Exponential stability of genetic regulatory networks with random delays

ARTICLE IN PRESS Neurocomputing 73 (2010) 759–769

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Exponential stability of genetic regulatory networks with random delays Xuyang Lou , Qian Ye, Baotong Cui School of Communication and Control Engineering, Jiangnan University, 1800 Lihu Road, Wuxi, Jiangsu 214122, China

a r t i c l e in fo

abstract

Article history: Received 17 January 2009 Received in revised form 13 October 2009 Accepted 26 October 2009 Communicated by H. Jiang Available online 17 November 2009

This paper on global exponential stability in the mean square sense of genetic regulatory networks (GRNs) is motivated by a practical consideration that different genes have different time delays for transcription and translation, and in some cases, each multimer is assigned to a randomly chosen gene promoter site as an activator or inhibitor. One important feature of the obtained results reported here is that the time-varying delays are assumed to be random and their probability distributions are known a priori. By employing the information of the probability distributions of the time delays, we present some stability criteria for the uncertain delayed genetic networks with SUM regulatory logic where each transcription factor acts additively to regulate a gene. The effects of both variation range and distribution probability of the time delays are taken into account in the proposed approach. Another feature of the results is that a novel Lyapunov functional dependence on auxiliary delay parameters is exploited, which renders the results to be potentially less conservative and allows the time-varying delays to be not differentiable. The theoretical findings are illustrated and verified with two examples. & 2009 Elsevier B.V. All rights reserved.

Keywords: Genetic regulatory network Random delays Exponential stability Probability distribution

1. Introduction A great number of genes and proteins either directly or indirectly interact with one another in living cells. Such interactions make up of a dynamic genetic regulatory network (GRN) which acts as a complex dynamic system for controlling cellular functions. In the area of GRNs, insights into the nature and function of these networks are of interest to many researchers, as it has been proved that many diseases stem from the malfunction of GRNs of the corresponding cell lines [1,2]. As an example, a gene regulatory network published by Ronen et al. [3] for the SOS DNA Repair network of the Escherichia coli bacterium is shown in Fig. 1. Through theoretical analysis, several simple genetic networks have been successfully constructed by means of experiments, for example, genetic switches [4], a clustering-based approach [5] and a single negative feedback loop network [6]. These results intrinsically show that mathematical modeling of genetic networks as dynamical system models can be a powerful tool for studying gene regulation processes in living organisms, since genetic networks are biochemically dynamical systems. Generally speaking, there are two types of genetic network models [7–13]: Boolean or logical model and differential equation or dynamical system model. (1) For the first type, only two states (that is, ON and OFF) are used to express the activity of each gene, and the state of a gene is determined by a Boolean function of the states of other related genes. (2) For the second type, the differential equation is utilized to describe the whole network, with the state variables describing the

 Corresponding author.

E-mail address: [email protected] (X. Lou). 0925-2312/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2009.10.006

concentrations of gene products, such as mRNAs and proteins. Due to continuity of the states in the second type model, it can be more accurate and provide detailed understanding of the nonlinear dynamical behavior exhibited by biological systems. In the design and application of networks, such as GRNs and complex dynamical networks, it is important to analyze deeply their dynamical behaviors like stability. There have been some studies on the stability or synchronization analysis for complex dynamical networks [30–33]. Recently, mathematical GRN models defined as dynamical systems have been studied extensively for stability [6], synchronization [14,15], and periodic oscillation [16] of GRNs. For instance, Li et al. [15] provided a theoretical method for analyzing synchronization of coupled non-identical genetic oscillators and established sufficient conditions for the synchronization and the estimation of the bound of the synchronization error are also derived, without considering the effects of time delay, which exist in the real-life genetic regulatory network. However, it should be noted that time delay may play an important role in dynamics of genetic networks, and theoretical models without consideration of these factors may even provide wrong predictions [11,17]. But due to the incorporation of the time delay in the genetic networks, the dynamics will be more complicated. For instance, Chen and Aihara [11] presented a model for GRNs with constant time delays and proposed successfully necessary and sufficient conditions for the simplified GRNs. Subsequently, they explained periodic oscillations which are mainly generated by nonlinearly negative and positive feedback loops in gene regulatory systems, and explored effects of time delay on stability region of the oscillations in [16]. In [19], the authors revealed that the genetic network can exhibit Hopf bifurcation (oscillation appears) as the sum of delays or transcriptional rate passes through some critical values.

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Nomenclature

AgBð!BÞ ABg0ð!0Þ

lmax ðPÞ (lmin ðPÞ) the maximum (minimum) eigenvalue of the Rn Rnm Rþ

the n-dimensional real space the set of real n  m matrices the set of non-negative real numbers | the empty set transpose of matrix A AT inverse of matrix A A1 Pg0 ðP!0Þ P is a positive (negative) definite symmetric matrix

Nevertheless, among all of these works, little study has been performed on GRNs when considering unavoidable uncertainties or external perturbations, but in the applications and designs of networks, such as genetic networks and neural networks, there are often some unavoidable uncertainties such as modeling errors, external perturbations, and parameter fluctuations, which may cause the networks to be unstable. Therefore, it is essential to investigate the global robust stability of such networks with errors and perturbations. In order to synthetically investigate the stability of GRNs, some excellent works have been carried out. In [12], the authors derived delay-dependent robust asymptotic stability criteria for a class of GRNs with time-varying delays and parameter uncertainties via Lyapunov method and linear matrix inequality (LMI) approach. For discrete-time versions of the continuous-time GRNs with SUM regulatory functions, some sufficient conditions are presented to ensure the global exponential stability of the networks in Ref. [18]. Li et al. [13] presented a nonlinear model for GRNs with SUM regulatory functions and established sufficient conditions of the cases of genetic networks with time-varying delays and stochastic perturbations. In the sequel, they further extended the results on the stochastic stability of the genetic networks with disturbance attenuation by using Lyapunov method and the Lur’e system approach in [20]. Despite the fruitful contributions in this area, the reported results have two main limitations as follows (in view of the goal in our work):

 First, only the deterministic time delay case was concerned. However, it is mentioned in [21] that the time-varying delays in GRNs are often existent in a random fashion, and their

I diagðÞ JJ ½AB C  Efg %



real symmetric matrix P identity matrix of appropriate dimension a block diagonal matrix the Euclidean vector norm or spectral norm as appropriate T % represents the symmetric form of matrix; i.e., % ¼ B the mathematical expectation

probabilistic characteristics, such as Poisson distribution or normal distribution, can often be obtained by statistical methods. Hence, it is necessary to consider random-delay effects in GRNs. Second, these results mainly aim at asymptotic stability but not exponential stability and are more or less conservative, which makes it difficult to apply them to practical problems. By reason of rigorous restriction on time-varying delays, that is, either their derivatives are less than 1 or less information of the variation range of the delays is used, the above treatments may lead to conservative results. In addition, in many evolutionary processes, e.g., optimal control models and flying object motion, there are many signal functions which are not monotone-non-decreasing and even do not satisfy Lipschitz condition. Thus it is also reasonable to consider nonlinear function, i.e., the feedback regulation of the protein for GRNs, to be non-monotonic or non-Lipschitz continuous.

