Stability analysis for switched genetic regulatory networks: An average dwell time approach

Journal of the Franklin Institute 348 (2011) 2718–2733 www.elsevier.com/locate/jfranklin

Stability analysis for switched genetic regulatory networks: An average dwell time approach$ Yingtao Yao, Jinling Liang, Jinde Cao Department of Mathematics, Southeast University, Nanjing 210096, PR China Received 26 July 2010; received in revised form 27 March 2011; accepted 23 April 2011 Available online 27 July 2011

Abstract In this correspondence, the problem of exponential stability for switched genetic regulatory networks (GRNs) with time delays is investigated. The GRNs are composed of N modes and the network switches from one mode to another. By employing the piecewise Lyapunov functional method combined with the average dwell time approach and by using a novel Lyapunov–Krasovskii functional (LKF), sufficient criteria are given to ensure the exponential stability for the switched GRNs with constant and time-varying delays, respectively. These criteria are proved to be much less conservative than the most recent results, since the results reported in this paper not only depend on the delay bounds, but also depend on the partitioning. All the conditions presented here are in the form of matrix inequalities which are easy to be verified via the Matlab toolbox. Two examples are provided in the end of this paper to illustrate the effectiveness of the obtained theoretical results. & 2011 Published by Elsevier Ltd. on behalf of The Franklin Institute.

1. Introduction Genetic regulatory networks have become an important new area of research in the biological and biomedical sciences in the passed few years. Theoretical studies on genetic networks may not only contribute to the understanding of the gene functions, but also

$

This work was supported by the Teaching and Research Fund for Excellent Young Teachers at Southeast University of China and the National Natural Science Foundation of China under Grant 60804028. Corresponding author. Tel.:þ862583792315; fax:þ862583792316. E-mail address: [email protected] (J. Liang). 0016-0032/$32.00 & 2011 Published by Elsevier Ltd. on behalf of The Franklin Institute. doi:10.1016/j.jfranklin.2011.04.016

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have potential significance on engineering applications, such as developing circuits and systems with biotechnological design principles of synthetic genetic regulatory networks and new kinds of integrated circuits like neurochips learnt from biological neural networks [1–3]. Therefore, a great number of results on stability of GRNs have been proposed in the literature [4–6]. In practice, when we deal with the dynamic behaviors of GRNs, the time delay is often unavoidable because of the finite speed of information processing, which may cause oscillation and instability [7–13]. In addition, as shown in [14–19], a network may exhibit a special characteristic called network mode switching. In other words, the network switch from one mode to another in accordance with uncertain transition probabilities. Therefore, it is essential to investigate the delayed GRNs with switching parameters. Recent studies on GRNs with or without switching parameters are fruitful, and many important results have been reported in the literature. In [20], both monostability and multistability are analyzed in a unified framework by applying control theory and mathematical tools; the genetic regulatory networks with multiple time-varying delays and different types of regulation functions are considered. A hybrid GRN model based on Markov chain is proposed in [21], which takes into account the structure variations at discrete time instances during the process of gene regulation. In [22], the robust state estimation problem is concerned for uncertain time-delay Markovian jumping GRNs with SUM logic, where the uncertainties enter into both the network parameters and the mode transition rate. It should be noted that the mode transition rate related in [21,22] are known. For more information of the GRNs, we refer to Refs. [23–27]. However, to the best of the authors’ knowledge, there have been very few results on the exponential stability problem for delayed GRNs with switching parameters with unknown mode transition rate by using the average dwell time approach, and the purpose of this paper is therefore to shorten such a gap. Motivated by the above discussion, in this paper, we are concerned with the exponential stability of delayed GRNs, where the parameter values switch from one mode to another. By utilizing the Lyapunov functional approach and average dwell time approach and by using a novel Lyapunov–Krasovskii functional, it is shown that the exponential stability problem is solvable if a set of linear matrix inequalities (LMIs) are feasible. The features of this paper can be summarized as follows: the piecewise Lyapunov functional is used; the average dwell time approach is concerned; all the conditions obtained depend not only upon the time delays, but also upon the partitioning. Two examples are provided in the end of the paper to show the effectiveness of the proposed criteria. The rest of this paper is organized as follows: In Section 2, preliminaries and problem formulation are given. In Section 3, some conditions are established to ensure the exponential stability of the genetic system. In Section 4, two examples are illustrated to show the effectiveness of the obtained theoretical results. And finally, conclusions are given in Section 5. Notations: I and O denotes, respectively, the identity matrix and the zero matrix with appropriate dimensions. diagf   denotes the diagonal matrix. The notation P4Q (respectively PZQ), where P and Q are symmetric matrices, means that the matrix PQ is positive definite (respectively, positive semi-definite). The superscript T stands for matrix transposition. J  J denotes the Euclidean norm in Rn , J  J1 denotes the spectral norm for matrices. If A is a square matrix, lmax ðAÞ (respectively, lmin ðAÞ) is used to represent the largest (respectively, smallest) eigenvalue of A, sym(A) is used to represent A þ AT .

