σ)−b−

ll (βl−σ)]

= βl − σ + g2(σ)/b−ll

< βl − σ

which contradicts (52). Therefore, (54) holds and the proof ofLemma 7 is complete.

Acknowledgment

The authors would like to thank Prof. M. Forti at theUniversity of Siena, Siena, Italy, very much for his helpfulcomments on an earlier version of this paper. They are alsovery grateful to the associate editor and the five anonymousreviewers, whose valuable comments and suggestions helpedto improve this paper.

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Wu-Hua Chen received the B.S. degree in math-ematics from Hubei Normal University, Huangshi,China, the M.S. degree in basic mathematics fromGuangxi Normal University, Guilin, China, and thePh.D. degree in control theory and control engineer-ing from the Huazhong University of Science andTechnology, Wuhan, China, in 1988, 1991, and 2004,respectively.

From 1991 to 2001, he was with the Guangxi Uni-versity for Nationalities, Nanning, China. In 2001,he joined Guangxi University, Nanning, Guangxi,

China, where he is currently a Professor with the College of Mathematicsand Information Science. From April 2005 to October 2005, July 2007 toJanuary 2008, and September 2008 to July 2009, he was a Visiting Fellowwith the University of Western Sydney, Sydney, NSW, Australia. His currentresearch interests include time-delay systems, robust control, neural networks,impulsive systems, and stochastic systems.


Wei Xing Zheng (M’93–SM’98) was born in Nan-jing, China. He received the B.S. degree in appliedmathematics, and the M.S. and Ph.D. degrees inelectrical engineering, in 1982, 1984, and 1989,respectively, all from Southeast University, Nanjing,China.

From 1984 to 1991, he was with the Institute ofAutomation, Southeast University, first as a Lecturerand later as an Associate Professor. From 1991 to1994, he was a Research Fellow with the Departmentof Electrical and Electronic Engineering, Imperial

College of Science, Technology and Medicine, London, U.K., with theDepartment of Mathematics, University of Western Australia, Perth, Australia,and with the Australian Telecommunications Research Institute, Curtin Uni-versity of Technology, Perth, Australia. Since 1994, he has been with theUniversity of Western Sydney, Sydney, NSW, Australia, where he is currentlya Full Professor with School of Computing and Mathematics. He has alsoheld various visiting positions with the Institute for Network Theory andCircuit Design, Munich University of Technology, Munich, Germany, with theDepartment of Electrical Engineering, University of Virginia, Charlottesville,and with the Department of Electrical and Computer Engineering, Universityof California, Davis. He co-authored the book Linear Multivariable Systems:Theory and Design (Nanjing, China: SEU Press, 1991). His current researchinterests include systems and controls, signal processing, and communications.

Dr. Zheng has received several science prizes, including the Chinese Na-tional Natural Science Prize awarded by the Chinese Government in 1991. Hehas served on the Technical Program or Organizing Committee of numerousinternational conferences, including the 42nd IEEE International Symposiumon Circuits and Systems (ISCAS) in 2009, the 15th International Federationof Automatic Control (IFAC) Symposium on System Identification (SYSID)in 2009, and the 49th IEEE Conference on Decision and Control (CDC) in2010. He has also served on several IEEE or IFAC technical committees,and has been the Chair of IEEE Circuits and Systems Society’s TechnicalCommittee on Neural Systems and Applications, the Chair of IEEE Circuitsand Systems Society’s Technical Committee on Blind Signal Processing,a member of IEEE Control Systems Society’s Technical Committee onSystem Identification and Adaptive Control, and a member of IFAC TechnicalCommittee on Modeling, Identification, and Signal Processing. He has servedas an Associate Editor for four flagship journals: IEEE Transactions on

Circuits and Systems-I: Fundamental Theory and Applications from2002 to 2004, IEEE Transactions on Automatic Control from 2004 to2007, IEEE Transactions on Circuits and Systems-II: Express Briefs

from 2008 to 2009, and IEEE Signal Processing Letters from 2007to present. Since 2000, he has also been an Associate Editor of the IEEEControl Systems Society’s Conference Editorial Board. Currently, he servesas a Guest Associate Editor of Special Issue on Blind Signal Processing andits Applications for the IEEE Transactions on Circuits and Systems-I:Regular Papers.

a new method for complete stability analysis of cellular neural networks with time delay

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