In order to overcome the two limitations, we adopt a delaydistribution-dependent scheme to discuss global robust exponential stability (GRES) in the mean square sense (MSS) of GRNs with random delays. The developed criteria depend on not only the variation range of delays but also their probability distributions. To the best of our knowledge, few contributions have addressed such stability problems involving random or stochastic delays [22–26], which are the focus of this work. Different from the common assumptions on the time delay in existing literature, it is assumed in this paper that the probability distribution of the delay appearing in some intervals can be observed and treated as a function of the stochastic variable satisfying Bernoulli random binary distribution. A novel Lyapunov functional with auxiliary delay parameters combining with relaxation matrix approach and LMI techniques is explored to derive some criteria guaranteeing the GRES in MSS of uncertain GRNs in the presence of random delays for the entire uncertainty domain. Moreover, we also give the maximal exponential convergence rate. Unlike some results in the literature, none of the established results depends on the derivatives of the time-varying delays. Therefore, the results are suitable to the cases with very fast-varying delays. Briefly, the paper is organized as follows. Section 2 introduces the problem formulation related to the GRN models and gives some preliminaries used in this paper. In Section 3 and 4, some mild criteria ensuring the exponential mean-square stability are presented for GRNs without uncertainties or with norm-bounded parameter uncertainties, respectively. In Section 5, two examples are provided to illustrate the effectiveness and advantages of the proposed results. Finally, conclusions follow in Section 6.

2. Model description and preliminaries Fig. 1. An example of genetic regulatory network—the SOS DNA repair network. Inhibitions are represented by ‘‘ ’’ ; while activations are represented by ‘‘) ’’ :

In a genetic regulatory network, a number of genes interact and regulate the expression of other genes by proteins, the gene

ARTICLE IN PRESS X. Lou et al. / Neurocomputing 73 (2010) 759–769

derivatives. The change in expression of a gene is governed by the stimulation and inhibition of proteins in transcriptional, translational, and post-translational processes. The following differential equations have been used recently to describe the genetic regulatory networks [13]: 8 _ > < M i ðtÞ ¼ ai Mi ðtÞ þ Gi ðP1 ðtsðtÞÞ; P2 ðtsðtÞÞ; . . . ; Pn ðtsðtÞÞÞ; ð1Þ > :_ P i ðtÞ ¼ ci Pi ðtÞ þ di Mi ðttðtÞÞ; where i ¼ 1; 2; . . . ; n, Mi ðtÞ; Pi ðtÞ A R are the concentrations of mRNA and protein of the ith node, respectively. The parameters ai and ci are the degradation rates of mRNA and protein, respectively; di is the translation rate, and the functions Gi represent the feedback regulation of the i gene, which is generally a nonlinear function but has a form of monotonicity with each variable [7,8]; sðtÞ 4 0 and tðtÞ 40 are inter- and intra-node random delays, respectively. For the structure and regulation mechanism of the genetic network, one can refer to [11,13]. Here, we focus on the case that each transcription factor acts additively to regulate the ith gene. That is, the regulatory function is of the P form Gi ðP1 ðtÞ; P2 ðtÞ; . . . ; Pn ðtÞÞ ¼ nj¼ 1 Gij ðPj ðtÞÞ, which is also called SUM logic [13,27]. As described in [13], the function Gij ðPj ðtÞÞ is generally expressed by a monotonic function of the Hill form [14]: if transcription factor j is an activator of gene i, then Gij ðPj ðtÞÞ ¼ gij

ðPj ðtÞ=dj ÞHj 1 þ ðPj ðtÞ=dj ÞHj

;

if transcription factor j is a repressor of gene i, then Gij ðPj ðtÞÞ ¼ gij

1 1 þ ðPj ðtÞ=dj ÞHj

;

where Hj is the Hill coefficient, dj is a positive constant, and gij is the dimensionless transcriptional rate of transcription factor j to gene i, which is a bounded constant. We can also rewrite (1) into the following form: 8 n X > > _ i ðtÞ ¼ ai Mi ðtÞ þ > : P_ ðtÞ ¼ c P ðtÞ þ d M ðttðtÞÞ; i ¼ 1; 2; . . . ; n; i i i i i P where fj ðsÞ ¼ ðs=dj ÞHj =ð1þ ðs=dj ÞHj Þ, Ui ¼ j A Ii gij and Ii is the set of nn is defined as follows: repressors of gene i, B ¼ ðbij Þ A R 8 g if transcription factor j is anactivator of gene i; > < ij if there is no link from node j to i; bij ¼ 0 ð3Þ > : g if transcription factor j is arepressor of gene i: ij

One can rewrite system (2) into the following compact matrix form: ( _ MðtÞ ¼ AMðtÞ þ Bf ðPðtsðtÞÞÞ þ U; ð4Þ _ PðtÞ ¼ CPðtÞ þ DMðttðtÞÞ; where

 T MðtÞ ¼ M1 ðtÞ; M2 ðtÞ; :::; Mn ðtÞ  T PðtÞ ¼ P1 ðtÞ; P2 ðtÞ; :::; Pn ðtÞ ; f ðPðtsðtÞÞÞ ¼ ðf1 ðP1 ðtsðtÞÞÞ; ðf2 ðP2 ðtsðtÞÞÞ; MðttðtÞÞ ¼ ðM1 ðttðtÞÞ; M2 ðttðtÞÞ;

g ¼ ðg1 ; g2 ; :::; gn ÞT ; A ¼ diag ða1 ; a2 ; :::; an Þ; C ¼ diag ðc1 ; c2 ; :::; cn Þ; D ¼ diag ðd1 ; d2 ; :::; dn Þ:

:::; fn ðPn ðtsðtÞÞÞÞT ;

:::; Mn ðttðtÞÞÞT ;

761

Vector ðMT ; P T ÞT is called an equilibrium point of the system (4) if they satisfy ( AM  þ Bf ðP  Þ þ U ¼ 0; ð5Þ CP  þ DM ¼ 0; Assume that an intended equilibrium of system (4) is ðM T ; PT ÞT . Using the transformation xðtÞ ¼ MðtÞM ; yðtÞ ¼ PðtÞP ; the system (4) can be transformed into the following form: ( _ ¼ AxðtÞ þBgðyðtsðtÞÞÞ; xðtÞ _ ¼ CyðtÞ þ DxðttðtÞÞ; yðtÞ

ð6Þ

where gðyðtÞÞ ¼ ðg1 ðy1 ðtÞÞ; g2 ðy2 ðtÞÞ; . . . ; fn ðPn ðtÞÞÞT with gi ðyi ðtÞÞ ¼ fi ðyi ðtÞ þ Pi Þfi ðPi Þ. Throughout the paper, we need the following assumption: Assumption 1. For i A f1; 2; . . . ; ng, each function gi ðÞ satisfies u i r

gi ðs1 Þgi ðs2 Þ ruiþ ; s1 s2

8s1 ; s2 A R;

ð7Þ

þ where u i and ui are some constants.

Remark 1. Obviously, the monotonically increasing function fi in (2) or (4) satisfies (7) with u i ¼ 0. However, Assumption 1 endows with even milder restriction than monotonically increasþ are allowed to be ing condition since the constants u i ; ui positive, negative, or zero. Although the above system (6) is formed by analyzing genetic networks of the form of (1), the above derivation is also applicable to genetic networks with random delays and parameter uncertainties. In analyzing the stability of an equilibrium point ðM T ; PT ÞT , it is equivalent of studying systems (1) and (6). In the following, we consider the following uncertain genetic networks with random delays: ( _ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞgðyðtsðtÞÞÞ; xðtÞ ð8Þ _ ¼ ðC þ DCðtÞÞyðtÞ þ ðD þ DDðtÞÞxðttðtÞÞ; yðtÞ where DAðtÞ, DBðtÞ, DCðtÞ and DDðtÞ are time-varying parameter uncertainties, which are assumed to satisfy ½DAðtÞDBðtÞDCðtÞDDðtÞ ¼ HFðtÞ½E1 E2 E3 E4 ;

