Delay-probability-distribution-dependent robust stability analysis for stochastic neural networks with time-varying delay

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www.elsevier.com/locate/pnsc

Progress in Natural Science 19 (2009) 1333–1340

Delay-probability-distribution-dependent robust stability analysisfor stochastic neural networks with time-varying delay

Jie Fu, Huaguang Zhang *, Tiedong Ma

School of Information Science and Engineering, Northeastern University, Box 134, Shenyang 110004, China

Received 8 October 2008; received in revised form 7 November 2008; accepted 10 November 2008

Abstract

The delay-probability-distribution-dependent robust stability problem for a class of uncertain stochastic neural networks (SNNs) withtime-varying delay is investigated. The information of probability distribution of the time delay is considered and transformed intoparameter matrices of the transferred SNNs model. Based on the Lyapunov–Krasovskii functional and stochastic analysis approach,a delay-probability-distribution-dependent sufficient condition is obtained in the linear matrix inequality (LMI) format such that delayedSNNs are robustly globally asymptotically stable in the mean-square sense for all admissible uncertainties. An important feature of theresults is that the stability conditions are dependent on the probability distribution of the delay and upper bound of the delay derivative,and the upper bound is allowed to be greater than or equal to 1. Finally, numerical examples are given to illustrate the effectiveness andless conservativeness of the proposed method.� 2009 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in

China Press. All rights reserved.

Keywords: Delay-probability-distribution-dependent; Stochastic neural networks; Time-varying delay; Linear matrix inequality

1. Introduction

The stability analysis problem for delayed NNs hasreceived considerable research attention in the last decade,see Refs. [1–3], where the delay type can be constant, timevarying or distributed, and the stability criteria can bedelay dependent or delay independent. Since delay-depen-dent methods make use of information on the length ofdelays, they are generally less conservative than delay-inde-pendent ones. However, in a real system, time delay oftenexists in a random form, that is, if some values of the timedelay are very large but the probability of the delay takingsuch large values is very small, it may lead to a more con-servative result if only the information of variation range ofthe time delay is considered. In addition, its probabilisticcharacteristic, such as the Bernoulli distribution and the

1002-0071/$ - see front matter � 2009 National Natural Science Foundation o

and Science in China Press. All rights reserved.

doi:10.1016/j.pnsc.2008.11.012

* Corresponding author. Tel.: +86 24 83687762; fax: +86 24 83689605.E-mail address: [email protected] (H. Zhang).

Poisson distribution, can also be obtained by statisticalmethods. Therefore, it is necessary and realizable to inves-tigate the probability-distribution delay. Recently, the sta-bility of discrete NNs and discrete SNNs with probability-distribution delay are investigated in Refs. [4,10], respec-tively. But neither of them considers the information ofthe delay derivative.

It has been known that there are two kinds of distur-bances that are unavoidable to be considered when onemodels the NNs. One is parameter uncertainty, the otheris stochastic disturbance. For the stability analysis of SNNswith parameter uncertainty, some results related to thisproblem have recently been published, see Refs. [5–12].However, Refs. [5,6,10,11] do not consider the informationof the delay derivative. In Refs. [7–9], the information ofthe derivative is taken into consideration, but the upperbound l of the derivative must be smaller than 1. In thecase of l P 1 the results in the aforementioned literatureseither cannot be applicable [9] or discard the information

f China and Chinese Academy of Sciences. Published by Elsevier Limited

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1334 J. Fu et al. / Progress in Natural Science 19 (2009) 1333–1340

of the derivative of time delays [7,8], which is obviouslyunreasonable. Therefore, it is essential and significant toinvestigate the problem of how to eliminate the constrainton the upper bound of the delay derivative in SNNs.

In this paper, we investigate the robust stability problemfor a class of uncertain SNNs with time-varying delay. Theinformation of delay-probability distribution is introducedinto the SNNs model and a new method is proposed toeliminate the constraint on the upper bound of the delayderivative. Less conservative stability criteria are presentedsuch that the SNNs with probability distribution delay isrobustly, globally, asymptotically stable in the mean-square sense for all admissible uncertainties. All resultsare established in the form of LMI and can be solved easilyusing the interior algorithms [13]. Numerical examplesshow less conservativeness of our results.

2. Problem formulation and preliminaries

Notation. For s > 0;Cð½�s; 0�; RnÞ denotes the family ofcontinuous functions / from ½�s; 0� to Rn with the normk/k ¼ sup�s6h60j/ðhÞj. ðX;F; fFtgt>0;PÞ be a completeprobability space with a filtration fFtgt>0 satisfying theusual conditions. Let L2

F0ð½�s; 0�; RnÞ be the family of F0

measurable Cð½�s; 0�; RnÞ-valued random variablesn ¼ fnðhÞ : �s 6 h 6 0g such that sup�s6h60EjnðhÞj

2 <1,where Ef�g stands for the mathematical expectation oper-ator with respect to the given probability measure P. I isthe identity matrix of appropriate dimensions.

