New delay-dependent stability criteria for uncertain stochastic neural networks with discrete interval and distributed delays

Neurocomputing 101 (2013) 1–9

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

New delay-dependent stability criteria for uncertain stochastic neural networks with discrete interval and distributed delays$ Huabin Chen Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, PR China

a r t i c l e i n f o

abstract

Article history: Received 31 December 2011 Received in revised form 23 March 2012 Accepted 12 June 2012 Communicated by H. Jiang Available online 4 September 2012

This paper studies the globally robustly asymptotical stability in mean square of uncertain stochastic neural networks with discrete interval and distributed time-varying delays. By constructing an augmented Lyapunov–Krasovskii functional, some delay-dependent criteria for the globally robustly asymptotical stability of such systems are formulated in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to illustrate the effectiveness of the obtained results. & 2012 Elsevier B.V. All rights reserved.

Keywords: Uncertain stochastic neural networks Lyapunov–Krasovskii functional Linear matrix inequalities (LMIs) Interval time-varying delay Distributed delay

1. Introduction Over the past decades, neural networks have been widely considered by many authors due to their wider applications in a variety of areas, such as signal processing, pattern recognition, static image processing, associative memory, combinatorial optimization and many other fields. These applications are largely dependent upon the stability of the equilibrium of neural networks, that is, stability is of much importance in dynamical properties about neural networks when neural networks are designed. In practice, time-delay is often encountered in various engineering, biological and economic systems. For the finite speed of information processing, the existence of time-delay can usually bring oscillation, divergence, or even instability of neural networks. As we know, the previous stability criteria for delayed neural networks can be classified into two types: delay-independent criteria [7–9,29,31,32,52] and delay-dependent criteria [6,14,15,19,21,22,24–26,35–37,39,40,42–44]. Generally speaking, delay-dependent stability criteria are usually less conservative than delay-independent ones especially when the size of the delay is small. Thus, obtaining the delay-dependent stability criteria is not only of theoretical importance, but also of practical value. However, most of delayed neural network models proposed and discussed above are deterministic and are only applicable to the $ This work was supported by the National Natural Science Foundations of China under Grant No. 11126278, the Natural Science Foundation of Jiangxi Province in China under Grant No. 20114BAB211001. E-mail address: [email protected]

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.06.010

case when no perturbation is produced by stochastic phenomenon. In practice, stochastic neural networks with time-delay could well reflect the reality. Recently, there were many valuable results on stability analysis for stochastic delayed neural networks [4,5,10,11,16–18,20,23,27,29,33,34,38,41,45,47–52]. For example, in [10,11,22,34,48], the stability analysis for uncertain stochastic neural networks with discrete and distributed delays has been discussed. When the exponential stability problem for the stochastic neural networks with time-varying delays is considered, one always assumes that the derivative of time-varying delay is less than one, see [10,11]. As a result, the obtained results in [10,11] are invalid when the derivative of time-varying delay equals to or is greater than one. In order to overcome this shortage in [10,11], the mean-square robustly stability for stochastic Hopfield neural networks with time-varying delay and distributed delay has been investigated in [18,20,23,27,45] by utilizing the free-weighting matrix technique. On the other hand, the foregoing stability criteria for stochastic neural networks with time-varying delay are only applied into the case when the lower bound of the delay is zero. In practice, there exists a special type of time delay in practical engineering systems, i.e., interval time-varying delay, hm rhðtÞ r hM and hm is not restricted to be zero, which commonly exists in networked control systems, and the delay-dependent stability of neural networks with interval time-varying delay was widely studied in [6,36,46] and the references therein. The delay-dependent stability criteria for uncertain stochastic neural networks with discrete interval and distributed delays were obtained in [2,3,30] by constructing a modified Lyapunov–Krasovskii functional and using the

2

H. Chen / Neurocomputing 101 (2013) 1–9

free-weighting matrix technique, but the bounding technique is frequently employed to deal with the discrete delay items. It is worth mentioning that the excessive use of the bounding technique can usually bring much conservatism while using the free-weighting matrices approach. Although the asymptotical stability analysis for uncertain stochastic neural networks with discrete interval and distributed delays is also considered in [22], some non-negative items are ignored. Thus, the obtained results in [2,3,22,30] are much more conservative. Thus, how to obtain the less conservative stability criteria for uncertain stochastic neural networks with discrete interval and distributed delays still remains a challenging problem, which motivates the present study. Inspired by the statements above, we consider the problem of the globally robustly asymptotical stability in mean square for uncertain stochastic neural networks with discrete interval and distributed delays. By constructing an augmented Lyapunov– Krasovskii functional, the LMI-based sufficient conditions ensuring the globally robustly asymptotical stability in mean square for such systems can be derived by using the free-weighting matrices, which are less conservative than some existing reports. And what is more, in contrast to the results in [7–9,29,31,32,52], the proposed LMI-based ones are computationally efficient as they can be solved numerically by employing the LMI toolbox in Matlab. Finally, two illustrative examples are provided to show the effectiveness of the given results in this paper. Notation. In this paper, Rn and Rmn denote the n-dimensional Euclidean space and the set of all m  n real matrices, respectively. E stands for the identity matrix with appropriate dimensions; for two matrices X and Y, the notation X 4 Y (respectively, X Z Y) means that the XY denotes positive definite (respectively, positive semi-definite); J  J denotes the Euclidean vector norm. ‘T’ denotes the transpose for a matrix or a vector. ðO,I,fIgt Z 0 ,PÞ is a probability space with a filtration fIgt Z 0 satisfying the usual conditions (i.e. the filtration contain all P-null sets and is right continuous). Denote by L2I0 ð½r,0; Rn Þ the family of all I0 -measurable Cð½-r,0; Rn Þ-valued random variables x ¼ fxðyÞ : r r y r 0g such that supy A ½r,0 EJxðyÞJ2 o þ 1, where EðÞ stands for the mathematical expectation. Matrices, it is not explicitly stated, are assumed to have compatible dimensions.

2. Problem formulation Consider the following uncertain stochastic neural networks with discrete interval and distributed time-varying delays:  dxðtÞ ¼ AðtÞxðtÞ þW 0 ðtÞf ðxðtÞÞ þW 1 ðtÞf ðxðthðtÞÞÞ   Z t þ W 2 ðtÞ f ðxðsÞÞ ds dt þ CðtÞxðtÞ þ DðtÞxðthðtÞÞ ttðtÞ

þ B0 ðtÞf ðxðtÞÞ þ B1 ðtÞf ðxðthðtÞÞÞþ B2 ðtÞ

Z

t

 f ðxðsÞÞ ds dwðtÞ,

y A ½t ,0, t ¼ maxfhM , tg,

ð2:2Þ

n

t Z0, where xðtÞ A R is the state vector associated with the neurons; j A Cð½t ,0; Rn Þ is the initial function. f ðxðtÞÞ ¼ ½f 1 ðx1 ðtÞÞ,f 2 ðx2 ðtÞÞ, . . . ,f n ðxn ðtÞÞ denotes the neuron activation function. And wðtÞ ¼ ½w1 ðtÞ,w2 ðtÞ, . . . ,wn ðtÞT A Rn is an n-dimensional Brownian motion defined on a complete probability space ðO,I, fIt gt Z 0 ,PÞ. The delays h(t), tðtÞ satisfy the following assumptions: 0 r hm rhðtÞ rhM ,