ð9Þ

where H; E1 ; E2 ; E3 ; E4 are known constant matrices with appropriate dimensions and FðtÞ is an unknown time-varying matrix with Lebesque measurable elements bounded by F T ðtÞFðtÞ%I. sðtÞ A ½sm ; sM  and tðtÞ A ½tm ; tM  are the time-varying delays, where sm , sM , tm , tM are positive constants. In this paper, sðtÞ and tðtÞ change randomly, and their probability distributions can be observed. Remark 2. In what follows, in order to transform system (8) with random delays into an equivalent system which dependents on distributed sequences, similar analysis in [24] can also be carried out for random delays tðtÞ and sðtÞ. First, by taking values of sðtÞ in ½sm ; s0  or ðs0 ; sM , one can define two sets O1 and O2 . Then, we can define a stochastic variable aðtÞ which is a Bernoulli distributed sequence with ( ProbfaðtÞ ¼ 1g ¼ EfaðtÞg ¼ a0 ; ð10Þ ProbfaðtÞ ¼ 0g ¼ 1EfaðtÞg ¼ 1a0 ; where 0 r a0 r 1 is a constant. Also similar to [24], two functions s1 ðtÞ, s2 ðtÞ can be defined. In the same way, suppose tðtÞ take values in ½tm ; t0  or ðt0 ; tM , and accordingly one can derive t1 ðtÞ, t2 ðtÞ, and stochastic variable bðtÞ. Therefore, based on the analysis in Remark 2, system (8) will be replaced equivalently in what follows by the following system

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X. Lou et al. / Neurocomputing 73 (2010) 759–769

Proof. It follows from (7) that

which dependents on distributed sequences (

_ xðtÞ _ yðtÞ

¼

ðA þ DAðtÞÞxðtÞ þ aðtÞðB þ DBðtÞÞ  gðyðts1 ðtÞÞÞþ ð1aðtÞÞ  ðB þ DBðtÞÞgðyðts2 ðtÞÞÞ;

¼

ðC þ DCðtÞÞyðtÞ þ bðtÞðD þ DDðtÞÞ  xðtt1 ðtÞÞ þð1bðtÞÞ  ðD þ DDðtÞÞxðtt2 ðtÞÞ:

For brevity of the following analysis, we denote xðtÞ, yðtÞ, xðtt1 ðtÞÞ, xðtt2 ðtÞÞ, gðyðts1 ðtÞÞÞ, gðyðts2 ðtÞÞÞ, DAðtÞ, DBðtÞ, DCðtÞ and DDðtÞ by xt ; yt ; xðt1 Þ; xðt2 Þ; gðyðs1 ÞÞ; gðyðs2 ÞÞ, DAt , DBt , DCt and DDt , respectively. Thus, system (11) can further be expressed as (

x_ t y_ t

which yields

ðA þ DAt Þxt þ a0 ðB þ DBt Þgðyðs1 ÞÞ þð1a0 ÞðB þ DBt Þgðyðs2 ÞÞ þ ðaðtÞa0 ÞðB þ DBt Þðgðyðs1 ÞÞgðyðs2 ÞÞÞ;

¼

ðC þ DCt Þyt þ b0 ðD þ DDt Þxðt1 Þ þ ð1b0 ÞðD þ DDt Þxðt2 Þ þðbðtÞb0 ÞðD þ DDt Þðxðt1 Þxðt2 ÞÞ:

Before giving the main results, we need the following definitions and lemmas: Definition 1. The origin of system (12) is said to be globally robustly exponentially stable in the mean square sense, if there exist constants r^ 40 and e 4 0 such that for t Z 0 2

2

et

EfJxt J þJyt J g r r^ e ( sup tM r s r 0

E

_ ðsÞJ2 Þ þ ðJfðsÞJ2 þ Jf

sup s M r s r 0

Definition 2. For a given function VðtÞ : R-R, its infinitesimal operator L is defined as ð14Þ

Lemma 1 (Xu et al. [28]). Let S1 , S2 , S3 be real matrices of appropriate dimensions with S3 g0. Then for any vectors x and y with appropriate dimensions, 2xT ST1 S2 y rxT ST1 S3 S1 x þyT ST2 S1 3 S2 y:

ð15Þ

Lemma 2 (Gu et al. [29]). For any positive definite matrix M 40, scalar 0 o gðtÞ o g, vector function o : ½0; g-Rn such that the integrations concerned are well defined, then T Z gðtÞ  Z gðtÞ oðsÞ ds M oðsÞ ds r gðtÞ ðoT ðsÞM oðsÞÞ ds: 0

di ðgi ðxi ðtÞÞuiþ xi ðtÞÞðgi ðxi ðtÞÞu i xi ðtÞÞ r 0:

i¼1

This inequality can be written as g T ðxðtÞÞDgðxðtÞÞ þ xT ðtÞU 1 DU 2 xðtÞxT ðtÞDðU 1 þ U 2 ÞgðxðtÞÞ r 0: The proof is completed.

&

3. Exponential stability of GRNs without uncertainties In this section, we analyze the global exponential stability in the MSS of GRNs without uncertainties, that is, DAt ¼ DBt ¼ DCt ¼ DDt ¼ 0. In this case, the system (12) becomes (

x_ t y_ t

¼ ¼

Axt þ a0 Bgðyðs1 ÞÞ þ ð1a0 ÞBgðyðs2 ÞÞ þ ðaðtÞa0 ÞBðgðyðs1 ÞÞgðyðs2 ÞÞÞ; Cyt þ b0 Dxðt1 Þ þ ð1b0 ÞDxðt2 Þ þ ðbðtÞb0 ÞDðxðt1 Þxðt2 ÞÞ:

We give our first main result as follows:

where fðtÞ and cðtÞ are the initial functions of xt and yt , respectively.

1 LVðtÞ ¼ limþ EðVðt þhÞVðtÞÞ: h-0 h

n X

ð12Þ

ð17Þ

_ ðsÞJ2 Þg; ðJcðsÞJ2 þ Jc ð13Þ

0

ðgi ðxi ðtÞÞuiþ xi ðtÞÞðgi ðxi ðtÞÞu i xi ðtÞÞ r0;

¼

Remark 3. It is worth pointing out that by introducing the new functions t1 ðtÞ; t2 ðtÞ; s1 ðtÞ; s2 ðtÞ, Eq. (8) is transformed into (12). In (12), the probabilistic effects of the time delay have been translated into the parameter matrices of the transformed system. Under this transition, it will be shown that the GRE criteria in MSS can be derived, which exhibit the relationship between the stability of the system and the probability distribution parameters a0 , b0 and the variation range of the time delay.

Z gðtÞ

ð11Þ

0

Theorem 1. The origin of the system (17) is said to be globally exponentially stable in the MSS, if there exist two diagonal matrices D1 g0, D2 g0, positive definite matrices Q1 , Q2 , Z1 , Z2 , Ri ði ¼ 1; 2; 3; 4Þ, and arbitrary matrices Ni , Wi ði ¼ 1; 2; 3; 4Þ with appropriate dimensions, respectively, such that the following LMIs: 2

Y11

6 Z A 6 1 6 6 NT þ N2 1 X1 ¼ 6 6 6 N3T þ N4 6 6 N1T 4

%

%

%

%

%

Y22

%

%

%

%

0

Y33

%

%

%

0 0

0 N2T

Y44

%

%

0

R1

%

0

0

N4T

0

R3

N3T

2

^ 11 Y 6 6 Z2 C 6 6 W T þ W2 6 1 6 6 W T þ W4 3 6 X2 ¼ 6 6 0 6 6 6 0 6 6 W1T 4 W3T

%

%

%

%

3 7 7 7 7 7 7!0; 7 7 7 5

%

ð18Þ

%

%

^ 22 Y

%

%

%

%

%

%

0

^ 33 Y

%

%

%

%

%

0

0

^ 44 Y

%

%

%

%

0

D1 ðU 1 þ U 2 Þ

0

^ 55 Y

%

%

%

0

0

D2 ðU 1 þ U 2 Þ

0

^ 66 Y

%

%

0

W2T

0

0

0

R2

%

0

0

W4T

0

0

0

R4

3 7 7 7 7 7 7 7 7 7!0; 7 7 7 7 7 7 5

ð19Þ Lemma 3. Suppose that Assumption 1 holds. Then we claim g T ðxt ÞDgðxt Þ rxTt DðU 1 þ U 2 Þgðxt ÞxTt U 1 DU 2 xt ;

hold, where ð16Þ

for any diagonal positive definite matrix D ¼ diagðd1 ; d2 ; . . . ; dn Þg0, þ þ   þ where U 1 ¼ diagðu 1 ; u2 ; . . . ; un Þ, U 2 ¼ diagðu1 ; u2 ; . . . ; un Þ.