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2. Problem formulation and preliminaries In this letter, we will consider the following nonlinear GRNs [7]: ( _ ¼ AmðtÞ þ Bf ðpðttÞÞ þ L, mðtÞ _ ¼ CpðtÞ þ DmðtsÞ, pðtÞ

ð1Þ

where mðtÞ ¼ ½m1 ðtÞ,m2 ðtÞ, . . . ,mn ðtÞT ,pðtÞ ¼ ½p1 ðtÞ,p2 ðtÞ, . . . ,pn ðtÞT ,mi ðtÞ,pi ðtÞ 2 R are the concentrations of mRNA and protein of the ith node, respectively; A ¼ diagða1 ,a2 , . . . ,an Þ and C ¼ diagðc1 ,c2 , . . . ,cn Þ, ai , ci are the rates of degradation; D ¼ diagðd1 ,d2 , . . . ,dn Þ,di is the translation rate; B ¼ ðbij Þnn is the coupling matrix of genetic network which is defined as follows: if transcription factor j is an activator of gene i, bij ¼ lij ; if transcription factor j is a repressor of gene i, bij ¼ lij ; if there is no link from gene j to gene i, bij ¼ 0; t and s are the time-varying delays. Furthermore, nonlinear function f ðÞ is generally a nonlinear function which has a form of monotonicity with each parameter [28] and represents the feedback regulation of the protein on the transcription. Usually fi ðÞ is of the Michaelis–Menten type or hj hj of the Hill form. In this paper, Pwe take fj ðxÞ ¼ x =ð1 þ x Þ, where hj is the Hill coefficient. T L ¼ ½l1 ,l2 , . . . ,ln  , where li ¼ j2Ii lij and Ii set of all j nodes which are repressors of gene i. Let ðmn ,pn Þ be an equilibrium of Eq. (1), i.e., it satisfies Amn þ Bf ðpn Þ þ L ¼ 0 and Cpn þ Dmn ¼ 0. Denote xðtÞ ¼ mðtÞmn and yðtÞ ¼ pðtÞpn , we shift the equilibrium point of Eq. (1) to the origin and have ( _ ¼ AxðtÞ þ BgðyðttÞÞ, xðtÞ ð2Þ _ ¼ CyðtÞ þ DxðtsÞ, yðtÞ where gðyðtÞÞ ¼ f ðyðtÞ þ pn Þf ðpn Þ. Since f is a monotonically increasing function with saturation 0r

fi ðaÞfi ðbÞ rki ab

ð8a,b 2 R,aabÞ

the function g satisfies the following sector condition: gi ðaÞðgi ðaÞki aÞr0

ð8a 2 R=f0Þ:

ð3Þ

In practice, the gene networks may be a class of dynamical hybrid systems consisting of a family of continuous-time subsystems. In [21–23] the Markov jumping systems have been studied. However, we usually cannot know the transition probabilities. Therefore, in this paper we consider the following switching genetic network with unknown transition probabilities: ( _ ¼ AðaÞxðtÞ þ BðaÞgðyðttÞÞ, xðtÞ ð4Þ _ ¼ CðaÞyðtÞ þ DðaÞxðtsÞ, yðtÞ where ðAðaÞ,BðaÞ,CðaÞ,DðaÞÞ,ða ¼ 1,2, . . . ,NÞ are constant matrices of appropriate dimensions denoting the subsystems, and N is the number of subsystems. It is assumed that the value of aðtÞ is unknown, but its instantaneous value is available in real time. The initial condition of the GRN (4) is assumed to be xðtÞ ¼ j1 ðtÞ,

yðtÞ ¼ C1 ðtÞ,

rrtr0,

r ¼ maxfs,tg:

Remark 1. In this paper, we only know every subsystem of the GRN, but the relation of them (transition probabilities) are not known. Therefore it is less conservative than the

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most recent result. In addition, we wish to point out that the main criteria of this paper can be easily extended to the case where the time delays are different from each other. To obtain the main results of the paper, firstly we need to introduce the following lemma and definition. Lemma 1 (Gu [29]). For any constant matrix M 2 Rnn ,M ¼ M T 40, scalar s40, vector function w : ½0,s-Rn such that the integrations concerned are well defined, then Z s T Z s  Z s wðsÞ ds M wðsÞ ds rs wT ðsÞMwðsÞ ds: ð5Þ 0

0

0

For the switching signal aðtÞ, we visit the average dwell time property from the following definition. Definition 1 (Liberzon [30]). For any T2 4T1 Z0, let Na ðT1 ,T2 Þ denote the number of switchings of aðtÞ over ðT1 ,T2 Þ. If Na ðT1 ,T2 ÞrN0 þ ðT2 T1 Þ=Ta holds for Ta 40,N0 Z0, then Ta is called the average dwell time. Remark 2. In general, the stability of the subsystems themselves is not sufficient for the stability of the overall system. However if the switching signals such that the average time interval between consecutive switchings is at least the average dwell time Ta , then the overall system is stable [31,32]. What we need to do is how to specify the minimal Ta under some conditions. As commonly used in the literature, we choose N0 ¼ 0 in Definition 1. Definition 2. The system (4) is said to be globally exponentially stable, if there exist two constants m40 and l40 such that JzðtÞJrmJzt0 J1C elðtt0 Þ

ð6Þ

holds for all tZ0, where zðtÞ ¼ ½xðtÞ,yðtÞ and JzðtÞJC 1 9suprryr0 fJzðt þ yÞJ,J_z ðt þ yÞJ. T

Remark 3. In this paper, the average dwell time approach is used for the first time in the research of GRNs. With this method, less information is needed about the switched GRNs than the previous work (i.e., only information of each subsystem is to be known, and that the switched probabilities among subsystems are not needed), and we might get a better result. The number of the LMIs in this paper is N and not N 2 as those in the previous literatures, and so it is easier to be solved and the calculation is reduced. Furthermore, the results reported in this paper depend not only on the delay bounds, but also on the partitioning number. By choosing different partitioning numbers, we might obtain a smaller average dwell time Tan . From these points of view, the criteria acquired in this paper are less conservative than most of the recent results. 3. Stability analysis 3.1. Constant time-delay case In this section, we present our new delay-dependent exponential stability criterion for gene network (4) with constant time-delay. The sufficient conditions are summarized in the following theorem.