We consider uncertain SNN with time-varying delay asfollows:

dxðtÞ ¼ ½�AðtÞxðtÞ þW0ðtÞf ðxðtÞÞ þW 1ðtÞf ðxðt � sðtÞÞÞ�dt

þ ½CxðtÞ þDxðt � sðtÞÞ�dwðtÞxðtÞ ¼ nðtÞ; t 2 ½�sM ; 0�

ð1Þwhere xðtÞ ¼ ½x1ðtÞ; x2ðtÞ; . . . ; xnðtÞ�T 2 Rn is the neural statevector associated with the neurons, f ðxðtÞÞ ¼½f1ðx1ðtÞÞ; f2ðx2ðtÞÞ; . . . ; fnðxnðtÞÞ�T 2 Rn is the neuron activa-tion function. AðtÞ ¼ Aþ DAðtÞ;W0ðtÞ ¼W0 þ DW 0ðtÞ;W1ðtÞ ¼ W1 þ DW1ðtÞ;A ¼ diagða1; a2; . . . ; anÞ is a diago-nal matrix with positive entries ai > 0;W 0 2 Rn�n andW1 2 Rn�n are the connection weight matrix and the delayedconnection weight matrix, respectively, C 2 Rn�n andD 2 Rn�n are known real constant matrices. DAðtÞ;DW0ðtÞand DW1ðtÞ represent the parameter uncertainties. wðtÞ is aone-dimension Brownian motion. The time-varying delaysðtÞ satisfies

0 6 sðtÞ 6 sM

Assumption 1. Considering the information of probabilitydistribution of the time delay sðtÞ, two sets and functionsare defined.

X1 ¼ ft : sðtÞ 2 ½0; s0Þg and X2 ¼ ft : sðtÞ 2 ½s0; sM �g ð2Þ

s1ðtÞ¼sðtÞ; for t2X1;

�s1; for t2X2;

�and s2ðtÞ¼

sðtÞ; for t2X2

�s2; for t2X1

�ð3Þ

_s1ðtÞ 6 l1 <1; _s2ðtÞ 6 l2 <1 ð4Þwhere s0 2 ½0; sM �, �s1 2 ½0; s0Þ and �s2 2 ½s0; sM �.

It is easy to know t 2 X1 means the event sðtÞ 2 ½0; s0Þoccurs and t 2 X2 means the event sðtÞ 2 ½s0; sM � occurs.Therefore, a stochastic variable aðtÞ can be defined as

aðtÞ ¼1; t 2 X1

0; t 2 X2

�ð5Þ

Assumption 2. aðtÞ is a Bernoulli distributed sequence with

ProbfaðtÞ ¼ 1g ¼ EfaðtÞg ¼ a0;

ProbfaðtÞ ¼ 0g ¼ 1� EfaðtÞg ¼ 1� a0

where 0 6 a0 6 1 is a constant and EfaðtÞg is the expecta-tion of aðtÞ.

Remark 1. From Assumption 2, it is easy to knowEfaðtÞ � a0g ¼ 0; EfðaðtÞ � a0Þ2g ¼ a0ð1� a0Þ.

By Assumptions 1 and 2, the system (1) can be rewrittenas

dxðtÞ¼ ½�AðtÞxðtÞþW0ðtÞf ðxðtÞÞþaðtÞW1ðtÞf ðxðt� s1ðtÞÞÞþð1�aðtÞÞW1ðtÞf ðxðt� s2ðtÞÞÞ�dtþ½CxðtÞþaðtÞDxðt� s1ðtÞÞþð1�aðtÞÞDxðt� s2ðtÞÞ�dwðtÞ

xðtÞ¼ nðtÞ; t2 ½�sM ;0�

ð6Þ

which is equivalent to

dxðtÞ¼ ½�AðtÞxðtÞþW0ðtÞf ðxðtÞÞþa0W1ðtÞf ðxðt� s1ðtÞÞÞþð1�a0ÞW1ðtÞf ðxðt� s2ðtÞÞÞþðaðtÞ�a0ÞðW1ðtÞf ðxðt� s1ðtÞÞÞ�W1ðtÞf ðxðt� s2ðtÞÞÞÞ�dt

þ½CxðtÞþa0Dxðt� s1ðtÞÞþð1�a0ÞDxðt� s2ðtÞÞþðaðtÞ�a0ÞðDxðt� s1ðtÞÞ�Dxðt� s2ðtÞÞÞ�dwðtÞ

xðtÞ¼ nðtÞ; t2 ½�sM ;0�

ð7Þ

Assumption 3. The neural activation function fiðxiÞ satisfies

l�i 6fiðxiÞ� fiðyiÞ

xi� yi

6 lþi 8xi;yi 2R; xi–yi; i¼ 1;2; . . . ;n ð8Þ

which implies that

ðfiðxiÞ � lþi xiÞðfiðxiÞ � l�i xiÞ 6 0 ð9Þwhere l�i ; l

þi are some constants.