Y11 ¼ Q1 AAQ 1 þ Q1 þN1 þ N1T þ N3 þ N3T ; Y22 ¼ Z1 þ t20 R1 þ t2M R3 ;

ARTICLE IN PRESS X. Lou et al. / Neurocomputing 73 (2010) 759–769

Y33 ¼ b0 DT ðQ2 þZ2 ÞDN2 N2T ;

sM

Y44 ¼ ð1b0 ÞDT ðQ2 þ Z2 ÞDN4 N4T ; ^ 11 ¼ Y

Q2 CCQ 2 þQ2 þ W1 þ W1T

Z

t

tsM

763

Z _ ds r y_ T ðsÞR4 yðsÞ

t

ts2 ðtÞ

T Z _ ds R4 yðsÞ

t

 _ ds : yðsÞ

ts2 ðtÞ

ð25Þ

þW3 þ W3T ;

Making use of Lemma 1, we derive

^ 22 ¼ Z2 þ s2 R2 þ s2 R4 ; Y 0 M

2a0 xTt Q1 Bgðyðs1 ÞÞ r a0 xTt Q1 xt þ a0 g T ðyðs1 ÞÞBT Q1 Bgðyðs1 ÞÞ;

^ 33 ¼ 2U 1 D1 U 2 W2 W T ; Y 2

2ð1a0 ÞxTt Q1 Bgðyðs2 ÞÞ r ð1a0 ÞxTt Q1 xt þ ð1a0 Þg T ðyðs2 ÞÞBT Q1 Bgðyðs2 ÞÞ;

ð27Þ

^ 44 ¼ 2U 1 D2 U 2 W4 W T ; Y 4 2b0 yTt Q2 Dxðt1 Þ r b0 yTt Q2 yt þ b0 xT ðt1 ÞDT Q2 Dxðt1 Þ;

^ 55 ¼ a0 BT ðQ1 þZ1 ÞB2D1 ; Y

ð29Þ

Moreover, we have

For positive definite matrices Z1 and Z2 , it is clearly true that

EfJxt J2 þJyt J2 g r r^ eet E ( tM r s r 0

_ ðsÞJ2 Þ þ ðJfðsÞJ2 þ Jf

sup

0 ¼ 2x_ Tt Z1 x_ t þ 2x_ Tt Z1 x_ t ¼ 2x_ Tt Z1 x_ t þ 2x_ Tt Z1 ½Axt þ a0 Bgðyðs1 ÞÞ

_ ðsÞJ2 Þg; ðJcðsÞJ2 þ Jc

s M r s r 0

þð1a0 ÞBgðyðs2 ÞÞ þ ðaðtÞa0 ÞBðgðyðs1 ÞÞgðyðs2 ÞÞÞ:

with the maximal exponential convergence rate   l1 l2 ; ; e ¼ min lmax ðQ1 Þ lmax ðQ2 Þ

0 ¼ 2y_ Tt Z2 y_ t þ 2y_ Tt Z2 ½Cyt þ b0 Dxðt1 Þ

Proof. Consider the following Lyapunov functional: Z t Z t _ x_ T ðvÞR1 xðvÞ dv ds VðtÞ ¼ xTt Q1 xt þ t0 þ tM

Z

þ s0

t

Z

ttM s Z t t

ts0

tt0 t T

s

s

Z

t tsM

¼ þ ð1b0 ÞDxðt2 Þ þðbðtÞb0 ÞDðxðt1 Þxðt2 ÞÞ:

2a0 x_ Tt Z1 Bgðyðs1 ÞÞ r a0 x_ Tt Z1 x_ t þ a0 g T ðyðs1 ÞÞBT Z1 Bgðyðs1 ÞÞ; Z

t

ð31Þ

Using Lemma 1 again, the third and fourth terms in (30) and (31) can be further derived as

_ x_ ðvÞR3 xðvÞ dv dsþ yTt Q2 yt

_ y_ T ðvÞR2 yðvÞ dv ds þ sM

ð30Þ

Similarly, one can get

where l1 ¼ lmin fX1 g; l2 ¼ lmin fX2 g, r^ 40 will be defined in (49).

Z

ð28Þ

2ð1b0 ÞyTt Q2 Dxðt2 Þ r ð1b0 ÞyTt Q2 yt þ ð1b0 ÞxT ðt2 ÞDT Q2 Dxðt2 Þ:

^ 66 ¼ ð1a0 ÞBT ðQ1 þ Z1 ÞB2D2 : Y

sup

ð26Þ

_ y_ T ðvÞR4 yðvÞ dv ds;

s

ð32Þ

2ð1a0 Þx_ Tt Z1 Bgðyðs2 ÞÞ rð1a0 Þx_ Tt Z1 x_ t þ ð1a0 Þg T ðyðs2 ÞÞBT Z1 Bgðyðs2 ÞÞ;

ð33Þ

ð20Þ and where Qi ¼ QiT g0, i ¼ 1; 2, Rj ¼ RTj g0, j ¼ 1; 2; 3; 4. Using the infinitesimal operator (14), it follows that

2b0 y_ Tt Z2 Dxðt1 Þ r b0 y_ Tt Z2 y_ t þ b0 xT ðt1 ÞDT Z2 Dxðt1 Þ;

LVðtÞ ¼ 2xTt Q1 ½Axt þ a0 Bgðyðs1 ÞÞ þ ð1a0 ÞBgðyðs2 ÞÞ Z t _ dsþ t2M x_ Tt R3 x_ t tM þ t20 x_ Tt R1 x_ t t0 x_ T ðsÞR1 xðsÞ

ð35Þ

tt0

Z

t ttM

_ dsþ 2yTt Q2 ½Cyt þ b0 Dxðt1 Þ þð1b0 ÞDxðt2 Þ x_ T ðsÞR1 xðsÞ

þ s20 y_ Tt R2 y_ t s0 Z

t tsM

Z

t

ts0

_ ds þ s2M y_ Tt R4 y_ t sM y_ T ðsÞR2 yðsÞ ð21Þ

tt0

tt1 ðtÞ

tt1 ðtÞ

ð22Þ Z

t ttM

_ dsr  x_ T ðsÞR3 xðsÞ

Z

t tt2 ðtÞ

T Z _ ds R3 xðsÞ

t

 _ ds ; xðsÞ

tt2 ðtÞ

ð23Þ s0

Z

t

ts0

_ ds r y_ T ðsÞR2 yðsÞ

respectively. By introducing relaxation matrices Ni , Wi ði ¼ 1; 2; 3; 4Þ with appropriate dimensions, we obtain the following zero equations:   Z t _ ds ¼ 0; xðsÞ ð36Þ 2ðxTt N1 þ xT ðt1 ÞN2 Þ xt xðt1 Þ tt1 ðtÞ

_ ds: y_ T ðsÞR4 yðsÞ

According to definitions t1 ðtÞ; t2 ðtÞ; s1 ðtÞ and s2 ðtÞ, note that t1 ðtÞ r t0 , t2 ðtÞ r tM , s1 ðtÞ r s0 , s2 ðtÞ r sM . Thereby, from Lemma 2, we have Z t T Z t  Z t _ dsr  _ ds R1 _ ds ; t0 x_ T ðsÞR1 xðsÞ xðsÞ xðsÞ

tM

ð34Þ

2ð1b0 Þy_ Tt Z2 Dxðt2 Þ rð1b0 Þy_ Tt Z2 y_ t þð1b0 ÞxT ðt2 ÞDT Z2 Dxðt2 Þ;

