Applied Mathematics and Computation 188 (2007) 850–854

www.elsevier.com/locate/amc

Further note on global exponential stability of uncertaincellular neural networks with variable delays

Ju H. Park

Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea

Abstract

For cellular neural networks with time-varying delays and uncertainties, a recent work for exponential stability withconvergence rate k of the networks is extended. A new linear matrix inequality criterion for the stability, which give infor-mation on the delay-dependent property, is derived.� 2006 Elsevier Inc. All rights reserved.

Keywords: Neural networks; Exponential stability; LMI; Time-varying delays; Uncertainties

1. Introduction

The dynamical behaviors of neural networks have attracted increasing attention due to their applicability insome optimization problems and pattern recognition problems. In particular, the problem of stability analysisof delayed neural networks has been a focused topic of theoretical and practical importance since time-delay iscommonly encountered in biological and artificial neural networks [1–7] and its existence is frequently a sourceof oscillation and instability [8–12]. Recently, using various analyzing methods, many criteria for global sta-bility of neural networks with constant delays or time-varying delays have been presented [13–18].

As we have known, fast convergence of a system is essential for real-time computation, and the exponen-tially convergence rate is generally used to determine the speed of neural computations. Thus, global exponen-tial stability for neural networks without delays or with delays has been also investigated [19–23] in very recentyears.

In this note, we extend the recent result [23] for the exponential stability and estimate the exponential con-vergence rates of neural networks with time-varying delays to the network with uncertainties. Based on thelinear matrix inequality (LMI) criterion given in [23], a extended delay-dependent criterion for exponentialstability of the system is derived by simple procedure. The advantage of the proposed approach is that theresulting stability criterion can be used efficiently via existing numerical convex optimization algorithms suchas the interior-point algorithms for solving LMIs [24].

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.10.036

E-mail address: jessie@yu.ac.kr

J.H. Park / Applied Mathematics and Computation 188 (2007) 850–854 851

Throughout the note, Rn denotes the n dimensional Euclidean space, and Rn�m is the set of all n · m realmatrices. kxk denotes the norm of vector x. w denotes the elements below the main diagonal of a symmetricblock matrix. diag{Æ} denotes the block diagonal matrix. For symmetric matrices X and Y, the notation X > Y

(respectively, X P Y) means that the matrix X � Y is positive definite, (respectively, nonnegative).

2. Main results

Consider a continuous neural networks with time-varying delays can be described by the following stateequations:

_yiðtÞ ¼ �ðai þ DaiÞyiðtÞ þXn

j¼1

ðwij þ DwijÞfjðyjðtÞÞ þXn

j¼1

ðw1ij þ Dw1

ijÞfjðyjðt � hðtÞÞÞ þ bi; i ¼ 1; 2; . . . ; n;

ð1Þ

or equivalently

_yðtÞ ¼ �ðAþ DAÞyðtÞ þ ðW þ DW Þf ðyðtÞÞ þ ðW 1 þ DW 1Þf ðyðt � hðtÞÞÞ þ b; ð2Þ

where yðtÞ ¼ ½y1ðtÞ; y2ðtÞ; . . . ; ynðtÞ�T 2 Rn is the neuron state vector, f ðyðtÞÞ ¼ ½f1ðy1ðtÞÞ; . . . ; fnðynðtÞÞ�

T 2 Rn

is the activation functions, f ðyðt � hðtÞÞÞ ¼ ½f1ðy1ðt � hðtÞÞÞ; f2ðy2ðt � hðtÞÞÞ; . . . ; fnðynðt � hðtÞÞÞ�T 2 Rn,b = [b1,b2, . . .,bn]T is a constant input vector, A = diag(ai) is a positive diagonal matrix, W = (wij)n·n andW 1 ¼ ðw1

ijÞn�n are the interconnection matrices representing the weight coefficients of the neurons, the time de-lays h(t) is bounded nonnegative functions satisfying 0 6 hðtÞ 6 �h with _hðtÞ < hd < 1, and DA, DW, DW1 areparametric uncertainties and it is assumed that the continuous matrix-valued functions DA(t), DW(t), DW1(t)are of the form:

DAðtÞ ¼ H 1F 1ðtÞE1; DW ðtÞ ¼ H 2F 2ðtÞE2; DW 1ðtÞ ¼ H 3F 3ðtÞE3;

where H1, H2, H3, E1, E2, E3 are known constant matrices of appropriate dimensions, and F1(t), F2(t), F3(t) areunknown time-varying matrix with Lebesgue measurable elements bounded by:

F Ti ðtÞF iðtÞ 6 I ; i ¼ 1; 2; 3:

In this note, it is assumed that the activation function f(y) is nondecreasing, bounded and globally Lipschitz;that is,

0 6fiðn1Þ � fiðn2Þ

n1 � n2

6 ri; i ¼ 1; 2; . . . ; n: ð3Þ

Then, by using the well-known Brouwer’s fixed point theorem [3], one can easily prove that there exists at leastone equilibrium point for Eq. (2).