_ r m, hðtÞ

and

1

1

1

8 > < W 2 ðtÞ ¼ W 2 þ DW 2 ðtÞ, CðtÞ ¼ C þ DCðtÞ, > : DðtÞ ¼ D þ DDðtÞ,

8 > < B0 ðtÞ ¼ B0 þ DB0 ðtÞ, B1 ðtÞ ¼ B1 þ DB1 ðtÞ, > : B ðtÞ ¼ B þ DB ðtÞ, 2

2

ð2:4Þ

2

where A, W 0 , W 1 , W 2 , C, D, B0 , B1 and B2 are known real constant matrices, and DAðtÞ, DW 0 ðtÞ, DW 1 ðtÞ, DW 2 ðtÞ, DCðtÞ, DDðtÞ, DB0 ðtÞ, DB1 ðtÞ, DB2 ðtÞ are unknown matrices representing time-varying parameter uncertainties in system model. We assume that the uncertainties are norm-bounded and can be described as ½DAðtÞ DW 0 ðtÞ DW 1 ðtÞ DW 2 ðtÞ DCðtÞ DDðtÞ DB0 ðtÞ DB1 ðtÞ DB2 ðtÞ ¼ MFðtÞ½N1 N2 N3 N4 N 5 N 6 N 7 N 8 N9 ,

ð2:5Þ

where F(t) are unknown real and possibly time-varying matrix for any given t and satisfy F T ðtÞFðtÞ rI,

ð2:6Þ

and M, N1 , N2 , N3 , N 4 , N 5 , N 6 , N 7 , N 8 , and N9 are some known real matrices with appropriate dimension. It is assumed that the elements of F(t) are Lebesgue measurable. When FðtÞ ¼ 0, systems (2.1) have the following nominal case:   Z t dxðtÞ ¼ AxðtÞ þW 0 f ðxðtÞÞ þ W 1 f ðxðthðtÞÞ þW 2 f ðxðsÞÞ ds dt ttðtÞ

 þ CxðtÞ þDxðthðtÞÞ þB0 f ðxðtÞÞ þ B1 f ðxðthðtÞÞÞ Z þB2

t

 f ðxðsÞÞ ds dwðtÞ,

t Z0:

ð2:7Þ

ttðtÞ

To obtain the main results, we need the following assumption: (H) The activation function f(x) is bounded and satisfy the following Lipschitz condition: 9f ðu1 Þf ðu2 Þ9 r K9u1 u2 9,

8u1 ,u2 A Rn ,

and f ð0Þ ¼ 0, where K ¼ diagfk1 ,k2 , . . . ,kn g is a positive definite diagonal matrix. Remark 1. Obviously, under the condition (H), the stochastic neural networks with discrete interval and distributed delays (2.7) have one trivial solution when the initial value j ¼ 0, we can refer to [28]. Lemma 2.1 (Gu et al. [12,13]). For any constant matrix S A Rnn , S ¼ ST 4 0, a scalar g 40 and a vector function o : ½0, g-Rn such that the integrations are well defined, the following inequality holds: Z g T Z g  Z g oðsÞ ds S oðsÞ ds r g oT ðsÞSoðsÞ ds: 0

0

0

ttðtÞ

ð2:1Þ xðyÞ ¼ jðyÞ,

uncertainties, that is, 8 > < AðtÞ ¼ Aþ DAðtÞ, W 0 ðtÞ ¼ W 0 þ DW 0 ðtÞ, > : W ðtÞ ¼ W þ DW ðtÞ,

0 r tðtÞ r t,

ð2:3Þ

where hm , hM , m and t are constants; AðtÞ, W 0 ðtÞ, W 1 ðtÞ, W 2 ðtÞ, CðtÞ, DðtÞ, B0 ðtÞ, B1 ðtÞ and B2 ðtÞ are matrix functions with time-varying

Lemma 2.2 (Boyd et al. [1]). Given matrices Q ¼ Q T , M,N with appropriate dimensions, then Q þ MFðtÞN þ NT F T ðtÞM T o 0, for all F(t) satisfying F T ðtÞFðtÞ r I, if and only if there exists e 40 such that Q þ e1 MT M þ eN T N o 0: Lemma 2.3 (Boyd et al. [1]). (Schur complement) For a given matrix " # S11 S12 S¼ T S12 S22

H. Chen / Neurocomputing 101 (2013) 1–9

3

with S11 ¼ ST11 , S22 ¼ ST22 , then the following conditions are equivalent:

Ok37 ¼ vk ½Q 112 þ Q 212 þ Q 312 , Ok38 ¼ Ok39 ¼ 0,

(1) S o0; T (2) S22 o0,S11 S12 S1 22 S12 o 0; T 1 (3) S11 o0,S22 S12 S11 S12 o 0.

Ok44 ¼ vk ½Q 311 , Ok45 ¼ Ok46 ¼ Ok47 ¼ 0,

Ok310 ¼ vk ½P 24 þ P34 , Ok311 ¼ vk ½P25 þ P 35 , Ok48 ¼ vk ½Q 312 , Ok49 ¼ 0, Ok410 ¼ vk ½P 34 , Ok411 ¼ vk ½P 35 , Ok55 ¼ vk ½Q 9 þ QW 0 þ W T0 Q T þ Q 122 þ h2m Q 7 þ h2Mm Q 8 þ t2 Q 4 , Ok56 ¼ vk ½QW 1 , Ok57 ¼ Ok58 ¼ 0, Ok59 ¼ vk ½QW 2 , Ok510 ¼ vk ½W T0 P 14 þP 44 , Ok511 ¼ vk ½W T0 P 15 þ P45 ,

3. Main results In this section, the sufficient conditions ensuring the globally asymptotical stability for the stochastic neural networks with discrete interval and distributed delays (2.7) are firstly obtained by constructing an augmented Lyapunov–Krasovskii functional and using the free weighting matrices. Before the results are given, the notations are needed as follows: Z t yðtÞ ¼ AxðtÞ þ W 0 f ðxðtÞÞ þW 1 f ðxðthðtÞÞÞ þW 2 f ðxðsÞÞ ds, ttðtÞ

and

s2 ¼

hMm ¼ hM hm .