Z

t ts1 ðtÞ

T Z _ ds R2 yðsÞ

t

 _ ds ; yðsÞ

 Z 2ðxTt N3 þ xT ðt2 ÞN4 Þ xt xðt2 Þ

 _ ds ¼ 0; xðsÞ

t

ð37Þ

tt2 ðtÞ

 Z 2ðyTt W1 þ yT ðs1 ÞW2 Þ yt yðs1 Þ  Z 2ðyTt W3 þ yT ðs2 ÞW4 Þ yt yðs2 Þ

t

 _ ds ¼ 0; yðsÞ

ð38Þ

 _ ds ¼ 0: yðsÞ

ð39Þ

ts1 ðtÞ t ts2 ðtÞ

In addition, in view of Lemma 3, it is easy to obtain that 2g T ðyðs1 ÞÞD1 gðyðs1 ÞÞ r 2yT ðs1 ÞD1 ðU 1 þ U 2 Þgðyðs1 ÞÞ 2yT ðs1 ÞU 1 D1 U 2 yðs1 Þ;

ð40Þ

2g T ðyðs2 ÞÞD2 gðyðs2 ÞÞ r 2yT ðs2 ÞD2 ðU 1 þ U 2 Þgðyðs2 ÞÞ

ts1 ðtÞ

ð24Þ

2yT ðs2 ÞU 1 D2 U 2 yðs2 Þ:

ð41Þ

ARTICLE IN PRESS 764

X. Lou et al. / Neurocomputing 73 (2010) 759–769

Substituting (22)–(41) into (21), we can rearrange (21) as follows:

Choose e r minfl1 =lmax ðQ1 Þ; l2 =lmax ðQ2 Þg which implies EfYðtÞg EfYð0Þg r 0, i.e.,

LVðtÞ r xTt ½Q1 AAQ 1 þ Q1 þN1 þ N1T þ N3 þ N3T xt

eet EfVðtÞg r EfVð0Þg r r1

þ x_ Tt ðZ1 þ t20 R1 þ t2M R3 Þx_ t 2x_ Tt Z1 Axt þ 2xT ðt1 ÞðN1T þ N2 Þxt

þ r2

þ 2xT ðt2 ÞðN3T þ N4 Þxt þ xT ðt1 Þðb0 DT ðQ2 þZ2 ÞDN2 N2T Þxðt1 Þ þ xT ðt2 ÞðN4 N4T þð1b0 ÞDT ðQ2 þZ2 ÞDÞxðt2 Þ Z t _ dsþ 2ðxTt N3 þxT ðt2 ÞN4 Þ þ 2ðxTt N1 þxT ðt1 ÞN2 Þ xðsÞ Z

Z _ ds xðsÞ

t

tt2 ðtÞ

Z 

tt1 ðtÞ

t tt1 ðtÞ

t tt2 ðtÞ

T Z _ ds R1 xðsÞ

T Z _ ds R3 xðsÞ

tt2 ðtÞ

 _ ds þ yTt ½Q2 CCQ 2 þ Q2 xðsÞ

þ W1 þ W1T þ W3 þW3T yt þ y_ Tt ðZ2 þ 20 R2 þ 2M R4 Þy_ t 2y_ Tt Z2 Cyt þ 2yT ð 1 ÞðW1T þ W2 Þyt þ2yT ð 2 ÞðW3T þW4 Þyt þ yT ð 1 ÞðW2 W2T Þyð 1 Þ þ yT ð 2 ÞðW4 W4T Þyð 2 Þ þ 0 g T ðyð 1 ÞÞBT ðQ1 þ Z1 ÞBgðyð 1 ÞÞ þð1 0 Þg T ðyð 2 ÞÞBT Z t _ ds ðQ1 þ Z1 ÞBgðyð 2 ÞÞ þ2ðyTt W1 þyT ð 1 ÞW2 Þ yðsÞ ts1 ðtÞ Z t T Z t _ ds _ ds yðsÞ yðsÞ þ 2ðyTt W3 þ yT ð 2 ÞW4 Þ ts2 ðtÞ ts1 ðtÞ Z t  Z t T Z t 

s s

s

s

s

a

s

s s

s

s

s s

a

s

s

R2

_ ds  yðsÞ

_ ds yðsÞ

ts1 ðtÞ

_ ds yðsÞ

R4

ts2 ðtÞ T

T

b0 ÞDðxðt1 Þxðt2 ÞÞ ¼ x1 ðtÞX1 x1 ðtÞ þ x2 ðtÞX2 x2 ðtÞ þ 2ðaðtÞa0 Þx_ Tt Z1 Bðgðyðs1 ÞÞgðyðs2 ÞÞÞÞ þ 2ðbðtÞb0 Þy_ Tt Z2 Dðxðt1 Þxðt2 ÞÞ:

ð42Þ

T

ð43Þ

where "

Z

T  Z _ ds xðsÞ

t tt1 ðtÞ

t

T # _ ds ; xðsÞ

tt2 ðtÞ

EfVðtÞg ZEflmin ðQ1 ÞJxt J2 þ lmin ðQ2 ÞJyt J2 g:

ð48Þ

Combining with (47) and (48) leads to ( EfJxt J2 þ Jyt J2 g r r^ eet E þ

sup

_ ðsÞJ2 Þ ðJfðsÞJ2 þJf

sup

t M r s r 0

_ ðsÞJ2 Þg; ðJcðsÞJ2 þJc

ð49Þ

sM r s r 0

where r^ ¼ maxfr1 ; r2 g=rm , proof is completed. &

rm ¼ minflmin ðQ1 Þ; lmin ðQ2 Þg. The

 2 T T (1) Y11 [Y11 ¼ Q1 AAQ 1 þða0 þ 1a0 ÞQ1 þ N1 þN1 þN3 þ N3 ;

^ are unchanged; all the rest Yii and Y jj  T T (2) Y11 [Y11 ¼ Q1 AAQ 1 þð12a0 ÞQ1 þ N1 þ N1 þ N3 þN3 ; 

^ ¼ BT Q1 B þ a0 BT Z1 B2D1 ; ^ 55 [Y Y 55 

^ ¼ BT Q1 B þ ð1a0 ÞBT Z1 B2D2 ; ^ 66 [Y Y 66

h xT2 ðtÞ ¼ yTt y_ Tt yT ðs1 Þ yT ðs2 Þ g T ðyðs1 ÞÞ g T ðyðs2 ÞÞ Z t T Z t T # _ ds _ ds : yðsÞ yðsÞ ts1 ðtÞ

Meanwhile, it follows from (20)



EfLVðtÞg r Efx1 ðtÞX1 x1 ðtÞ þ x2 ðtÞX2 x2 ðtÞg;

xT1 ðtÞ ¼ xTt x_ Tt xT ðt1 Þ xT ðt2 Þ

r2 ¼ maxflmax ðQ2 Þ; s20 lmax ðR2 Þ þ s2M lmax ðR4 Þg:

^ ¼ BT Q1 B þ a0 BT Z1 B2D1 ; ^ 55 [Y Y 55

Taking expectation on both side of (42) yields T

ð47Þ

Corollary 1. The origin of the system (17) is said to be globally exponentially stable in the MSS, if there exist two diagonal matrices D1 g0, D2 g0, positive definite matrices Q1 , Q2 , Z1 , Z2 , Ri ði ¼ 1; 2; 3; 4Þ, an arbitrary matrices Ni , Wi ði ¼ 1; 2; 3; 4Þ with appropriate dimensions, respectively, such that LMIs (18) and (19) ^ ðj ¼ 1; 2; 3; 4; 5; 6Þ replaced by one of hold with Yii ði ¼ 1; 2; 3; 4Þ, Y jj the following denotations:

ts2 ðtÞ

2g T ðyðs1 ÞÞD1 gðyðs1 ÞÞ þ2y ðs1 ÞD1 ðU 1 þ U 2 Þgðyðs1 ÞÞ 2yT ðs1 ÞU 1 D1 U 2 yðs1 Þ2g T ðyðs2 ÞÞD2 gðyðs2 ÞÞ þ 2yT ðs2 ÞD2 ðU 1 þ U 2 Þgðyðs2 ÞÞ2yT ðs2 ÞU 1 D2 U 2 yðs2 Þ þ 2x_ Tt Z1 ðaðtÞa0 ÞBðgðyðs1 ÞÞgðyðs2 ÞÞÞÞ þ 2y_ Tt Z2 ðbðtÞ T

_ ðsÞJ2 g; EfJcðsÞJ2 þJc

r1 ¼ maxflmax ðQ1 Þ; t20 lmax ðR1 Þ þ t2M lmax ðR3 Þg;

tt1 ðtÞ

t

sup sM r s r 0

_ ðsÞJ2 g EfJfðsÞJ2 þJf

where

 _ ds xðsÞ

t

sup tM r s r 0

^ are unchanged; all the rest Yii and Y jj  2 T T (3) Y11 [Y11 ¼ Q1 AAQ 1 þð1 þ a0 ÞQ1 þ N1 þ N1 þ N3 þN3 ;

ts2 ðtÞ



^ ¼ BT Q1 B þ a0 BT Z1 B2D1 ; ^ 55 [Y Y 55

Therefore, we have EfLVðtÞg r l1 EfJxt J2 þJx_ t J2 gl2 EfJyt J2 þJy_ t J2 g;



ð44Þ

^ ¼ ð1a0 Þ2 BT Q1 B þ ð1a0 ÞBT Z1 B2D2 ; ^ 66 [Y Y 66 ^ are unchanged; all the rest Yii and Y jj

where l1 ¼ lmin fX1 g; l2 ¼ lmin fX2 g. Now let us define YðtÞ ¼ eet VðtÞ. Its infinitesimal operator L is given by LYðtÞ ¼ eeet VðtÞ þ eet LVðtÞ:

ð45Þ

Y33 [Y33 ¼ DT Q2 D þ b0 DT Z2 DN2 N2T ; 

^ ¼ Q2 CCQ þðb2 þ1b ÞQ2 þ W1 þW T þ W3 þ W T ; ^ 11 [Y Y 2 0 11 0 1 3

Integrating both sides of (45) and taking expectation gives EfYðtÞgEfYð0Þg ¼

 2 T T (4) Y11 [Y11 ¼ Q1 AAQ 1 þða0 þ 1a0 ÞQ1 þ N1 þN1 þN3 þ N3 ;



Z

t 0

r

^ ¼ BT Q1 B þ a0 BT Z1 B2D1 ; ^ 55 [Y Y 55

Efeees VðsÞ þees LVðsÞgds

Z

t 0

ees E

^ are unchanged; all the rest Yii and Y jj

 ðelmax ðQ1 Þl1 Þsup JxðsÞJ2 s40 2

þ ðelmax ðQ2 Þl2 Þsup JyðsÞJ g ds: s40

 T T (5) Y11 [Y11 ¼ Q1 AAQ 1 þð12a0 ÞQ1 þ N1 þ N1 þ N3 þN3 ;

ð46Þ

Y33 [Y33 ¼ DT Q2 D þ b0 DT Z2 DN2 N2T ;

ARTICLE IN PRESS X. Lou et al. / Neurocomputing 73 (2010) 759–769

Y44 [Y44 ¼ DT Q2 Dþ ð1b0 ÞDT Z2 DN4 N4T ; ^ ^ 11 [Y Y 11

such that the following LMIs: 2

¼ Q2 CCQ 2 þ ð12b0 ÞQ2 þ W1 þ W1T þW3 þ W3T ;



^ ¼ BT Q1 B þ ð1a0 ÞBT Z1 B2D2 ; ^ 66 [Y Y 66 ^ are unchanged; all the rest Yii and Y jj (6)

P11

6 Z A 1 6 6 6 N1T þ N2 6 6 NT þ N 4 6 3 6 N1T X^ 1 ¼ 6 6 6 N3T 6 6 6 HT Q1 6 6 0 4



^ ¼ BT Q1 B þ a0 BT Z1 B2D1 ; ^ 55 [Y Y 55

Y11 [Y11

765

2 T T 0 ÞQ1 þ N1 þ N1 þN3 þ N3 ;

¼ Q1 AAQ 1 þ ð1 þ a

%

%

%

%

%

%

%

P22

%

%

%

%

%

%

%

0

P33

%

%

%

%

%

%

0 0

0 N2T

P44

%

%

%

%

%

0

R1

%

%

%

%

0

0

N4T

0

R3

%

%

%

HT Z1 0

0

0 0

0 0

0 0

k1 I 0

%

%

C1

P55

%

0

0

C2

0

0

0

0

P66

0

Y33 [Y33 ¼ DT Q2 Dþ b0 DT Z2 DN2 N2T ; 2

^ 11 P

%

3 7 7 7 7 7 7 7 7 7!0; 7 7 7 7 7 7 7 5

ð51Þ 3

6 6 Z2 C 6 6 6 W1T þ W2 6 6 W T þ W 6 4 3 6 6 0 6 X^ 2 ¼ 6 6 0 6 6 W1T 6 6 6 W3T 6 6 6 HT Q2 6 6 0 4 0

%

%

%

%

%

%

%

%

%

%

^ 22 P

%

%

%

%

%

%

%

%

%

0

^ 33 P

%

%

%

%

%

%

%

%

0

0

^ 44 P

%

%

%

%

%

%

%

0

D1 ðU 1 þU 2 Þ

0

^ 55 P

%

%

%

%

%

%

%

%

%

%

%

0

0

D2 ðU 1 þ U 2 Þ

0

^ 66 P

0 0

W2T

0 W4T

0 0

0 0

R2 0

%

%

%

%

R4

%

%

%

0

HT Z2

0

0

0

0

0

0

k4 I

%

%

0

0

0

^1 C

0

0

0

^ 77 P

%

0

0

0

0

0 ^2 C

0

0

0

0

^ 88 P

7 7 7 7 7 7 7 7 7 7 7 7 7!0; 7 7 7 7 7 7 7 7 7 7 5

Y44 [Y44 ¼ ð1b0 Þ2 DT Q2 Dþ ð1b0 ÞDT Z2 DN4 N4T ;

hold, where

^ ^ 11 [Y Y 11

P11 ¼ Q1 AAQ 1 þ Q1 þN1 þ N1T þN3 þ N3T þ k1 ET1 E1 ;

2

¼ Q2 CCQ 2 þ ð1 þ b0 ÞQ2 þ W1 þ W1T þ W3 þ W3T ;



P22 ¼ Z1 þ t20 R1 þ t2M R3 ;



P33 ¼ b0 DT ðQ2 þ Z2 ÞDN2 N2T þ k2 ET4 E4 ;

^ ¼ BT Q1 B þ a0 BT Z1 B2D1 ; ^ 55 [Y Y 55 ^ ¼ ð1a0 Þ2 BT Q1 B þð1a0 ÞBT Z1 B2D2 ; ^ 66 [Y Y 66 ^ are unchanged; all the rest Yii and Y jj