Assumption 4. The parameter uncertainties DAðtÞ;DW0ðtÞand DW1ðtÞ are of the forms:

½DAðtÞDW0ðtÞDW1ðtÞ� ¼ EFðtÞ½H1H2H3� ð10Þwhere E;H1;H2 and H3 are known constant matrices, FðtÞsatisfies FTðtÞFðtÞ 6 I ; for 8t 2 R.

Let V ðxt; tÞ 2 C2;1ðRn � RþÞ be a positive functionwhich is continuously twice differentiable in x and once

J. Fu et al. / Progress in Natural Science 19 (2009) 1333–1340 1335

differentiable in t. Thus, the stochastic derivative operatorL acting on V ðxt; tÞ is defined by

LV ðxt; tÞ ¼ V tðxt; tÞ þ Vxðxt; tÞgðtÞ þ1

2½rTðtÞVxxðxt; tÞrðtÞ�

where

V tðxt; tÞ ¼oV ðxt; tÞ

ot; Vxðxt; tÞ ¼

oV ðxt; tÞox1

; . . . ;oV ðxt; tÞ

oxn

� �;

Vxxðxt; tÞ ¼o2V ðxt; tÞ

oxioxj

� �n�n

;

gðtÞ ¼ � AxðtÞ þW0f ðxðtÞÞ þ a0W1f ðxðt � s1ðtÞÞÞþ ð1� a0ÞW1f ðxðt � s2ðtÞÞÞ þ ðaðtÞ � a0Þ� ðW 1f ðxðt � s1ðtÞÞÞ �W1f ðxðt � s2ðtÞÞÞÞ;

rðtÞ ¼ CxðtÞ þ a0Dxðt � s1ðtÞÞ þ ð1� a0ÞDxðt � s2ðtÞÞþ ðaðtÞ � a0ÞðDxðt � s1ðtÞÞ �Dxðt � s2ðtÞÞÞ:

Definition 1. For system (7) and any n 2 L2F0ð½�s; 0�; RnÞ,

the trivial solution is robustly, globally, asymptotically sta-ble in the mean-square sense for all admissible uncertain-ties, if

limt!1

Ejxðt; nÞj2 ¼ 0 ð11Þ

where xðt; nÞ is the solution of system (7) at time t under theinitial state n.

Lemma 1. [14] For any G 2 Rn�n;G > 0, scalars b and k >0, and vector function x : ½b� k; b� ! Rn such that the inte-gration in the following is well defined, thenZ b

b�kxðsÞds

� �T

G

Z b

b�kxðsÞds 6 k

Z b

b�kxTðsÞGxðsÞds

Lemma 2. [15] Let U ;VðtÞ;W and Z be real matrices of

appropriate dimensions with Z satisfying Z ¼ ZT, then

Z þUVðtÞW þWTVTðtÞUT < 0; for all VTðtÞVðtÞ 6 I

if and only if there exists a scalar e > 0 such that

Z þ e�1UUT þ eWTW < 0

3. Main results

For presentation convenience, in the following, we denote

L1 ¼ diagðlþ1 l�1 ; . . . ; lþn l�n Þ, L2 ¼ diaglþ1þl�12; . . . ; lþn þl�n

2

� �.

We firstly investigate the stability of nominal SNN with-out parameter uncertainties.

Theorem 1. For given scalars s0 P 0; sM > 0; l1; 0 < a0 < 1satisfying a0l1 < 1, the SNN described by (7) is globally

asymptotically stable in the mean-square sense, if there existpositive matrices P > 0;Qj > 0 ðj ¼ 1; 2; 3Þ;R1 > 0;R2 > 0;S1 > 0;S2 > 0, positive diagonal matrices K j > 0 ðj ¼

1; 2; 3Þ and real matrices M i;N i ði ¼ 1; 2; . . . 6Þ of appropri-

ate dimensions, such that the following LMI holds:

N¼

W1 M �M ~M N �N �A

� �S1 0 0 0 0 0

� � �S1 0 0 0 0

� � � �S1 0 0 0

� � � � �S2 0 0

� � � � � �S2 0

� � � � � � ��P

2666666666664

3777777777775< 0 ð12Þ

where

W1 ¼

W1;1 W1;2 0 0 0 0 W1;7 W1;8 0 0 0 0 W1;13 W1;14 W1;15

� W2;2 W2;3 0 0 0 0 W2;8 W2;9 0 0 0 0 0 0

� � W3;3 W3;4 W3;5 0 0 0 W3;9 W3;10 0 0 0 W3;14 0

� � � W4;4 W4;5 0 0 0 0 W4;10 W4;11 0 0 0 0

� � � � W5;5 W5;6 0 0 0 0 W5;11 W5;12 0 0 W5;15

� � � � � W6;6 0 0 0 0 0 W6;12 0 0 0

� � � � � � W7;7 0 0 0 0 0 W7;13 W7;14 W7;15

� � � � � � � W8;8 0 0 0 0 0 0 0

� � � � � � � � W9;9 0 0 0 0 0 0

� � � � � � � � � W10;10 0 0 0 0 0

� � � � � � � � � � W11;11 0 0 0 0

� � � � � � � � � � � W12;12 0 0 0

� � � � � � � � � � � � W13;13 0 0

� � � � � � � � � � � � � W14;14 0

� � � � � � � � � � � � � � W15;15

266666666666666666666666666666664

377777777777777777777777777777775

W1;1 ¼ Q1 þQ2 þQ3 þM1 þMT1 �N5A� ATNT

5 � K1L1

W1;2 ¼ �M1 þMT2

W1;7 ¼ P � ATNT6 �N5

W1;8 ¼ �M1

W1;13 ¼ N5W 0 þ K1L2

W1;14 ¼ a0N5W 1

W1;15 ¼ ð1� a0ÞN5W1

W2;2 ¼ �ð1� a0l1ÞQ1 �M2 �MT2 þM3 þMT

3

W2;3 ¼ �M3 þMT4 ; W2;8 ¼ �M2; W2;9 ¼ �M3

W3;3 ¼ a0ð1� a0ÞDT �PD�M4 �MT4 þM5 þMT

5 � K2L1

W3;4 ¼ �M5 þMT6

W3;5 ¼ �a0ð1� a0ÞDT �PD

W3;9 ¼ �M4; W3;10 ¼ �M5

W3;14 ¼ K2L2

W4;4 ¼ �Q2 �M6 �MT6 þN1 þNT

1

W4;5 ¼ �N1 þNT2 ; W4;10 ¼ �M6; W4;11 ¼ �N1

W5;5 ¼ a0ð1� a0ÞDT �PD�N2 �NT2 þN3 þNT

3 � K3L1

W5;6 ¼ �N3 þNT4 ; W5;11 ¼ �N2; W5;12 ¼ �N3

W5;15 ¼ K3L2

W6;6 ¼ �Q3 �N4 �NT4

W6;12 ¼ �N4

W7;7 ¼ s0R1 þ ðsM � s0ÞR2 �N6 �NT6


W7;13 ¼ N6W 0

W7;14 ¼ a0N6W1

W7;15 ¼ ð1� a0ÞN6W1

W8;8 ¼ �1

a0s0

R1

W9;9 ¼ �1

s0ð1� a0ÞR1

W10;10 ¼ �1

s0

R1

W11;11 ¼ W12;12 ¼ �1

sM � s0

R2

W13;13 ¼ �K1; W14;14 ¼ �K2; W15;15 ¼ �K3

�P ¼ P þ s0S1 þ ðsM � s0ÞS2

MT ¼ ½MT1 MT

2 0 0 0 0 0 0 0 0 0 0 0 0 0��MT ¼ ½0 MT

3 MT4 0 0 0 0 0 0 0 0 0 0 0 0�

~MT ¼ ½0 0 MT5 MT

6 0 0 0 0 0 0 0 0 0 0 0�

NT ¼ ½0 0 0 NT1 NT

2 0 0 0 0 0 0 0 0 0 0��NT ¼ ½0 0 0 0 NT

3 NT4 0 0 0 0 0 0 0 0 0�

�AT ¼ ½�PC 0 a0�PD 0 ð1� a0Þ�PD 0 0 0 0 0 0 0 0 0 0�

Proof. Choose a Lyapunov–Krasovskii functional candi-date V ðxt; tÞ ¼

P4i¼1V iðxt; tÞ, where

V 1ðxt; tÞ ¼ xTðtÞPxðtÞ

V 2ðxt; tÞ ¼Z t

t�a0s1ðtÞxTðsÞQ1xðsÞdsþ

Z t

t�s0

xTðsÞQ2xðsÞds

þZ t

t�sM

xTðsÞQ3xðsÞds

V 3ðxt; tÞ ¼Z �0

�s0

Z t

tþsgTðhÞR1gðhÞdhdsþ

Z �s0

�sM

�Z t

tþsgTðhÞR2gðhÞdhds

V 4ðxt; tÞ ¼Z �0

�s0

Z t

tþsrTðhÞS1rðhÞdhdsþ

Z �s0

�sM

�Z t

tþsrTðhÞS2rðhÞdhds

Then, the stochastic differential of V ðxt; tÞ (see [16]) alongwith (7) can be obtained as

dV ðxt; tÞ ¼LV ðxt; tÞdt þ 2xTðtÞPrðtÞdwðtÞ ð13Þ

Furthermore, we can get

LV 1ðxt; tÞ ¼ 2xTðtÞPgðtÞ þ rTðtÞPrðtÞ ð14Þ

LV 2ðxt; tÞ 6 xTðtÞðQ1 þQ2 þQ3ÞxðtÞ� ð1� a0l1ÞxTðt � a0s1ðtÞÞQ1xðt � a0s1ðtÞÞ� xTðt � s0ÞQ2xðt � s0Þ� xTðt � sMÞQ3xðt � sMÞ ð15Þ