Theorem 3.1. Let a A ð0,1Þ, for the delays h(t), tðtÞ satisfying the condition (2.3), and the condition (H) is satisfied, if there exist matrices: P11 4 0, P22 4 0, P33 4 0, P44 4 0, P55 4 0, P12 , P13 , P14 , i

P 15 , P 23 , P24 , P 25 , P 34 , P35 , P 45 , Q i ¼ ½ðQQi 11ÞT 12

Q i12  40 Q i22

Ok77 ¼ vk ½Q 122 þ Q 222 þ Q 322 , Ok78 ¼ Ok79 ¼ 0, Ok710 ¼ vk ½P 44 þ PT45 , Ok711 ¼ vk ½P45 þ P 55 , Ok88 ¼ vk ½Q 322 , Ok89 ¼ 0, Ok810 ¼ vk ½P T45 , Ok811 ¼ vk ½P 55 , Ok99 ¼ vk ½Q 4 , Ok910 ¼ vk ½W T2 P 14 , Ok911 ¼ vk ½W T2 P 15 , Ok1010 ¼ vk ½Q 7 , Ok1011 ¼ 0, Ok1111 ¼ vk ½Q 8 , P24 þ P 25 P 25 PT12 W 2 0 0T ,

ttðtÞ

2 2 ðhM hm Þ=2,

Ok610 ¼ vk ½W T1 P 14 , Ok611 ¼ vk ½W T1 P15 ,

^ ¼ ½P T A þ P22 H1 H2 P 22 þP 23 P23 P T W 0 þ P24 P T W 1 H 12 12 12

zðtÞ ¼ CxðtÞ þ DxðthðtÞÞ þ B0 f ðxðtÞÞ þ B1 f ðxðthðtÞÞÞ Z t þB2 f ðxðsÞÞ ds, 2 s1 ¼ hm =2,

Ok66 ¼ vk ½ð1mÞQ 222 Q 10 , Ok67 ¼ Ok68 ¼ Ok69 ¼ 0,

ði ¼ 1,2,3Þ,

L~ ¼ ½L1 L2 0 0 0 0 0 0 0 0 0T , ~ ¼ ½M 1 M2 0 0 0 0 0 0 0 0 0T , M ~J ¼ ½J J 0 0 0 0 0 0 0 0 0T , 1

Q l 4 0 ðl ¼ 4,5, . . . ,8Þ, Rk 40, Z j ðk,j ¼ 1,2,3,4Þ, Q ¼ diagfq1 ,q2 , . . . , 9

diagfq91 ,q92 ,

10

10 10 qn g 40, Q ¼ Q ¼ diagfq10 1 ,q2 , . . . ,qn g 4 0 T T T and some appropriately dimensional matrices: L ¼ ½L1 L2  , M ¼ ½M T1 M T2 T , J ¼ ½J T1 J T2 T , H ¼ ½HT1 HT2 T and I ¼ ½IT1 IT2 T , such that

2

P11

6 n 6 6 P ¼ ðPij Þ55 ¼ 6 6 n 6 n 4 n

. . . ,q9n g 40,

3

P12

P 13

P14

P 15

P22

P 23

P24

n

P 33

P34

n

n

P44

n

n

n

P 25 7 7 7 P 35 7 7 4 0, P 45 7 5 P 55

and the following linear matrix inequalities (LMIs) hold: " # " # O1 ðO1 Þ12 O2 ðO2 Þ12 2 o 0, o 0, X1 ¼ X ¼ ððO1 Þ12 ÞT ðO1 Þ22 ððO2 Þ12 ÞT ðO2 Þ22 " # O3 ðO3 Þ12 X3 ¼ o 0, ð3:1Þ ððO3 Þ12 ÞT ðO3 Þ22 where

Ok ¼ ðOkij Þ1111 , k ¼ 1,2,3, Ok11 ¼ vk ½P 11 AAT P 11 þ PT12 þP 12 þ Q 111 þ hm Q 5 þ hMm Q 6 T

þK Q 9 K þLT1 þ L1 þ hm HT1 þ hm H1 þhMm IT1 þ hMm I1 , Ok12 ¼ vk ½L2 MT1 þ JT1 þ hm H2 þ ðhM hm ÞI2 , Ok13 ¼ vk ½LT2 þ MT1 P 12 þ P13 ,

Ok14 ¼ vk ½K T4 M1 þ hm J T4 þ lHT4 P 13 , Ok15 ¼ vk ½P11 W 0 þ P14 AT Q T þ Q 112 , Ok16 ¼ vk ½P11 W 1 , Ok17 ¼ vk ½P 14 þ P15 , Ok18 ¼ vk ½P15 , Ok19 ¼ vk ½P11 W 2 , Ok110 ¼ vk ½AT P 14 þ P24 , Ok111 ¼ vk ½AT P 15 þ P25 , Ok22 Ok23 Ok26 Ok33

¼ ¼ ¼ ¼

vk ½ð1 ÞQ 211 þ LQ 13 LM T2 M T2 þ JT2 þ J 2 þK T Q 10 K, vk ½L2 þ M 2 , Ok24 ¼ vk ½J2 , Ok25 ¼ 0, vk ½ð1uÞQ 212 , Ok27 ¼ Ok28 ¼ Ok29 ¼ Ok210 ¼ Ok211 ¼ 0, vk ½Q 111 þQ 211 þ Q 311 , Ok34 ¼ Ok35 ¼ Ok36 ¼ 0,

2

~ ¼ ½H1 H2 0 0 0 0 0 0 0 0 0T , H I~ ¼ ½I1 I2 0 0 0 0 0 0 0 0 0T , I^ ¼ ½P T13 A þ P T23 I1 I2 PT23 þ P 33 P 33 PT13 W 0 þP 34 PT13 W 1 P34 þ P 35 P 35 PT13 W 2 0 0T , A~ ¼ ½A 0 0 0 W 0 W 1 0 0 W 2 0 0, C~ ¼ ½C D 0 0 B0 B1 0 0 B2 0 0, T

T

T

^ hm Ls ~ 1H ~ v1 C~ P 11 v1 C~ LQ hm A~ R1 ðO1 Þ12 ¼ ½hm H T T T s1 A~ R3 hm C~ Z 1 s1 C~ Z 3 v1 LT v1 M T v1 N T hm HT ,

~T ~T ~T ~ ~ ðO2 Þ12 ¼ ½hMm I^ hMm Ms 2 I v2 C P 11 v2 C LQ hMm A R2 T

T

T

s2 A~ R4 hMm C~ Z 2 s2 C~ Z 4 v2 LT v2 M T v2 N T s1 IT , ~ 2 I~ v2 C~ T P 11 v2 C~ T LQ hMm A~ T R2 ðO3 Þ12 ¼ ½hMm I^ hMm Ns ~ T R4 hMm C~ T Z 2 s2 C~ T Z 4 v2 LT v2 M T v2 N T s1 IT ,

s2 A 1 22

ðO Þ

¼ diagfhm Q 5 hm R1 s1 R3 v1 P 11 v1 LQ hm R1 s1 R3 hm Z 1 s1 Z 3 v1 Z 1 v1 Z 2 v1 Z 2 hm Z 3 ,

ðO2 Þ22 ¼ diagfhMm Q 6 hMm R2 s2 R4 v2 P 11 v2 LQ hMm R2 s2 R4 hMm Z 2 s2 Z 4 v2 Z 1 v2 Z 2 v2 Z 2 hMm Z 4 , ðO3 Þ22 ¼ diagfhMm Q 6 hMm R2 s2 R4 v3 P 11 v3 LQ hMm R2 s2 R4 hMm Z 2 s2 Z 4 v3 Z 1 v3 Z 2 v3 Z 2 hMm Z 4 , v1 ¼ a, v2 ¼ v3 ¼ 1a, and n means symmetric terms, then the stochastic neural networks with discrete interval and distributed delays (2.7) are globally stochastic asymptotically stable. Proof. Define an augmented Lyapunov–Krasovskii functional as follows: Vðt,xt Þ ¼ V 1 ðt,xt Þ þV 2 ðt,xt Þ þ V 3 ðt,xt Þ,