For the sake of simplicity in the stability analysis of system (2), we make the following transformation tosystem (2):

xð�Þ ¼ yð�Þ � y�;

where y� ¼ ðy�1; y�2; . . . ; y�nÞT is an equilibrium point of Eq. (2). Under the transformation, it is easy to see that

system (2) becomes

_xðtÞ ¼ �ðAþ DAÞxðtÞ þ ðW þ DW ÞgðxðtÞÞ þ ðW 1 þ DW 1Þgðxðt � hðtÞÞÞ; ð4Þ

where xðtÞ ¼ ½x1ðtÞx2ðtÞ . . . xnðtÞ�T 2 Rn is the state vector of the transformed system, g(x) = [g1(x),

g2(x), . . .,gn(x)]T and gjðxjðtÞÞ ¼ fjðxjðtÞ þ y�j Þ � fjðy�j Þ with gj(0) = 0, "j. It is noted that each activation func-tion gi(Æ) satisfies the following sector condition:

0 6giðn1Þ � giðn2Þ

n1 � n2

6 ri; i ¼ 1; 2; . . . ; n: ð5Þ

The following definition, facts and lemma will be used in main result.

852 J.H. Park / Applied Mathematics and Computation 188 (2007) 850–854

Definition 1. For system defined by (4), if there exist a positive constant k and c(k) > 0 such that

kxðtÞk 6 cðkÞe�kt sup��h6h60

kxðhÞk 8t > 0;

then, the origin of system (4) is exponentially stable where k is called the convergence rate (or degree) of expo-nential stability.

Fact 1 (Schur complement). Given constant symmetric matrices R1, R2, R3 where R1 ¼ RT1 and 0 < R2 ¼ RT

2 ,then R1 þ RT

3 R�12 R3 < 0 if and only if

R1 RT3

R3 �R2

" #< 0; or

�R2 R3

RT3 R1

� �< 0:

Fact 2. For any z; y 2 Rn�m and a positive scalar �, the following inequality

2zTy 6 �zTzþ ��1yTy

holds.

Lemma 1 [23]. For system (4) without uncertainties, i.e. DA = 0, DW = 0, DW1 = 0, the equilibrium point of (4) is

globally exponentially stable with convergence rate k if there exist positive definite matrices P, Q, R, and a posi-

tive diagonal matrix D = diag{d1,d2, . . ., dn} satisfying the following LMI:

P ¼P1 PW þ 4kD PW 1

H P2 DW 1

H H P3

264

375 < 0; ð6Þ

where R = diag{r1,r2, . . .,rn} is given positive diagonal matrix and

P1 ¼ 2kP � PA� ATP þ R;

P2 ¼ DW þ W TDþ Q� 2DAR�1;

P3 ¼ �ð1� hdÞe�2k�hðQþ R�1RR�1Þ:

Then, we have the following theorem.

Theorem 1. The equilibrium point of system (4) is globally exponenitally stable with convergence rate k if there

exist positive definite matrices P, Q, R, a positive diagonal matrix D = diag{d1,d2, . . ., dn}, and six positive scalars�i (i = 1,2, . . ., 6), satisfying the following LMI:

P1 þ �1ET1 E1 PW þ 4kD PW 1 P6 0

H P�2 DW 1 0 P7

H H P3 þ ð�3 þ �6ÞET3 E3 0 0

H H H �P4 0

H H H H �P5

26666664

37777775< 0; ð7Þ

where

P�2 ¼ P2 þ ð�2 þ �5ÞET2 E2 þ �4R

�1ET1 E1R

�1;

P4 ¼ diagf�1; �2; �3g;P5 ¼ diagf�4; �5; �6g;P6 ¼ PH 1 PH 2 PH 3½ �;P7 ¼ DH 1 DH 2 DH 3½ �:

J.H. Park / Applied Mathematics and Computation 188 (2007) 850–854 853

Proof. By Lemma 1, the system (4) is globally exponentially stable with convergence rate k if the followinginequality hold:

Pþ X1F 1ðtÞXT2 þ X2F T

1 ðtÞXT1 þ X3F 2ðtÞXT

4 þ X4F T2 ðtÞXT

3 þ X5F 3ðtÞXT6 þ X6F T

3 ðtÞXT5 þ X7F 1ðtÞXT

8

þ X8F T1 ðtÞXT

7 þ X9F 2ðtÞXT10 þ X10F T

2 ðtÞXT9 þ X11F 3ðtÞXT

12 þ X12F T3 ðtÞXT

11 < 0; ð8Þ

where

X1 ¼�PH 1

0

0

264

375; X2 ¼

ET1

0

0

264

375; X3 ¼

PH 2

0

0

264

375; X4 ¼

0

ET2

0

264

375;

X5 ¼PH 3

0

0

264

375; X6 ¼

0

0

ET3

264

375; X7 ¼

0

�DH 1

0

264

375; X8 ¼

0

E1R�1

0

264

375;

X9 ¼0

DH 2

0

264

375; X10 ¼ X4; X11 ¼

0

DH 3

0

264

375; X12 ¼ X6:

By Fact 2, Eq. (8) holds if the following inequality satisfies

Pþ ��11 X1X

T1 þ �1X2X

T2 þ ��1

2 X3XT3 þ �2X4X

T4 þ ��1

3 X5XT5 þ �3X6X

T6 þ ��1

4 X7XT7 þ �4X8X

T8 þ ��1

5 X9XT9

þ �5X10XT10 þ ��1

6 X11XT11 þ �6X12X

T12 � Pþ X < 0; ð9Þ

where �i > 0, i = 1,2, . . ., 6, and

X ¼ð1; 1Þ 0 0

H ð2; 2Þ 0

H H ð�3 þ �6ÞET3 E3

264

375;

ð1; 1Þ ¼ ��11 PH 1H T

1 P þ ��12 PH 2H T

2 P þ ��13 PH 3HT

3 P þ �1ET1 E1;

ð2; 2Þ ¼ ��14 DH 1H T

1 Dþ ��15 DH 2H T

2 Dþ ��16 DH 3H T

3 Dþ ð�2 þ �5ÞET2 E2 þ �4R

�1ET1 E1R

�1:

Then, by Fact 1, inequality (9) is equivalent to LMI (7). This completes the proof. h

Remark 1. The criterion given in Theorem 1 is dependent on the bound, �h, of time-delay. It is well known thatthe delay-dependent criteria are less conservative than delay-independent criteria when the delay is small. Byiteratively solving the LMIs given in Theorem 1 with respect to �h, one can find the maximum allowable upperbound �h of time-delay h(t) for guaranteeing the exponential stability of system (4).

Remark 2. From Theorem 1, one can determine an upper bound of k such that system (4) is exponentiallystable. It requires solving the following optimization problem:

max k

subject to LMI ð7Þ:

This means that system (4) will be exponentially stable if k 6 �k where �k is the maximized value of k of theoptimization problem.

Example 1. Consider the following uncertain neural network:

_xðtÞ ¼ �ðAþ H 1F 1ðtÞE1ÞxðtÞ þ ðW þ H 2F 2ðtÞE2ÞgðxðtÞÞ þ ðW 1 þ H 3F 3ðtÞE3Þgðxðt � hðtÞÞÞ; ð10Þ

where �h ¼ 0:5, hd = 0.1 and

A ¼ I ; W ¼0:4 0:1

0:1 �0:5

� �; W 1 ¼

0:2 0:1

0 0:2

� �; H 1 ¼ 0; H 2 ¼ H 3 ¼ 0:05I ;

E1 ¼ 0; E2 ¼ E3 ¼ I ; giðxÞ ¼ 0:5ðjxþ 1j � jx� 1jÞ; F Ti ðtÞF iðtÞ 6 I ; 8i:

854 J.H. Park / Applied Mathematics and Computation 188 (2007) 850–854

For convergence rate k = 0.1, the LMI solutions of Theorem 1 for this example are

P ¼8:5068 2:2299

2:2299 3:6511

� �; Q ¼

0:2603 �0:2711

�0:2711 3:1996

� �;

R ¼4:5498 1:9580

1:9580 3:4164

� �; D ¼

6:5758 0

0 2:4943

� �;

�1 ¼ 2:8269; �2 ¼ 0:4409; �3 ¼ 0:5277;

�4 ¼ 2:8269; �5 ¼ 0:4241; �6 ¼ 0:5097;

which implies that the system is exponentially stable with the decay rate k = 0.1.

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