LV 3ðxt; tÞ ¼ gTðtÞ½s0R1 þ ðsM � s0ÞR2�gðtÞ

�Z t

t�a0s1ðtÞgTðsÞR1gðsÞds

�Z t�a0s1ðtÞ

t�s1ðtÞgTðsÞR1gðsÞds

�Z t�s1ðtÞ

t�s0

gTðsÞR1gðsÞds

�Z t�s0

t�s2ðtÞgTðsÞR2gðsÞds

�Z t�s2ðtÞ

t�sM

gTðsÞR2gðsÞds

6 gTðtÞ½s0R1 þ ðsM � s0ÞR2�gðtÞ �1

a0s0

�Z t

t�a0s1ðtÞgTðsÞdsR1

Z t

t�a0s1ðtÞgðsÞds

� 1

s0ð1� a0Þ

Z t�a0s1ðtÞ

t�s1ðtÞgTðsÞdsR1

�Z t�a0s1ðtÞ

t�s1ðtÞgðsÞds� 1

s0

�Z t�s1ðtÞ

t�s0

gTðsÞdsR1

Z t�s1ðtÞ

t�s0

gðsÞds

� 1

sM � s0

Z t�s0

t�s2ðtÞgTðsÞdsR2

�Z t�s0

t�s2ðtÞgðsÞds� 1

sM � s0

�Z t�s2ðtÞ

t�sM

gTðsÞdsR2

Z t�s2ðtÞ

t�sM

gðsÞds ð16Þ

LV 4ðxt; tÞ 6 rTðtÞ½s0S1 þ ðsM � s0ÞS2�rðtÞ

�Z t

t�a0s1ðtÞrTðsÞS1rðsÞds

�Z t�a0s1ðtÞ

t�s1ðtÞrTðsÞS1rðsÞds

�Z t�s1ðtÞ

t�s0

rTðsÞS1rðsÞds

�Z t�s0


�Z t�s2ðtÞ

t�sM

rTðsÞS2rðsÞds ð17Þ

For arbitrary matrices M i;N i ði ¼ 1; 2; . . . 6Þ with com-patible dimensions, we have


½xTðtÞM1þxTðt�a0s1ðtÞÞM2�

xðtÞ�xðt�a0s1ðtÞÞ�Z t

t�a0s1ðtÞgðsÞds�

Z t

t�a0s1ðtÞrðsÞdwðsÞ

" #¼ 0 ð18Þ

½xTðt�a0s1ðtÞÞM3þxTðt� s1ðtÞÞM4�

xðt�a0s1ðtÞÞ�xðt� s1ðtÞÞ�Z t�a0s1ðtÞ

t�s1ðtÞgðsÞds�

Z t�a0s1ðtÞ

t�s1ðtÞrðsÞdwðsÞ

" #¼ 0

ð19Þ

½xTðt� s1ðtÞÞM5þxTðt� s0ÞM6�

xðt� s1ðtÞÞ�xðt� s0Þ�Z t�s1ðtÞ

t�s0

gðsÞds�Z t�s1ðtÞ

t�s0

rðsÞdwðsÞ� �

¼ 0 ð20Þ

½xTðt� s0ÞN1þxTðt� s2ðtÞÞN2�

xðt� s0Þ�xðt� s2ðtÞÞ�Z t�s0

t�s2ðtÞgðsÞds�

Z t�s0


" #¼ 0 ð21Þ

½xTðt� s2ðtÞÞN3þxTðt� sM ÞN4�

xðt� s2ðtÞÞ�xðt� sM Þ�Z t�s2ðtÞ

t�sM

gðsÞds�Z t�s2ðtÞ

t�sM

rðsÞdwðsÞ� �

¼ 0 ð22Þ

½xTðtÞN5þgTðtÞN6�½�AxðtÞþW0f ðxðtÞÞþa0W1f ðxðt�s1ðtÞÞÞþð1�a0ÞW1f ðxðt� s2ðtÞÞÞþðaðtÞ�a0ÞðW1f ðxðt� s1ðtÞÞÞ�W1f ðxðt� s2ðtÞÞÞÞ�gðtÞ� ¼ 0 ð23Þ

For formulas (18)–(22), we further have

�2fTðtÞMZ t

t�a0s1ðtÞrðsÞdwðsÞ6 fTðtÞMS�1

1 MTfðtÞþRT1 S1R1

ð24Þ

�2fTðtÞ �M

Z t�a0

t�s1ðtÞs1ðtÞrðsÞdwðsÞ6 fTðtÞ �MS�1

1�MTfðtÞþRT

2 S1R2

ð25Þ

�2fTðtÞ ~M

Z t�s1ðtÞ

t�s0

rðsÞdwðsÞ6 fTðtÞ ~MS�11

~MTfðtÞþRT3 S1R3

ð26Þ

�2fTðtÞNZ t�s0

t�s2ðtÞrðsÞdwðsÞ 6 fTðtÞNS�1

2 NTfðtÞ þ RT4 S2R4

ð27Þ

�2fTðtÞ �NZ t�s2ðtÞ

t�sM

rðsÞdwðsÞ 6 fTðtÞ �NS�12

�NTfðtÞ þ RT5 S2R5

ð28Þ

where fTðtÞ ¼ ½xTðtÞxTðt � a0s1ðtÞÞxTðt � s1ðtÞÞxTðt � s0ÞxTðt � s2ðtÞÞxTðt � sMÞgTðtÞ