P44 ¼ ð1b0 ÞDT ðQ2 þ Z2 ÞDN4 N4T þ k3 ET4 E4 ; P55 ¼ b0 ET4 ðQ2 þ Z2 ÞE4 k2 I;

where ‘‘ [’’ means ‘‘is replaced by’’. Proof. (1) Choose the same Lyapunov functional used in (20) and the proof is similar to that in Theorem 1. The only difference is to deal with (26) as follows: 2xTt ða0 IÞðQ1 Bgðyðs1 ÞÞÞ r a20 xTt Q1 xt þ g T ðyðs1 ÞÞBT Q1 Bgðyðs1 ÞÞ:

ð50Þ

In the same way, we can easily prove (2)–(6) by coping with (26)–(29) in some proper different ways. & Remark 4. Note that one can also carry out many analogous results through dealing with (32)–(35) or various combinations with (26)–(29) in different ways with the help of Lemma 1. 4. Robust exponential stability of GRNs with parametric uncertainties In this section, using Lemma 4, we extend the result of the above section to the case with time-varying parametric uncertainties, that is, we consider the GRES in the MSS of GRN (12). Theorem 2. The origin of the system (12) is said to be globally robustly exponentially stable in the MSS, if there exist two diagonal matrices D1 g0, D2 g0, positive definite matrices Q1 , Q2 , Z1 , Z2 , Ri ði ¼ 1; 2; 3; 4Þ, an arbitrary matrices Ni , Wi ði ¼ 1; 2; 3; 4Þ with appropriate dimensions and positive scalars ki ði ¼ 1; 2; 3; 4; 5; 6Þ

P66 ¼ ð1b0 ÞET4 ðQ2 þ Z2 ÞE4 k3 I; C1 ¼ b0 ET4 ðQ2 þ Z2 ÞDT ; C2 ¼ ð1b0 ÞET4 ðQ2 þ Z2 ÞDT ; ^ 11 ¼ Q2 CCQ þ Q2 þ W1 þW T þ W3 þ W T þ k4 ET E3 ; P 2 1 3 3 ^ 22 ¼ Z2 þ s2 R2 þ s2 R4 ; P 0 M ^ 33 ¼ 2U 1 D1 U 2 W2 W T þ k5 ET E2 ; P 2 2 ^ 44 ¼ 2U 1 D2 U 2 W4 W T þ k6 ET E2 ; P 4 2 ^ 55 ¼ a0 BT ðQ1 þ Z1 ÞB2D1 ; P ^ 66 ¼ ð1a0 ÞBT ðQ1 þ Z1 ÞB2D2 ; P ^ 77 ¼ a0 ET ðQ1 þZ1 ÞE2 k5 I; P 2 ^ 88 ¼ ð1a0 ÞET ðQ1 þ Z1 ÞE2 k6 I; P 2 ^ 1 ¼ a0 ET ðQ1 þ Z1 ÞBT ; C 2 ^ 2 ¼ ð1a0 ÞET ðQ1 þZ1 ÞBT : C 2

ð52Þ

ARTICLE IN PRESS 766

X. Lou et al. / Neurocomputing 73 (2010) 759–769

Let f ðsÞ ¼ 0:7s2 =ð1 þ s2 Þ, then it is easy to get U 1 ¼ diagð0; 0; 0Þ, U 2 ¼ diagð0:455; 0:455; 0:455Þ. And assume that time-varying delays tðtÞ A ½0; 0:4 and sðtÞ A ½0; 0:3 are not continuously differentiable. Clearly, tm ¼ 0, tM ¼ 0:4, sm ¼ 0, sM ¼ 0:3. Moreover, suppose both tðtÞ and sðtÞ are uniform distribution with t0 ¼ 0:1, b0 ¼ 0:7, s0 ¼ 0:1, a0 ¼ 0:8.

Moreover, we have EfJxt J2 þJyt J2 g r r^ eet E ( sup tM r s r 0

_ ðsÞJ2 Þ þ ðJfðsÞJ2 þ Jf

sup s M r s r 0

_ ðsÞJ2 Þg; ðJcðsÞJ2 þ Jc

with the maximal exponential convergence rate ( ) l^ 1 l^ 2 ; ; e ¼ min lmax ðQ1 Þ lmax ðQ2 Þ

Proof. Choose the same Lyapunov functional (20) in the proof of Theorem 1, and replace A, B, C and D by A þ DAt , B þ DBt , C þ DCt and Dþ DDt , respectively. In addition, using the condition F T ðtÞFðtÞ%I from (9), we have

Since the delays are not continuously differentiable, those stability criteria in [13] are not applicable here. Furthermore, there is no feasible solution by using LMI conditions presented in Theorem 1 of Ref. [12]. Therefore, we cannot affirm whether this genetic network is asymptotically stable or not. However, by utilizing Theorem 1 derived in this paper, we solve LMIs (18) and (19) via Matlab LMI toolbox and obtain a feasible solution given as follows:

0 r k1 xTt ET1 E1 xt k1 ðFðtÞE1 xt ÞT ðFðtÞE1 xt Þ;

Q1 ¼ diagð0:1017; 0:0831; 0:1052Þ;

0 r k2 xT ðt1 ÞET4 E4 xðt1 Þk2 ðFðtÞE4 xðt1 ÞÞT ðFðtÞE4 xðt1 ÞÞ;

Q2 ¼ diagð0:1017; 0:0831; 0:1052Þ;

^ 1 g; l^ 2 ¼ lmin fX ^ 2 g, r^ 40 is defined in (49). where l^ 1 ¼ lmin fX

T

T T 2 ÞE4 E4 xð 2 Þk3 ðFðtÞE4 xð 2 ÞÞ ðFðtÞE4 xð 2 ÞÞ;

0 r k3 x ð t

t

t

t

Z1 ¼ diagð0:0110; 0:0112; 0:0086Þ;

0 r k4 yTt ET3 E3 yt k4 ðFðtÞE3 yt ÞT ðFðtÞE3 yt Þ; 0 r k5 g T ðyðs1 ÞÞET2 E2 gðyðs1 ÞÞk5 ðFðtÞE2 gðyðs1 ÞÞÞT ðFðtÞE2 gðyðs1 ÞÞÞ;

0.35

T T 2 ÞÞE2 E2 gðyð 2 ÞÞk6 ðFðtÞE2 gðyð 2 ÞÞÞ ðFðtÞE2 gðyð 2 ÞÞÞ:

T

0 r k6 g ðyðs

s

s

s

0.3

Calculating EfLVðtÞg together with the above inequalities, after some arrangements, we obtain ð53Þ

where "

zT1 ðtÞ ¼ xTt x_ Tt xT ðt1 ÞxT ðt2 Þ Z

t tt2 ðtÞ

Z

T _ ds xðsÞ

t tt1 ðtÞ

# T T T T _xðsÞ ds ðFðtÞE1 xt Þ ðFðtÞE4 xðt1 ÞÞ ðFðtÞE4 xðt2 ÞÞ ;

0.25 Variation of σ (t)

T ^ 1 z ðtÞ þ zT ðtÞX ^ 2 z ðtÞg; EfLVðtÞg r Efz1 ðtÞX 1 2 2

0.05

h

T Z _ ds yðsÞ

0

T _ ds ðFðtÞE3 yt ÞT yðsÞ ts1 ðtÞ ts2 ðtÞ i ðFðtÞE2 gðyðs1 ÞÞÞT ðFðtÞE2 gðyðs2 ÞÞÞT : t

t

0

A ¼ diagð5:1; 4:6; 5:5Þ;

C ¼ diagð7:1; 7:8; 7:3Þ;

D ¼ diagð0:8; 0:9; 0:7Þ; 0

6 B ¼ 4 2:5 0

0

2:5

0

0

2:5

0

3 7 5;

2

2:5

3

7 U¼6 4 2:5 5: 2:5

15

20

0.45 0.4 0.35 Variation of τ (t)

5. Two examples

Example 1. Consider a three-node genetic network (4) without parameter uncertainties:

10

Fig. 2. The variation of sðtÞ with s0 ¼ 0:1 and a0 ¼ 0:8.