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Theorem 1. Given an integer mZ1 and a constant b40, the GRN (4) is exponentially stable for any switching signal with average dwell time satisfying Ta 4Tan ¼ ln m=b, if there exist matrices P1 ðaÞ40, P2 ðaÞ40, Q1 ðaÞ40, Q2 ðaÞ40, Q3 ðaÞ40, R1 ðaÞ40, R2 ðaÞ40, and the diagonal matrix G ¼ diagfg1 ,g2 , . . . ,gn 40 satisfying XðaÞ ¼ X1 ðaÞ þ X2 ðaÞ þ X3 ðaÞ þ symðW1T O1 ðaÞW2 þ W1T O2 ðaÞW6 þ 2W2T O3 ðaÞW3 þW2T O4 ðaÞW5 þ W3T O5 ðaÞW4 þ 1=2bW5T GW3 Þo0,

ð7Þ

where a ¼ 1,2, . . . ,N, mZ1 satisfies that 8i,j ¼ 1,2, . . . ,N, P1 ðiÞrmP1 ðjÞ,

P2 ðiÞrmP2 ðjÞ,

Q1 ðiÞrmQ1 ðjÞ,

Q2 ðiÞrmQ2 ðjÞ,

Q3 ðiÞrmQ3 ðjÞ, R1 ðiÞrmR1 ðjÞ, R2 ðiÞrmR2 ðjÞ,  1 X1 ðaÞ ¼ diag Q1 ðaÞ, ebs R1 ðaÞ þ tDT R2 ðaÞD,Q2 ðaÞ, s  1  ebt R2 ðaÞ,Q3 ðaÞ,sBT R1 ðaÞB , t

ð8Þ

X2 ðaÞ ¼ diagfOnn ,ebs=m Q1 ðaÞ,Onn ,ebt=m Q2 ðaÞ,Onn ,ebt=m Q3 ðaÞg,  1 X3 ðaÞ ¼ diag 2P1 ðaÞA þ bP1 ðaÞ þ sAT R1 ðaÞA ebs R1 ðaÞ,Omnmn , s  1 1 T bt 2P2 ðaÞC þ bP2 ðaÞ þ tC R2 ðaÞC e R2 ðaÞ,Omnmn ,GCK ,Omnmn , t W1 ¼ ½Inn ,Onð3mnþ2nÞ ,

W2 ¼ ½Onmn ,Inn ,Onð2mnþ2nÞ ,

W3 ¼ ½OnðmnþnÞ ,Inn ,Onð2mnþnÞ , W5 ¼ ½Onð2mnþ2nÞ ,Inn ,OnðmnÞ , O1 ðaÞ ¼

1 bs e R1 ðaÞ, s

W4 ¼ ½Onð2mnþnÞ ,Inn ,OnðmnþnÞ , W6 ¼ ½Onð3mnþ2nÞ ,Inn ,

O2 ðaÞ ¼ P1 ðaÞBsAT R1 ðaÞB,

O3 ðaÞ ¼ DT P2 ðaÞtDT R2 ðaÞC,

1 T T 1 D G , O5 ðaÞ ¼ ebt R2 ðaÞ: 2 t lt Moreover JzðtÞJrZe Jzð0ÞJC 1 where    1=2 1 ln m l2 b Z1, l1 ¼ min ðlmin ðP1 ðiÞÞ,lmin ðP2 ðiÞÞÞ, l¼ , Z¼ i ¼ 1,2,...,N 2 Ta l1 O4 ðaÞ ¼

l2 ¼ þ

max ðlmax ðP1 ðiÞÞ þ lmax ðP2 ðiÞÞ þ slmax ðQ1 ðiÞÞ þ tlmax ðQ2 ðiÞÞ þ tlmax ðQ3 ðiÞÞ

i ¼ 1,2,...,N

 s2 t2 lmax ðR1 ðiÞÞ þ lmax ðR2 ðiÞÞ þ JGKJ1 : 2 2

ð9Þ

Proof. To prove the theorem, choose a Lyapunov–Krasovskii functional candidate as V ðt,aÞ ¼ V1 ðt,aÞ þ V2 ðt,aÞ þ V3 ðt,aÞ þ V4 ðt,aÞ,

tZ0,

ð10Þ

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where V1 ðt,aÞ ¼ xT ðtÞP1 ðaÞxðtÞ þ yT ðtÞP2 ðaÞyðtÞ, Z