R tt�a0s1ðtÞ g

TðsÞdsR t�a0

t�s1ðtÞ s1ðtÞgT

ðsÞdsR t�s1ðtÞ

t�s0gTðsÞds

R t�s0

t�s2ðtÞ gTðsÞds

R t�s2ðtÞt�sM

gTðsÞdsf TðxðtÞÞf Tðxðt � s1ðtÞÞÞf Tðxðt � s2ðtÞÞÞ�, R1 ¼

R tt�a0s1ðtÞ rðsÞdwðsÞ,

R2 ¼R t�a0s1ðtÞ

t�s1ðtÞ rðsÞdwðsÞ, R3 ¼R t�s1ðtÞ

t�s0rðsÞdwðsÞ, R4 ¼R t�s0

t�s2ðtÞ rðsÞdwðsÞ, R5 ¼R t�s2ðtÞ

t�sMrðsÞdwðsÞ.

From (9), for any matrices K i ¼ diagðki1; ki2; . . . ;kinÞP 0; i ¼ 1; 2; 3, it is easy to obtain2 3
�Xn
j¼1

k1j

xðtÞ

f ðxðtÞÞ

" #T lþj l�j ejeTj � lþj þl�j

2ejeT

j

� lþj þl�j2

ejeTj ejeT

j

64 75

�xðtÞ

f ðxðtÞÞ

" #�X2

i¼1

Xn

j¼1

kðiþ1Þjxðt � siðtÞÞ

f ðxðt � siðtÞÞÞ

" #T

�lþj l�j ejeT

j � lþj þl�j2

ejeTj

� lþj þl�j2

ejeTj ejeT

j

264

375 xðt � siðtÞÞ


" #

¼xðtÞ

f ðxðtÞÞ

" #T �K1L1 K1L2

K1L2 �K1

" #xðtÞ

f ðxðtÞÞ

" #

þX2

i¼1

xðt � siðtÞÞ


" #T �Kiþ1L1 Kiþ1L2

Kiþ1L2 �Kiþ1

" #

�xðt � siðtÞÞ


" #P 0

ð29Þ

By Remark 1, it is easy to know

EfrTðtÞðP þ s0S1 þ ðsM � s0ÞS2ÞrðtÞg¼ Ef½CxðtÞ þ a0Dxðt � s1ðtÞÞ þ ð1� a0Þ�Dxðt � s2ðtÞÞ�T �P½CxðtÞ þ a0Dxðt � s1ðtÞÞþ ð1� a0ÞDxðt � s2ðtÞÞ� þ 2ðaðtÞ � a0Þ½CxðtÞþ a0Dxðt � s1ðtÞÞ þ ð1� a0ÞDxðt � s2ðtÞÞ�T

� �P½ðDxðt � s1ðtÞÞ �Dxðt � s2ðtÞÞÞ� þ ðaðtÞ � a0Þ2

� ½Dxðt � s1ðtÞÞ �Dxðt � s2ðtÞÞ�T �P½Dxðt � s1ðtÞÞ� �Dxðt � s2ðtÞÞ�g ¼ ½CxðtÞ þ a0Dxðt � s1ðtÞÞþ ð1� a0ÞDxðt � s2ðtÞÞ�T �P½CxðtÞ þ a0Dxðt � s1ðtÞÞþ ð1� a0ÞDxðt � s2ðtÞÞ� þ a0ð1� a0Þ½Dxðt � s1ðtÞÞ�Dxðt � s2ðtÞÞ�T �P½Dxðt � s1ðtÞÞ �Dxðt � s2ðtÞÞ� ð30Þ

Since

E

Z t

t�a0s1ðtÞrTðsÞdwðsÞS1

Z t

t�a0s1ðtÞrðsÞdwðsÞ

( )

¼ E

Z t

t�a0s1ðtÞrTðsÞS1rðsÞds

( )ð31Þ

E

Z t�a0s1ðtÞ

t�s1ðtÞrTðsÞdwðsÞS1

Z t�a0s1ðtÞ


( )

¼ E

Z t�a0s1ðtÞ


( )ð32Þ


E

Z t�s1ðtÞ

t�s0

rTðsÞdwðsÞS1

Z t�s1ðtÞ

t�s0

rðsÞdwðsÞ�

¼ E

Z t�s1ðtÞ

t�s0

rTðsÞS1rðsÞds�

ð33Þ

E

Z t�s0

t�s2ðtÞrTðsÞdwðsÞS2

Z t�s0


( )