Remark 5. As analysis in Corollary 1 and Remark 4, one can deduce a series of corollaries for Theorem 2.

In this section, two different genetic networks with random delays are proposed to illustrate the effectiveness and advantages of the obtained results.

5

time

The following proof for exponential convergence rate and exponential stability definition form is similar to those in Theorem 1. For the sake of simplicity, we omit it here. &

2

0.15 0.1

zT2 ðtÞ ¼ yTt y_ Tt yT ðs1 Þ yT ðs2 Þ g T ðyðs1 ÞÞ g T ðyðs2 ÞÞ Z

0.2

0.3 0.25 0.2 0.15 0.1 0.05 0

0

5

10 time

15

Fig. 3. The variation of tðtÞ with t0 ¼ 0:1 and b0 ¼ 0:7.

20

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0.8

Z2 ¼ diagð0:0097; 0:0072; 0:0099Þ; M1 M2 M3

0.75 0.7 mRNA concentrations

767

D1 ¼ diagð0:4050; 0:3416; 0:4081Þ; D2 ¼ diagð0:1234; 0:1019; 0:1257Þ;

0.65

R1 ¼ diagð0:1127; 0:1181; 0:0902Þ;

0.6 0.55

R2 ¼ diagð0:1781; 0:1387; 0:1808Þ;

0.5

R3 ¼ diagð0:0272; 0:0280; 0:0211Þ;

0.45

R4 ¼ diagð0:0320; 0:0250; 0:0326Þ:

0.4 0.35 0.3

0

5

10

15

20

time Fig. 4. Trajectories of M1 , M2 , M3 of the genetic network.

Therefore, all conditions in Theorem 1 are satisfied, which implies the above genetic network is globally exponentially stable in the MSS with equilibrium ðMT ; P T Þ ¼ ½0:4893; 0:5418; 0:4528; 0:0551; 0:0625; 0:0434. Figs. 2 and 3 show the variations of random delays sðtÞ and tðtÞ, respectively. Figs. 4 and 5 depict the trajectories of mRNA concentrations M1 , M2 , M3 and protein concentrations P1 , P2 , P3 under the initial values ½0:3; 0:8; 0:5T and ½0:2; 0:7; 0:5T , respectively. 0.4

0.8

0.3 Variation of τ (t)

0.6 protein concentrations

0.35

P1 P2 P3

0.4 0.2 0

0.25 0.2 0.15 0.1

−0.2

0.05

−0.4

0 −0.6

0 0

5

10

15

5

20

time

10 time

15

20

Fig. 7. The variation of tðtÞ with t0 ¼ 0:2 and b0 ¼ 0:5.

Fig. 5. Trajectories of P1 , P2 , P3 of the genetic network.

1.2

0.4 0.35

mRNA concentrations

Variation of σ (t)

0.3 0.25 0.2 0.15

0.8 0.6 0.4 0.2

0.1

0

0.05

−0.2

0

M1 M2 M3

1

−0.4 0

5

10

15

time Fig. 6. The variation of sðtÞ with s0 ¼ 0:2 and a0 ¼ 0:5.

20

0

5

10

15

time Fig. 8. Trajectories of M1 , M2 , M3 of the genetic network.

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X. Lou et al. / Neurocomputing 73 (2010) 759–769

0.8

6. Conclusions P1 P2 P3

protein concentrations

0.6

Several delay-distribution-dependent criteria are exploited for ensuring global exponential stability in the MSS and GRES in the MSS of a class of genetic networks with time-varying delays when the information of the probability distributions of the delays is known a priori. The merit of the proposed conditions lies in mild assumption of activation functions and their less conservativeness, which is achieved by utilizing much information of delays and introducing relaxation matrices. Moreover, the proposed method allows the time-varying delays to be not differentiable. For the reason, our results possess highly important significance. The usefulness and advantages of the obtained results are also illustrated through two examples.

0.4 0.2 0 −0.2 −0.4 −0.6 0

5

10 time

15

20

Fig. 9. Trajectories of P1 , P2 , P3 of the genetic network.

Example 2. Consider a five-node genetic network (12) with random delays and parameter uncertainties: A ¼ 6I;

C ¼ 5I;

2

0 6 1 6 6 B¼6 6 0 6 4 1

D ¼ 0:3I;

1

1

0

0

0

1

1

0

0

0

1

0

0

1

1

0

0

3

1 7 7 7 1 7 7; 7 0 5 0

þ U ¼ ½1:92; 2:88; 0:96; 1:92; 0:96T , fi ðsÞ ¼ s2 =ð1 þ s2 Þ with u i ¼ 0; ui

¼ 0:65; i ¼ 1; 2; 3; 4; 5. Uncertain parameters are given by H ¼ 0:1I; 2

E2 ¼ 0:02I;

0:3 6 0:2 6 6 E1 ¼ 6 6 0:1 6 4 0 0:1 2

0:1

6 0 6 6 E3 ¼ 6 6 0:1 6 4 0 0:3

E4 ¼ 0:01I;

0:2

0:1

0:1

0:3

0:2

0:1

0:2 0:2

0:4 0:3

0:2 0:2

0

0:1

0:2

0:2

0

0

0

0:1

0 0:2

0

0:1

0:2

0

0

0

0:2

0:1

0:2

3

0:2 7 7 7 0:1 7 7; 7 0 5 0:3 0

3

0:2 7 7 7 0 7 7; 7 0:1 5 0:1

FðtÞ ¼ diagðsinðtÞ; cosðtÞ; sinðt 2 Þ; cosð3tÞ; cosð2tÞÞ: Moreover, both tðtÞ and sðtÞ are uniform distribution within [0, 0.4] and both of them exist in the first half interval with 0.5 probability, that is, t0 ¼ 0:2, b0 ¼ 0:5, s0 ¼ 0:2, a0 ¼ 0:5. By solving LMIs (51) and (52) via Matlab LMI toolbox, we get feasible solutions which will not be shown here for simplicity. Therefore, all conditions in Theorem 2 are satisfied, which implies the above genetic network is globally exponentially stable in the MSS with equilibrium ðM T ; P T Þ ¼ ½0:3212; 0:4786; 0:1600; 0:0193; 0:0287; 0:0096. The computational simulation results of random delays and trajectories of concentrations can be seen in Figs. 6–9 with the initial values ½1:2; 0:3; 0:3; 0:2; 0:7; 0:5T .

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Xuyang Lou received the B.S. degree from Zhejiang Ocean University, China, in 2004 and visited CSIRO Division of Mathematical and Information Sciences, Waite Campus of Adelaide University from October 2007 to October 2008. He is now pursuing the Ph.D. degree in School of Communication and Control Engineering, Jiangnan University, China. His interests lie in the general area of mathematical analysis of dynamical systems.

769 Qian Ye was born in 1985. She is now pursuing the Ph.D. degree in College of Communication and Control Engineering, Jiangnan University, China, and received the M.Sc. degree from Jiangnan University, China, in June 2009. Her current research interests include networked control systems and genetic regulatory networks.

Baotong Cui received the Ph.D. degree in control theory and control engineering from the College of Automation Science and Engineering, South China University of Technology, China, in July 2003. He was a post-doctoral fellow at Shanghai Jiaotong University, China, from July 2003 to September 2005 and a visiting scholar at Department of Electrical and Computer Engineering, National University of Singapore from August 2007 to February 2008. He became an associate professor in December 1993 and a full professor in November 1995 at Department of Mathematics, Binzhou University, Shandong, China. He joined the College of Communication and Control Engineering, Jiangnan University, China, in June 2003, where he is a full professor for the College of Communication and Control Engineering. His current research interests include systems analysis, stability theory, impulsive control, artificial neural networks, and chaos synchronization.