t

ebðstÞ jT ðsÞQ1 ðaÞjðsÞ ds þ

V2 ðt,aÞ ¼

Z

ts=m

Z

t

ebðstÞ fT ðsÞQ2 ðaÞfðsÞ ds

tt=m

t

ebðstÞ cT ðsÞQ3 ðaÞcðsÞ ds,

þ tt=m

Z

Z

0

t

_ ds dy þ ebðstÞ x_ T ðsÞR1 ðaÞxðsÞ

V3 ðt,aÞ ¼ tþy

s

V4 ðt,aÞ ¼

Z

Z

n X

0 t

yi ðtÞ

t

_ ds dy, ebðstÞ y_ T ðsÞR2 ðaÞyðsÞ

tþy

V4 ðt,aÞrgT ðyðtÞÞGyðtÞ,

gi ðsÞ ds,

gi

Z

0

i¼1

   T  s m1 s , , . . . ,x t jðtÞ9 xðtÞ,x t m m    T  t m1 t , fðtÞ9 yðtÞ,y t , . . . ,y t m m     T   t  m1 t cðtÞ9 gðyðtÞÞ,g y t : , . . . ,g y t m m For a fixed a, the derivatives of Vi ðt,aÞ,i ¼ 1,2,3,4 are given by _ þ 2yT ðtÞP2 ðaÞyðtÞ, _ V_ 1 ðt,aÞ ¼ 2xT ðtÞP1 ðaÞxðtÞ V_ 2 ðt,aÞ ¼ b

Z

e Z

Z

t bðstÞ

T

t

ebðstÞ fT ðsÞQ2 ðaÞfðsÞ ds

j ðsÞQ1 ðaÞjðsÞ dsb

ts=m t

tt=m

e

bðstÞ

T

T

c ðsÞQ3 ðaÞcðsÞ ds þ j ðtÞQ1 ðaÞjðtÞ  s s eðbs=mÞ jT t Q1 ðaÞj t m  mt   t T bt=m T þf ðtÞQ2 ðaÞfðtÞe f t Q2 ðaÞf t þ cT ðtÞQ3 ðaÞcðtÞ m m  t  t ebt=m cT t Q3 ðaÞc t , m m Z 0Z t Z 0Z t _ ds dyb _ ds dy ebðstÞ x_ T ðsÞR1 ðaÞxðsÞ ebðstÞ y_ T ðsÞR2 ðaÞyðsÞ V_ 3 ðt,aÞ ¼ b b

tt=m

s



tþy

t

Z

tþy

t

_ _ ds þ ty_ T ðtÞR2 ðaÞyðtÞ _ þsx_ T ðtÞR1 ðaÞxðtÞ ebðstÞ x_ T ðsÞR1 ðaÞxðsÞ ts Z t _ ds  ebðstÞ y_ T ðsÞR2 ðaÞyðsÞ tt Z 0Z t _ ds dy ebðstÞ x_ T ðsÞR1 ðaÞxðsÞ rb s

tþy

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Z

0

Z

t

_ ds dy ebðstÞ y_ T ðsÞR2 ðaÞyðsÞ Z t bs _ _ ds þ ty_ T ðtÞR2 ðaÞyðtÞ _ þsx_ T ðtÞR1 ðaÞxðtÞe x_ T ðsÞR1 ðaÞxðsÞ ts Z t _ ds, ebt y_ T ðsÞR2 ðaÞyðsÞ

b

t

tþy

tt

V_ 4 ðt,aÞ ¼

n X

_ gi gi ðyi ðtÞÞy_ i ðtÞ ¼ gT ðyðtÞÞGyðtÞ:

ð11Þ

i¼1

From Lemma 1, it is obvious that Z Z t Z t 1 t _ dsr _ ds R1 ðaÞ _  x_ T ðsÞR1 ðaÞxðsÞ xðsÞ xðxÞ ds, s ts ts ts Z Z t Z t 1 t _ dsr _ ds R2 ðaÞ _ ds:  y_ T ðsÞR2 ðaÞyðsÞ yðsÞ yðsÞ t tt tt tt