ð3:2Þ

where T

V 1 ðt,xt Þ ¼ x ðtÞPxðtÞ þ2

m

V 2 ðt,xt Þ ¼

Z

n X i¼1

t

Z

xi 0

jT ðsÞQ 1 jðsÞ ds þ

thm

Z

qi

thm

þ thM

f i ðsÞ ds, Z

thm

jT ðsÞQ 2 jðsÞ ds

thðtÞ

jT ðsÞQ 3 jðsÞ dsþ t

Z

0

t

Z

t tþy

T

f ðxðsÞÞQ 4 f ðxðsÞÞ ds

4

H. Chen / Neurocomputing 101 (2013) 1–9

Z

Z

0

t tþy

hm

þ hm

Z

t

Z

hm

Z

t

t

y ðsÞR1 yðsÞ ds dy þ

Z

tþy

V 3 ðt,xt Þ ¼

Z

0

Z

0

y

hm

Z

t

hM 0

Z

0

t

t

þ 2xT ðthM Þ½P33 

yT ðsÞR4 yðsÞ ds dl dy

Z

t

hm

Z

0

y

thm

f ðxðsÞÞ ds

thm

xðsÞ ds

ttðtÞ

þ 2f

T

ðxðtÞÞ½W T0 P 14 þP 44 

T

t

þ 2f ðxðtÞÞ½W T0 P 15 þP 45 

T

z ðsÞZ 4 zðsÞ ds dl dy tþl

T

þ 2f ðxðtÞÞ½W T0 P 12 þP T24 

and

T

"

Z

T

xðtÞ ¼ x ðtÞ

t

T

x ðsÞ ds

Z

Z

T

x ðsÞ ds thM

thm

T

T

thm

t

T

f ðxðsÞÞ ds

Z

thm

þ 2f ðxðtÞÞ½W T0 P 13 þP T34 

#T T

f ðxðsÞÞ ds

,

thM

thm

T

þ 2f ðxðthðtÞÞÞ½W T1 P 14 

T

jðtÞ ¼ ½x ðtÞf ðxðtÞÞ :

T

From Ito’s formula, the stochastic differential of Vðt,xt Þ is obtained as follows: " dVðt,xt Þ ¼ LVðt,xt Þdt þ 2 x ðtÞP 11 zðtÞ þ f ðxðtÞÞQzðtÞ t

T

x ðsÞ

þ Z

thm t

T dsP12 zðtÞ þ

T

f ðxðsÞÞ

þ thm

þ 2f ðxðthðtÞÞÞ½W T1 P 15  T

þ 2f ðxðthðtÞÞÞ½W T1 P 12 

T

T

Z

Z

thm

T

x ðsÞ

thM

Z

T dsP 14 zðtÞ þ

xðsÞ ds

thm

T

z ðsÞZ 3 zðsÞ ds dl dy

þ hM

f ðxðsÞÞ ds thm

þ 2f ðxðtÞÞ½QW 1 f ðxðthðtÞÞÞ Z t T f ðxðsÞÞ ds þ 2f ðxðtÞÞ½QW 2 

T

Z

xðsÞ ds

T

tþl

y

hm

thm

t

thM Z t

Z

xðsÞ ds thm

þ f ðxðtÞÞ½QW 0 þW T0 Q f ðxðtÞÞ

zT ðsÞZ 2 zðsÞ ds dy

0Z

Z

t

thM

Z Z

Z

thM

tþy

þ Z

þ 2xT ðthM Þ½PT23 

zT ðsÞZ 1 zðsÞ ds dy

Z

hm

0

y ðsÞR2 yðsÞ ds dy,

þ 2xT ðthM Þ½P35 

tþy

tþl

t

þ Z

T

tþy

hm

hM

t

y ðsÞR3 yðsÞ ds dl dy

þ Z

Z

T

y

Z

þ 2xT ðthM Þ½P34 

tþl

Z

hm

hm

hM

þ Z

Z

þ 2xT ðthm Þ½P22 þ PT23  þ 2xT ðthm Þ½P23 þ P33 

f ðxðsÞÞQ 8 f ðxðsÞÞ ds dy

T

þ hm

tþy

xT ðsÞQ 6 xðsÞ ds dy

T

tþy

hM

Z

0

t

f ðxðsÞÞQ 7 f ðxðsÞÞ ds dy

þ ðhM hm Þ Z

Z

T

tþy

hm

hm hM

Z

0

Z

xT ðsÞQ 5 xðsÞ ds dy þ

þ

thm

T

þ 2f ðxðthðtÞÞÞ½W T1 P 12 

T dsP 13 zðtÞ

f ðxðsÞÞ

thM

T dsP14 zðtÞ

t

f ðxðsÞÞ ds

thm

Z

thm

f ðxðsÞÞ ds

thM t

Z

xðsÞ ds

thm

Z Z

thm

xðsÞ ds

thM t

f ðxðsÞÞ ds

thm

Z

thm

f ðxðsÞÞ ds

thM t

Z

xðsÞ ds

thm

Z

thm

T

dwðtÞ,

T

þ 2f ðxðthm ÞÞ½P 45 þP 55 

ð3:3Þ T

þ 2f ðxðthm ÞÞ½P T24 þP T25 

where LVðt,xt Þ ¼ LV 1 ðt,xt Þ þ LV 2 ðt,xt Þ þ LV 3 ðt,xt Þ,

ð3:4Þ

T

þ 2f ðxðthm ÞÞ½P T34 þP T35 

T

LV 1 ðt,xt Þ ¼ x ðtÞ½P11 AA P T11 þP T12 þP 12 xðtÞ þ2xT ðtÞ½P12 þ P13 xðthm Þ þ2xT ðtÞ½P13 xðthM Þ þ2xT ðtÞ½P11 W 0 þ P14 AT Q T f ðxðtÞÞ Z t f ðxðsÞÞ þ2xT ðtÞP 11 W 1 f ðxðthðtÞÞÞþ 2xT ðtÞP 11 W 2 ttðtÞ T