¼ E

Z t�s0


( )ð34Þ

E

Z t�s2ðtÞ

t�sM

rTðsÞdwðsÞS2

Z t�s2ðtÞ

t�sM

rðsÞdwðsÞ�

¼ E

Z t�s2ðtÞ

t�sM

rTðsÞS2rðsÞds�

ð35Þ

Substituting (14)–(17) into (13), adding (18)–(23) and (29)to (13), and taking expectation on both sides of (13), thenusing (24)–(28) and (30)–(35), we can get

EdV ðxt; tÞ ¼ ELV ðxt; tÞ 6 fTðtÞWfðtÞ ð36Þwhere W¼W1þMS�1

1 MTþ �MS�11

�MTþ ~MS�11

~MTþNS�12 NT

þ �NS�12

�NTþ �A�P�1 �AT. By the Schur complement, it is easyto derive that (12) is equivalent to W<0. Letk0¼minfkminð�WÞg, then, by the generalized Ito formula[17], we have

EV ðxðtÞ; tÞ�EV 0ðxð0Þ;0Þ¼ E

Z t

0

LV ðxðsÞ;sÞds6�k0E

Z t

0

jxðsÞj2 ds

Moreover,

E

Z t

0

jxðsÞj2 ds 61

k0

EV 0ðxð0Þ; 0Þ; t P 0

It indicates that system (7) is globally asymptotically stablein the mean-square sense. h

Remark 2. In Refs. [7,8], when l P 1;Q will no longer behelpful to improve the stability condition since �ð1� lÞQis nonnegative definite. However, by Theorem 1 in thispaper, when l1 P 1, if a0l1 < 1 is satisfied, then�ð1� a0l1ÞQ1 is still negative definite. Therefore, the con-straint on l1 < 1 is eliminated.

Next, we will discuss the stability of SNN with param-eter uncertainties. Based on the results obtained in Theo-rem 1, the following stability criterion can be derived easily,which is robust for all admissible uncertainties.

Theorem 2. For given scalars s0 P 0; sM > 0; l1; 0 < a0 < 1satisfying a0l1 < 1, the SNN described by (7) is robustly,

globally, asymptotically stable in the mean-square sense for

all admissible uncertainties, if there exist positive matrices

P > 0;Qj > 0 ðj ¼ 1; 2; 3Þ;R1 > 0;R2 > 0;S1 > 0;S2 > 0,

positive diagonal matrices K j > 0 ðj ¼ 1; 2; 3Þ and realmatrices M i;N i ði ¼ 1; 2; . . . 6Þ of appropriate dimensions,

positive scalar c, such that the following LMI holds:

�W1 M �M ~M N �N �A R

� �S1 0 0 0 0 0 0

� � �S1 0 0 0 0 0

� � � �S1 0 0 0 0

� � � � �S2 0 0 0

� � � � � �S2 0 0

� � � � � � ��P 0

� � � � � � � �cI

266666666666666664

377777777777777775

< 0 ð37Þ

where �W1¼W1þdiag�

cHT1 H1;0;...0;|fflffl{zfflffl}

11

cHT2 H 2;cHT

3 H 3;cHT3 H 3

�;

q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þa2

0þð1�a0Þ2q

;RT¼"qETNT

5 0 . . . 0|fflffl{zfflffl}5

qETNT6 0 . . . 0|fflffl{zfflffl}

8

#;

M; �M ; ~M;N ; �N ; �A; �P are defined as in Theorem 1.

Proof. Replace A;W0;W 1 in the LMI (12) withAþ DA;W0 þ DW 0;W1 þ DW 1, respectively, then theLMI (12) can be rewritten as

NþHP� þ � TPTHT < 0 ð38Þ

where H ¼ HT1 0 . . . 0|fflfflfflfflffl{zfflfflfflfflffl}

5

HT2 0 . . . 0|fflfflfflfflffl{zfflfflfflfflffl}

14

24

35T

, H1 ¼

"N 5E 0 . . . 0|fflfflffl{zfflfflffl}

11

N 5Ea0N 5Eð1 � a0ÞN 5E 0 . . . 0|fflfflffl{zfflfflffl}6

#, H2 ¼

"N 6E 0 . . . 0|fflfflffl{zfflfflffl}

11

N 6Ea0N 6Eð1 � a0ÞN 6E 0 . . . 0|fflfflffl{zfflfflffl}6

#and P ¼

diagðFðtÞ;FðtÞ;...;FðtÞÞ;�¼diag �H 1;0;...0;|ffl{zffl}11

H 2;H 3;H 30;...;0|ffl{zffl}6

0@

1A.

It follows from Lemma 2 that the matrix inequality (38)is equivalent to the following inequality.