ð12Þ

ð13Þ

The sector condition (3) ensures that for the positive diagonal matrix K ¼ diagfk1 , k2 , . . . ,kn , the following inequality holds: gT ðyðtÞÞGCðaÞyðtÞrgT ðyðtÞÞGCðaÞK 1 gðyðtÞÞ:

ð14Þ

Then from Eqs. (10) to (14) we have V_ ðt,aÞ þ bV ðt,aÞr2xT ðtÞP1 ðaÞ½AðaÞxðtÞ þ BðaÞgðyðttÞÞ þ 2yT ðtÞP2 ðaÞ½CðaÞyðtÞ þDðaÞxðtsÞ þ bxT ðtÞP1 ðaÞxðtÞ þ byT ðtÞP2 ðaÞyðtÞ þ jT ðtÞQ1 ðaÞjðtÞ  s  s Q1 ðaÞj t þ fT ðtÞQ2 ðaÞfðtÞ þ cT ðtÞQ3 ðaÞcðtÞ ebs=m jT t m m

þs½AðaÞxðtÞ þ BðaÞgðyðttÞÞT R1 ðaÞ½AðaÞxðtÞ þ BðaÞgðyðttÞÞ 1  ebs ½xðtÞxðtsÞT R1 ðaÞ½xðtÞxðtsÞ þ t½CðaÞyðtÞ þ DðaÞxðtsÞT R2 ðaÞ s

1 ½CðaÞyðtÞ þ DðaÞxðtsÞ ebt ½yðtÞyðttÞT R2 ðaÞ½yðtÞyðttÞ t

þgT ðyðtÞÞG½CðaÞK 1 gðyðtÞÞ þ DðaÞxðtsÞ þ bgT ðyðtÞÞGyðtÞ ¼ zT ðtÞXðaÞzðtÞ, where zðtÞ ¼ ½jT ðtÞ,xT ðtsÞ,fT ðtÞ,yT ðttÞ,cT ðtÞ,gT ðyðttÞÞT and XðaÞ is defined in Eq. (7). Therefore we have V_ ðt,aÞ þ bV ðt,aÞrzT ðtÞXðaÞzðtÞo0 ð15Þ for all nonzero zðtÞ. Now for any arbitrary piecewise constant switching signal a, and for any t40, we let 0 ¼ t0 ot1 o    otk o    ðk ¼ 1,2, . . .Þ denote the switching points of a over the interval ð0,tÞ. As mentioned earlier, the ik th subsystem is activated when t 2 ½tk ,tkþ1 Þ. Integrating the above inequality (15) from tk to t gives V ðt,aÞrebðttk Þ V ðtk ,aðtk ÞÞ:

ð16Þ

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Using Eqs. (8) and (16), at switching instant tk , we have  V ðtk ,aðtk ÞÞrmV ðt k ,aðtk ÞÞ:

ð17Þ

Therefore, it follows from (16), (17) and the relationship d ¼ Na ð0,tÞrðt0Þ=Ta that  d bðt0Þ V ðt0 ,að0ÞÞreðbln m=Ta Þt V ðt0 ,að0ÞÞ: V ðt,aÞrmebðttk Þ V ðt k ,aðtk ÞÞr    rm e