T

T

þ2x ðtÞ½A P 14 þ P 24 

Z

t

T

T

þ2x ðtÞ½A P 15 þ P 25  T

T

þ2x ðtÞ½A P 12 þ P 22  T

þ2x ðtÞ½A P 13 þ P 23  T

thm

þ2 þ2

thm

þ2 þ2

thm

f ðxðsÞÞ ds thM

Z

t

Z

ttðtÞ

f ðxðsÞÞ ds

thm

t

ttðtÞ t

xðsÞ ds

Z

Z

ttðtÞ

xðsÞ ds

thM Z t

þ2x ðthm Þ½P 25 þ P35 

T

f ðxðsÞÞ ds

thM t

þ2x ðthm Þ½P 24 þ P34  T

T

þ 2f ðxðthm ÞÞ½P T25  þ 2f ðxðthm ÞÞ½P T35 

thm

T

ds

f ðxðsÞÞ ds

Z Z

T

þ 2f ðxðthm ÞÞ½P 55 

T

thm

Z

T

þ 2f ðxðthM ÞÞ½PT45 

Z

t

ttðtÞ

Z Z

xðsÞ ds

thM

þ 2f ðxðthm ÞÞ½P 44 þP T45 

# T

Z

Z

f ðxðsÞÞ ds

thm

Z

thm

f ðxðsÞÞ ds

thM t

Z

xðsÞ ds

thm

Z

thm

xðsÞ ds

thM

t

f ðxðsÞÞ ds thm thm

f ðxðsÞÞ ds

thM t

Z

xðsÞ ds

thm

Z

thm

thM

T

f ðxðsÞÞ ds½W T2 P14  T

f ðxðsÞÞ ds½W T2 P15  T

f ðxðsÞÞ ds½W T2 P12  T

f ðxðsÞÞ ds½W T2 P13  T

t

þ zT ðtÞP 11 zðtÞ þf ðxðtÞÞQzðtÞ,

xðsÞ ds Z Z Z Z

t

f ðxðsÞÞ ds thm thm

f ðxðsÞÞ ds thM t

xðsÞ ds thm thm

xðsÞ ds thM

ð3:5Þ

H. Chen / Neurocomputing 101 (2013) 1–9

Z

LV 2 ðt,xt Þ r jT ðtÞQ 1 jðtÞ þ jT ðthðtÞÞ½ð1mÞQ 2 jðthðtÞÞ

5

thðtÞ

yðsÞ ds



þ jT ðthM Þ½Q 3 jðthM Þ þ jT ðthm Þ½Q 1

thM



2

þf ðxðtÞÞ½t2 Q 4 þ hm Q 7 þ ðhM hm ÞQ 8 f ðxðtÞÞ Z t T f ðxðsÞÞQ 4 f ðxðsÞÞ dsþ yT ðtÞ½hm R1 þ ðhM hm ÞR2 t tt Z t T þs1 R3 þ s2 R4 yðtÞhm f ðxðsÞÞQ 7 f ðxðsÞÞ ds Z

thM

Z

t

thm

Z

t

x ðsÞQ 5 xðsÞ ds

thm

"

thm

Z

T

"

þ

2 2 hM hm

2 Z

# Z 4 zðtÞ t

thm

t

zT ðsÞZ 2 zðsÞ ds

Z

thM

t

Z r

ttðtÞ

Z

t

thm

thm

"Z

ð3:8Þ

ttðtÞ

t

r

thm

f ðxðsÞÞ ds thM

Q8

(Z

Z2

thðtÞ

"Z

#T

Z

Z

0

hm

Z

0

ð3:9Þ

ð3:10Þ

thM

From the condition (H), we have ( T f ðxðtÞÞQ 9 f ðxðtÞÞ r xT ðtÞKQ 9 KxðtÞ,

#

thm

zðsÞ dwðsÞ ,

Z2

"Z

t

t

tþy

zðsÞ dwðsÞ ,

T Z zðsÞ dwðsÞ Z 3

 zðsÞ dwðsÞ dy,

t

And from Newton–Leibniz formula [14,15], it yields  Z t a1 ðtÞ :¼ 2ZT ðtÞLT xðtÞxðthm Þ yðsÞ ds thm  Z t zðsÞ dwðsÞ ¼ 0, 

Z

hm

Z

tþy

hM

hm

Z

t

t

zðsÞ dwðsÞ rðhM hm ÞZT ðtÞIT Z 1 4 IZðtÞ

T Z zðsÞ dwðsÞ Z 4

tþy

t

 zðsÞ dwðsÞ dy:

thm

ð3:11Þ

"Z

#T

thm

zðsÞ dwðsÞ

E

Z2

"Z

thðtÞ

thm

#

thm

zðsÞ dwðsÞ ¼ E

thðtÞ

 T

Z

thm

a3 ðtÞ :¼ 2ZT ðtÞJ xðthðtÞÞxðthM Þ

zT ðsÞZ 1 zðsÞ ds, ð3:23Þ

"Z ð3:12Þ

#T

thðtÞ

zðsÞ dwðsÞ

E

Z2

"Z

#

thðtÞ

zðsÞ dwðsÞ ¼ E thM

Z

thðtÞ

zT ðsÞZ 1 zðsÞ ds,

thM

ð3:24Þ #

zðsÞ dwðsÞ ¼ 0, thðtÞ

thm

thðtÞ



yðsÞ ds

Z

thðtÞ

thM

thm

ð3:21Þ

tþy

On the other hand, from the Itˆo isometry in [28], we can obtain Z t T Z t  Z t E zðsÞ dwðsÞ Z 1 zðsÞ dwðsÞ ¼ E zT ðsÞZ 1 zðsÞ ds,

thm

Z

ð3:20Þ

tþy

ð3:22Þ

T



ð3:19Þ

zðsÞ dwðsÞ rhm ZT ðtÞHT Z 1 3 H ZðtÞ

thm

f ðxðthðtÞÞÞQ 10 f ðxðthðtÞÞÞr xT ðthðtÞÞKQ 10 KxðthðtÞÞ:

a2 ðtÞ :¼ 2ZT ðtÞMT xðthm ÞxðthðtÞÞ

ð3:18Þ

#

thðtÞ

tþy

hM

)

ð3:17Þ

and

Z

f ðxðsÞÞ ds :

tþy

thM

þ

thm

zðsÞ dwðsÞ dy ¼ 0,

zðsÞ dwðsÞ r ZT ðtÞN T Z 1 2 N ZðtÞ

zðsÞ dwðsÞ

thm

T

)T

#

t

zðsÞ dwðsÞ r ZT ðtÞM T Z 1 2 M ZðtÞ

thðtÞ

2ZT ðtÞHT

f ðxðsÞÞQ 8 f ðxðsÞÞ ds

thM

(Z

Z

hm

 f ðxðsÞÞ ds ,

Z

 zðsÞ dwðsÞ ,

t

thM

 f ðxðsÞÞ ds ,

xðsÞ ds thM

thðtÞ

þ

and thm

Z

#T

thM

2ZT ðtÞIT Z

Z1

zðsÞ dwðsÞ

þ

T Z f ðxðsÞÞ ds Q 7

ðhM hm Þ

thm

thm

2ZT ðtÞNT

Z t

Z

þ

f ðxðsÞÞQ 7 f ðxðsÞÞ ds

Z r

T

thðtÞ

t

thðtÞ

thm

thðtÞ

"Z

zT ðsÞZ 3 zðsÞ ds dy

T

hm

zðsÞ dwðsÞ

ð3:7Þ

T Z f ðxðsÞÞ ds Q 4

t

t

2ZT ðtÞM T

From Lemma 2.1, it follows that Z t T f ðxðsÞÞQ 4 f ðxðsÞÞ ds t tt

Z

T

T

thm

tþy

hM

hm hM

thm

Z þ

zT ðsÞZ 1 zðsÞ ds

zT ðsÞZ 4 zðsÞ ds dy:



hm Z3 2

tþy

hm

hm Z

t

xðsÞ ds

where ZðtÞ ¼ ½x ðtÞ x ðthðtÞÞ . From the formula (3.12)–(3.16), we have Z t zðsÞ dwðsÞ r ZT ðtÞLT Z 1 2ZT ðtÞLT 1 LZðtÞ