Nþ c�1HHT þ c� T� < 0 ð39Þ

By the Schur complement, (37) is equivalent to (39) for ascalar c > 0. Then, similar to the proof of Theorem 1, theresults of Theorem 2 can be obtained. h

Remark 3. As a special case, when a0 ¼ 1 (or a0 ¼ 0Þ, by set-tingQ1 ¼ Q2 ¼ R2 ¼ S2 ¼ 0; sM ¼ s0 (or Q1 ¼ R1 ¼S1 ¼ 0Þin the Lyapunov–Krasovskii functional of Theorem 2, therobust stability criteria can be obtained easily and the corre-sponding proof is similar to Theorem 2, which are omitted.

4. Numerical examples

Example 1. Consider the uncertain SNN (7) with param-eters as follows (Example in Ref. [6]):

Table 1Allowable upper bound of sM for various l1.

Methods l1 ¼ 0:97 l1 ¼ 1 l1 ¼ 1:5 l1 ¼ 2 Unknown

[6] – – – – 0.419[7] 0.785 0.779 0.779 0.779 0.779[8] 0.771 0.746 0.746 0.746 0.746Theorem 2 a0 ¼ 0:2 1.294 1.294 1.292 1.291 1.279

a0 ¼ 0:4 1.338 1.337 1.324 1.299 1.281s0 ¼ 0:6 a0 ¼ 0:6 1.430 1.426 1.303 1.292 1.292

a0 ¼ 0:8 1.615 1.579 1.323 1.323 1.323

Table 2Allowable upper bound of sM for s0 ¼ 0:4.

Results l1 ¼ 0:2 l1 ¼ 0:6 l1 ¼ 1 l1 ¼ 1:5 l1 ¼ 2 l1 ¼ 2:5

a0 ¼ 0:2 0.972 0.972 0.971 0.970 0.968 0.967a0 ¼ 0:5 1.092 1.083 1.071 1.044 1.024 1.024a0 ¼ 0:8 1.545 1.490 1.342 1.242 1.242 1.242a0 ¼ 0:99 5.523 5.181 3.529 3.243 3.243 3.243


A¼4 0

0 5

� �; W0¼

0:4 �0:7

0:1 0

� �; W1¼

�0:2 0:6

0:5 �0:1

� �;

C ¼0:5 0

0 0:5

� �; D¼

0 �0:5

�0:5 0

� �; E¼ ½0:1 �0:1�T;

H1¼ ½0:2 0:3�; H2¼ ½0:2 �0:3�; H3¼ ½�0:2 �0:3�;L1¼ 0; L2¼ 0:25I

by Assumption 3, L1 = 0, L2 = 0.25I equivalent toL ¼ 0:5I in Ref. [6].

For various l1, the computed upper bound sM , whichguarantee the robust stability of system (7), are listed inTable 1. From Table 1, when the information of the delay-probability distribution is considered, for various a0 theallowable upper bound sM is larger than those in Refs. [6–8], where only the variation range of the delay is consid-ered. In addition, when l1 P 1, the stability criteria fail inRef. [6] and become derivative independent in Refs. [7,8].However, in this paper, the constraint l1 < 1 is eliminatedfor a0l1 < 1. Therefore, Theorem 2 in this paper is lessconservative than those in Refs. [6–8].

Example 2. Consider the uncertain SNN (7) with parame-ters as follows:

A ¼7 0

0 6

� �; W0 ¼

0:2 �4

0:1 0:3

� �; W 1 ¼

0:4 0:2

0:1 0:7

� �;

C ¼0:3 0

0 0:3

� �; D ¼

0:5 �0:1

�0:5 0

� �; E ¼ ½0:1 0:1�T;

H1 ¼ H2 ¼ H3 ¼ ½1 1�:

Take the activation function as: f1ðx1Þ ¼ tanhð�0:2x1Þ;f2ðx2Þ ¼ tanhðx2Þ. It is obvious that �0:2 6d

dx1tanhð�0:2x1Þ < 0; 0 < d

dx2tanhðx2Þ 6 1, so L1 ¼ diagð0;

0Þ;L2 ¼ diagð�0:1; 0:5Þ.For s0 ¼ 0:4, various l1 and delay probability distribu-

tion a0, the computed upper bound sM , which guaranteethe robust stability of system (7), are listed in Table 2.

From Table 2, this paper overcomes the constraint l1 < 1for a0l1 < 1.

5. Conclusions

The problem of robust stability for uncertain SNNs withprobability-distribution-dependent time-varying delay hasbeen addressed in this paper. Some new stability criteria havebeen proposed to guarantee the robust global asymptoticalstability of the SNNs. Probability distribution of time vary-ing delay is introduced into the stability criteria, and the newmethod eliminates the constraint that the derivative of thedelay must be smaller than 1. Numerical examples showthe effectiveness and less conservatism of the method.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (60534010, 60572070,60774048, 60804006 and 60728307), the Program for Che-ung Kong Scholars and Innovative Research Groups ofChina (60521003), the Research Fund for the DoctoralProgram of China Higher Education (20070145015) andthe National High Technology Research and DevelopmentProgram of China (2006AA04Z183).

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Delay-probability-distribution-dependent robust stability analysis for stochastic neural networks with time-varying delay