thm

Z

0

t

thm

Z

ð3:15Þ

tþy

ð3:16Þ

tþy

Z

yðsÞ ds dy

T

yT ðsÞR3 yðsÞ ds dy

yT ðsÞR4 yðsÞ ds dy

tþy

hM

 Z

Z

0

hm

Z

hm

 Z

Z

zðsÞ dwðsÞ dy ¼ 0,

ð3:6Þ

2

LV 3 ðt,xt Þ r y ðtÞ½s1 R3 þ s2 R4 yðtÞ þz ðtÞ hm Z 1 þ ðhM hm ÞZ 2 þ

t tþy

hM

yT ðsÞR2 yðsÞ ds,

T

Z

hm



and T

#

t

thðtÞ

thM

thm

Z

a5 ðtÞ :¼ 2Z ðtÞI ðhM hm ÞxðtÞ

x ðsÞQ 6 xðsÞ ds

thM

Z

yT ðsÞR1 yðsÞ ds



Z

Z

0

hm

T

T

T

T



yðsÞ ds dy tþy

hm

f ðxðsÞÞQ 8 f ðxðsÞÞ ds

ðhM hm Þ

xðsÞ ds Z

t



thm

thm

ð3:14Þ

t thm

Z

0

zðsÞ dwðsÞ ¼ 0, Z

a4 ðtÞ :¼ 2ZT ðtÞHT hm xðtÞ Z

#

thðtÞ

thM

þQ 2 þ Q 3 jðthm Þ þ xT ðtÞ½hm Q 5 þ ðhM hm ÞQ 6 xðtÞ T

Z

Z ð3:13Þ

0

Z

t

zðsÞ dwðsÞ

E hm

Z

tþy 0 Z t

¼E hm

tþy

T

Z3

Z

t

 zðsÞ dwðsÞ dy

tþy T

z ðsÞZ 3 zðsÞ ds dy,

ð3:25Þ

6

H. Chen / Neurocomputing 101 (2013) 1–9

and Z hm Z E hM

Z

k

t

zðsÞ dwðsÞ

tþy hm Z t

¼E hM

T

Z4

Z

t

~ ¼ v ½P11 W 0 þ P14 AT Q T þ Q 1 e NT N 2 þ e NT N 7 , O k 1k 1 2k 5 15 12

 zðsÞ dwðsÞ dy

k

~ ¼ v ½P11 W 1 e N T N3 þ e N T N8 , O k 1k 1 2k 5 16 k

~ ¼ v ½P11 W 2 e N T N4 þ e N T N9 , O k 1k 1 2k 5 19

tþy

zT ðsÞZ 4 zðsÞ ds dy:

ð3:26Þ

tþy

þ e2k NT6 N6 ,

Substituting (3.5)–(3.26) into (3.4), and then taking the mathematical expectation, it yields Z 0 Z t Z t 2 zT ðt, y,u,vÞX1 zðt, y,u,vÞ dv du dy ELVðt,xt Þ rE 3 hm hm t þ y thm Z hm Z t Z thm 1 þE zT ðt, y,u,vÞX2 zðt, y,u,vÞ dv du dy hMm s2 hM t þ y thðtÞ Z hm Z t Z thðtÞ 2 þE zT ðt, y,u,vÞX3 zðt, y,u,vÞ dv du dy, hMm s2 hM t þ y thM ð3:27Þ

T

T

T

ttðtÞ

¼ e2k N T6 N 9 , 2

¼ vk ½Q 9 þQW 0 þ W T0 Q T þ Q 122 þhm Q 7 þhMm Q 8 þ t2 Q 4  þe

T T 1k N 2 N 2 þ 2k N 7 N 7 ,

e

~ k ¼ v ½QW  þ e NT N3 þ e NT N8 , O k 1 1k 2 2k 7 56 k

~ ¼ v QW þ e NT N 4 þ e NT N 9 , O k 2 1k 2 2k 7 59 k

~ ¼ v ½ð1mÞQ 2 Q  þ e N T N3 þ e N T N8 , O k 1k 3 2k 8 10 66 22

k

~ Þ13 ¼ ½v M T P 11 0 0 0 v MT Q 0 0 0 0 v MT P14 v M T P 15 T , ðO k k k k

thm

thm

T

k

f ðxðsÞÞ dsx ðsÞyT ðsÞ

k

1

~ Þ231 ¼ ½hm MT P 12 0 0 0 hm MT R1 s1 M T R3 hm M T Z 1 s1 MT Z 3 0 0 0 0T , ðO

#T

T

k

~ Þ231 ðO ~ Þ232 , ~ Þ23 ¼ ½ðO ðO k

~ Þ232 ¼ ½0 0 v M T P 11 v MT QL 0 0 0 0 0 0 0 0T , ðO k k

:

thM

2

3

~ Þ231 ¼ ðO ~ Þ231 ¼ ½hMm M T P 12 0 0 0 hMm M T R2 s2 M T R4 hMm M T Z 2 ðO

s2 M T Z 4 0 0 0 0T , " # 0 e1k E ~ k Þ33 ¼ , ðO 0 e2k E

Consequently, it can be obtained from (3.1) that ELVðt,xt Þ rlEJxðtÞJ2 , 1

2

3

where l ¼ minflmin ðX Þ, lmin ðX Þ, lmin ðX Þg 4 0, which indicates that the system (2.7) is globally asymptotically stable in mean square. & Theorem 3.2. Let a A ð0,1Þ, for the delays h(t), tðtÞ satisfying the condition (2.3), and the condition (H) is satisfied, if there exist matrices: P 11 4 0, P22 4 0, P 33 40, P44 4 0, P 55 40, P12 , P13 , P 14 , i

P 15 , P23 , P24 , P25 , P34 , P 35 , P 45 , Q i ¼ ½ðQQi 11ÞT 12

Q i12  40 Q i22

ði ¼ 1,2,3Þ,

Q l 4 0 ðl ¼ 4,5, . . . ,8Þ, Rk 40 ðk ¼ 1,2,3,4Þ, Q ¼ diagfq1 ,q2 , . . . ,qn g 10 10 Q 9 ¼ diagfq91 ,q92 , . . . ,q9n g 4 0, Q 10 ¼ diagfq10 1 ,q2 , . . . ,qn g 4 0

some

appropriately

dimensional

matrices:

L ¼ ½LT1 LT2 T ,

M ¼ ½M T1 M T2 T , J ¼ ½J T1 J T2 T , H ¼ ½HT1 HT2 T , I ¼ ½IT1 IT2 T , and some positive scalars e1j 4 0, e2j 4 0ðj ¼ 1,2,3Þ, such that P ¼ ðPij Þ55 4 0 and the following linear matrix inequalities (LMIs) hold: 2 1 2 2 3 3 ~ 1 Þ13 ~ 2 Þ13 ~ ~ O ðO1 Þ12 ðO O ðO2 Þ12 ðO 6 6 7 7 6 02 ~ 1 Þ23 7 ~ 2 Þ23 7 X01 ¼ 6 6 n ðO1 Þ22 ðO 7 o 0, X ¼ 6 n ðO2 Þ22 ðO 7 o 0, 4 4 5 5 ~ 1 Þ33 ~ 2 Þ33 n n ðO n n ðO 2 3 3 ~ ~ 3 Þ13 O ðO3 Þ12 ðO 6 7 ~ 3 Þ23 7 X03 ¼ 6 ð3:28Þ 6 n ðO3 Þ22 ðO 7 o0, 4 5 3 33 ~ n n ðO Þ

k

~ ¼ v ½P 11 AAT P 11 þP T þP 12 þQ 1 þ K T Q K þhm Q þhMm Q O k 9 5 6 11 11 12 þ LT1 þL1 þ hm HT1 þ hm H1 þ hMm IT1 þ hMm I1  þ e1k NT1 N1 þ e2k NT5 N 5 , ~ 12 ¼ v ½L2 M T þ JT þ hm H2 þ ðhM hm ÞI2  þ e N T N6 , O 2i 5 k 1 1

v1 ¼ a,

v2 ¼ v3 ¼ 1a,

Proof. If A, W 0 , W 1 , W 2 , C, D, B0 , B1 and B2 in (3.1) are replaced with A þ DAðtÞ, W 0 þ DW 0 ðtÞ, W 1 þ DW 1 ðtÞ, W 2 þ DW 2 ðtÞ, C þ DCðtÞ, D þ DDðtÞ, B0 þ DB0 ðtÞ, B1 þ DB1 ðtÞ and B2 þ DB2 ðtÞ, where DAðtÞ, DW 0 ðtÞ, DW 1 ðtÞ, DW 2 ðtÞ, DCðtÞ, DDðtÞ, DB0 ðtÞ, DB1 ðtÞ and DB2 ðtÞ are described in (2.5), then (3.1)–(3.3) for the uncertain stochastic neural networks with discrete interval and distributed delays (2.1) are equivalent to the following conditions:

Xk þ Gk1d FðtÞðGk1e ÞT þ Gk1e F T ðtÞðGk1d ÞT þ Gk2d FðtÞðGk2e ÞT þ Gk2e F T ðtÞðGk2d ÞT o0,

ð3:29Þ

for k ¼ 1,2,3. where

G11d ¼ ½v1 P11 M 0 0 0 v1 QM 0 0 0 0 v1 MT P 14 v1 MT P 15 hm MT P 12 0 0 0 hm RT1 M s1 RT3 M 0 0 0 0 0 0T ,

G21d ¼ G31d ¼ ½v2 P 11 M 0 0 0 v2 QM 0 0 0 0 v2 MT P14 v2 MT P15 hMm M T P 12 0 0 0 hMm RT2 M s2 RT4 M 0 0 0 0 0 0T ,

G12d

¼ ½0 0 0 0 0 0 0 0 0 0 0 0 0 v1 P T11 M v1 LQM 0 0 hm Z 1 M s1 Z 3 M 0 0 0 0T , ¼ G32d ¼ ½0 0 0 0 0 0 0 0 0 0 0 0 0 v1 PT11 M v1 LQM 0 0 hm Z 1 M s1 Z 3 M 0 0 0 0T ,

k

~ Þ ~ ¼ ðO O k ¼ 1,2,3, ij 1111 ,

k ¼ 1,2,3,

and other items are given in Theorem 3.1, then the stochastic neural networks with mixed delays (2.1) are globally robustly asymptotically stable in mean square.

G22d

where k

¼ e2k N T6 N 8 ,

Ok99 ¼ Q 4 þ e1k N T4 N 4 þ e2k N T9 N 9 ,

T

f ðxðtÞÞ f ðxðthðtÞÞÞ f ðxðthm ÞÞ f ðxðthM ÞÞ Z t Z t T T f ðxðsÞÞ ds f ðxðsÞÞ ds

and

¼e

T 2k N 6 N 7 ,

k

 zðt,s,u,vÞ ¼ xT ðtÞ xT ðthðtÞÞ xT ðthm Þ xT ðthM Þ

40,

~k O 25 ~k O 26 ~k O 29 k ~ O 55

~ ¼ e NT N4 þ e NT N9 , O 1k 3 2k 8 69

where

Z

k

~ ¼ v ½ð1mÞQ 2 þ LQ LMT M T þJ T þ J þ K T Q K O k 13 10 2 11 2 2 2 22

Gi1e

¼ ½N T1 0 0 0 N T2 N T3 0 0 N T4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,

Gi2e ¼ ½NT5 NT6 0 0 NT7 NT8 0 0 NT9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, i ¼ 1,2,3: By Lemma 2.1, there exist some positive scalars e1k 4 0,

e2k 4 0 ðk ¼ 1,2,3Þ such that

H. Chen / Neurocomputing 101 (2013) 1–9 k k T k T k k T k T k 1 k Xk1 þ e1 1k G1d ðG1d Þ þ e2k ðG1e Þ G1e þ e2k G2d ðG2d Þ þ e2k ðG2e Þ G2e o 0,

k ¼ 1,2,3:

ð3:30Þ

Applying Lemma 2.2, we can obtain that (3.30) are equivalent to (3.28), respectively. Remark 2. Some sufficient conditions ensuring the globally robustly asymptotical stability of uncertain stochastic neural networks with discrete interval and distributed delays have been given in Theorems 3.1–3.2. In Theorems 3.1–3.2, the derivative value of the Lyapunov–Krasovskii functional can ultimately be written as the sum of three parts (3.27). This treatment is different from one involved in [22] and Theorems 3.1–3.2 can have the less conservatism than ones given in [22] since some important information neglected in [22] and the information R thðtÞ  th yT ðsÞR4 yðsÞ ds are fully considered in this paper. Remark 3. The augmented Lyapunov–Krasovskii functional Vðt,xt Þ, as pointed in [6,37,47], can play an important role in reducing the conservatism of our results. More specifically, by Rt R th Rt T taking the states thm xT ðsÞ ds, thMm xT ðsÞ ds, thm f ðxðsÞÞ ds, R hm R t R thm T thM f ðxðsÞÞ ds and hM t þ y yðsÞ ds dy as augmented variables, the stability in Theorems 3.1 and 3.2 sufficiently utilize more information on state variables, which can yield less conservatism. Remark 4. In contrast to the results obtained by using matrix theory technique [7–9,29,31,32,52], our proposed LMI-based ones are computationally efficient as they can be solved numerically by employing the LMI toolbox in Matlab. 4. Two illustrative examples Example 4.1. Consider uncertain system (2.1) with the following parameters:         0:2 4 0:4 0:2 2 0 1 0 A¼ , W0 ¼ , W1 ¼ , W2 ¼ , 0:1 0:3 0:1 0:7 0 3 0 1       0:3 0 0:2 0:6 0:5 0:1 C¼ , D¼ , , B0 ¼ 0 0:1 0:5 0:1 0:5 0       0:3 0:6 0:2 0 1 0 B1 ¼ , B2 ¼ , L¼ , M ¼ ½0:1 0:1, 0:2 0:1 0 0:2 0 1 N1 ¼ N 2 ¼ N3 ¼ N4 ¼ N 5 ¼ N6 ¼ N7 ¼ ½1 1,

P25 ¼ 1:0  e10 

N8 ¼ N9 ¼ ½0 0:

Let a ¼ 0:1, by using Theorem 3.2, for given m ¼ 0:6, hm ¼1, and t ¼ 1:6, it is concluded that this system is globally robustly asymptotical stable in mean square while hM is up to 7:0  108 . And when m ¼ 0:6, hm ¼1, hM ¼ 7:0  108 and t ¼ 1:6, by using the Matlab LMI Control Toolbox [1] to solve the LMIs in (3.28) in Theorem 3.2, we derive a set of feasible solutions as follows:   0:3799 0:0466 P 11 ¼ , P 12 ¼ 1:0  e3 0:0466 0:1970     0:2223 0:3306 0:0012 0:0018  , P22 ¼ , 0:2223 0:3306 0:0018 0:0138   0:0526 0:0914 P 13 ¼ 1:0  e10  , 0:0232 0:1938   0:0022 0:0030 , P 14 ¼ 0:0024 0:0037   0:0025 0:0712 P 15 ¼ 1:0  e9  , 0:0498 0:1309   0:0797 0:0680 , P 23 ¼ 1:0  e10  0:0695 0:5625   0:0010 0:0002 , P 33 ¼ 1:0  e10  0:0002 0:0011

P24 ¼



0:0026 0:0039

P34 ¼ 1:0  e10  P35 ¼ 1:0  e10 

7



0:1080

0:3057

0:2659  , 0:0089  0:1038

0:2859

 ,

0:0056



0:2987

 ,

0:2759

0:2866

0:2302

0:2704

 ,

0:2759 0:2866  0:0201 0:0022 P44 ¼ , 0:0022 0:0782   0:0203 0:0175 , P45 ¼ 1:0  e8  0:0256 0:1795   0:0318 0:0293 , P55 ¼ 1:0  e8  0:0293 0:2092     0:6240 0:0312 0:0430 0:7535 Q 111 ¼ , Q 112 ¼ , 0:0312 0:3636 0:0752 0:0131   0:3754 0:0042 , Q 122 ¼ 0:0042 1:4221     0:6090 0:0217 0:0906 0:6930 , Q 212 ¼ , Q 211 ¼ 0:0217 0:2219 0:0041 0:0671   0:0900 0:1278 , Q 222 ¼ 0:1278 0:8969     0:0078 0:0055 0:0246 0:0268 , Q 312 ¼ , Q 311 ¼ 0:0055 0:0808 0:0433 0:0272   0:1507 0:0523 , Q 322 ¼ 0:0523 0:2325     3:0451 0:0523 0:0020 0:0074 , Q5 ¼ , Q4 ¼ 0:0523 0:2325 0:0074 0:0518   0:0042 0:0165 , Q 6 ¼ 1:0  e8  0:0165 0:1129     0:0076 0:0233 0:0328 0:0325 , , Q 8 ¼ 1:0  e5 Q7 ¼ 0:0233 0:1208 0:0325 0:2133   9:3241 0:0000 Q9 ¼ , 0:0000 9:3241     0:8942 0:0000 0:0236 0:0000 , Q¼ Q 10 ¼ , 0:0000 0:8942 0:0000 0:0236   0:0011 0:0024 , R1 ¼ 0:0024 0:0135   0:1723 0:1830 , R2 ¼ 1:0  e10  0:1830 0:9494   0:1124 0:1484 , R4 ¼ 1:0  e18 0:1484 0:5848     0:0022 0:0060 0:0033 0:0026 , R3 ¼ , Z1 ¼ 0:0060 0:0252 0:0026 0:0120   0:0030 0:0071 , Z3 ¼ 0:0071 0:0283   0:0489 0:1260 , Z 2 ¼ 1:0  e10  0:1260 0:5449   0:0661 0:1760 , Z 4 ¼ 1:0  e18  0:1760 0:7589 

e11 ¼ 0:0726, e12 ¼ 0:0248, e21 ¼ 0:0075, e22 ¼ 0:0028, e31 ¼ 0:0075, e32 ¼ 0:0028: Remark 5. Obviously, the stability criteria given in [4,5,10,11,16– 18,20,23,27,29,33,34,38,41,45,47–52] cannot be used to dealt with this example.

8

H. Chen / Neurocomputing 101 (2013) 1–9

Table 1 Allowable upper bound of hM with given for unknown m in Example 4.2. Methods

hm ¼ 1:0

hm ¼ 1:5

hm ¼ 2:0

Li et al. [22] Theorem 3.1 ða ¼ 0:1Þ

1.7647 1.9913

2.2417 2.5463

2.7374 3.0907

Table 2 Allowable upper bound of hM with given for unknown m in Example 4.2. Methods

hm ¼ 1:0

hm ¼ 1:5

hm ¼ 2:0

Li et al. [22] Theorem 3.1 ða ¼ 0:1Þ

1.9482 2.0643

2.3740 2.6338

2.8722 3.1880

Example 4.2. Consider the following stochastic neural networks with interval time-varying delay [22]: dxðtÞ ¼ ½AxðtÞ þW 0 f ðxðtÞÞ þ W 1 f ðxðthðtÞÞÞ dt þ ½CxðtÞ þDxðthðtÞÞ þ B0 f ðxðtÞÞþ B1 f ðxðthðtÞÞÞ dwðtÞ, where      0:4 0:7 0:2 4 0 A¼ , W2 ¼ , W0 ¼ 0:1 0 0:5 0 5   0:5 0 C¼ , 0 0:5     0:1 0 0 0:5 , D¼ , B0 ¼ 0 0:1 0:5 0     0:1 0 0:5 0 , L¼ B1 ¼ : 0 0:1 0 0:5

0:6 0:1

 ,

For unknown m and given different lower values hm, by using Theorem 3.1, we can obtain the upper bounds hM, which guarantee the globally asymptotical stability of this system, are given in Table 1. When B0 ¼ B1 ¼ 0, Table 2 lists that the upper bounds hM for different value hm when unknown m. We can see that Theorem 3.1 is less conservative than ones proposed in [22].

5. Conclusion In this paper, the problem of the globally robustly asymptotical stability in mean square for uncertain stochastic neural networks with discrete interval and distributed delays is considered. By constructing an augmented Lyapunov–Krasovskii functional, the LMI-based sufficient conditions ensuring the globally robustly asymptotical stability in mean square for such systems can be derived by using the free-weighting matrices, which are much less conservative than some existing reports. And, in contrast to the results in [7–9,29,31,32,52], the proposed LMIbased ones are computationally efficient as they can be solved numerically by employing the LMI toolbox in Matlab. Finally, two illustrative examples are provided to show the effectiveness of the given results in this paper.

Acknowledgement The author would like to thank the anonymous referees for their very helpful comments and suggestions which can greatly improve this manuscript, and the editors for their carefully reading of this paper.

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Huabin Chen was born in Hubei Province, China. He received Ph.D. degree in 2009 from the School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei Province, China. Since July 2009, he has been working in the Department of Mathematics, School of Science, Nanchang University, Nanchang, Jiangxi Province, China. His current research interests include time-delay systems, stochastic systems and their application.