ð18Þ Notice from (10) that V ðt,aÞrl1 JzðtÞJ2 ,

V ðt0 ,að0ÞÞrl2 Jzð0ÞJ0C1 ,

ð19Þ

where l1 and l2 are defined as in (9). Combining (18) and (19) yields JzðtÞJ2 r

1 l2 V ðt,aÞr eðbln m=Ta Þt Jzð0ÞJ2C1 , l1 l1

which is (6). By Definition 2 with t0 ¼ 0, the switched GRNs in (4) is exponentially stable. The proof is completed. & Remark 4. Theorem 1 presents a new delay-dependent exponential stability criterion for GRNs by using the new Lyapunov–Krasovskii functional defined in (10). We can know that the stability of the subsystems themselves is not sufficient for the stability of the overall system. In [21,22] they all need N2 LMIs. However there are only N LMIs in (7) which can provide the stability of every subsystems. The average dwell time approach is applied to derive the results in this paper. In addition all the conditions obtained depend not only upon the time delays, but also upon the partitioning. Therefore we get a less conservative results. 3.2. Time-varying delay case In this section, we consider the following genetic network with time-varying delays: ( _ ¼ AðaÞxðtÞ þ BðaÞgðyðttðtÞÞÞ, xðtÞ ð20Þ _ ¼ CðaÞyðtÞ þ DðaÞxðtsðtÞÞ, yðtÞ where 0ototðtÞot and 0ososðtÞos are time-varying delays. We assume that _ t_ ðtÞrd1 o1 and sðtÞrd 2 o1. For this network, the main result is summarized in the following theorem. Theorem 2. Given an integer mZ1 and a constant b40, the GRN (20) is exponentially stable for any switching signal with average dwell time satisfying Ta 4Tan ¼ ln m=b, if there exist matrices P1 ðaÞ40, P2 ðaÞ40, Qi ðaÞ40, Mj ðaÞ40, i ¼ 1,2,3,4,5, j ¼ 1,2,3,4, and one diagonal matrix G ¼ diagfg1 ,g2 , . . . ,gn 40 satisfying FðaÞ ¼ F1 ðaÞ þ F2 ðaÞ þ F3 ðaÞ þ symðM1T L1 ðaÞM3 þ M1T L2 ðaÞM10 þ M2T L3 ðaÞM5 þ M2T L4 ðaÞM9 þM3T L5 ðaÞM4 þ M5T L6 ðaÞM7 þ M7T L7 ðaÞM8 þ 1=2bM9T GM5 Þo0,

ð21Þ

where a ¼ 1,2, . . . ,N, mZ1 satisfies that 8m,m ¼ 1,2, . . . ,N, P1 ðmÞrmP1 ðnÞ,

P2 ðmÞrmP2 ðnÞ,

Qi ðmÞrmQi ðnÞ,

Rj ðmÞrmRj ðnÞ

ð22Þ

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F1 ðaÞ ¼ diag Q1 ðaÞ,DT ðaÞðtR2 ðaÞ þ ðttÞR4 ðaÞÞDðaÞ,ebs Q4 ðaÞ 1 1 bs e R3 ðaÞ,  ebs R1 ðaÞ s ss 1 bs 1 1 bt e R3 ðaÞ,Q2 ðaÞOnn ,ebt Q5 ðaÞ ebt R2 ðaÞ e R4 ðaÞ, ebs Q4 ðaÞ ss t tt ebt Q5 ðaÞ

1 bt e R4 ðaÞ,Q3 ðaÞ,BT ðaÞðsR1 ðaÞ þ ðssÞR3 ðaÞÞBðaÞ, tt

F2 ðaÞ ¼ diag 2P1 ðaÞAðaÞ þ AT ðaÞðsR1 ðaÞ þ ðssÞR3 ðaÞÞAðaÞ 1  ebs R1 ðaÞbP1 ðaÞOðmnþ2nÞðmnþ2nÞ ,2P2 ðaÞCðaÞ s þ C T ðaÞðtR2 ðaÞ þ ðttÞR4 ðaÞÞCðaÞ 1  ebt R2 ðaÞ þ bP2 ðaÞ,Oðmnþ2nÞðmnþ2nÞ ,GCðaÞK 1 ,Omnmn , t      d2 d1 bs=m bt=m 1 1 F3 ðaÞ ¼ diag Onn ,e Q1 ðaÞ,O3n3n ,e Q2 ðaÞ, m m    d1 O3n3n ,ebt=m 1 Q3 ðaÞ m M1 ¼ ½Inn ,Onð3mnþ6nÞ ,

M2 ¼ ½Onmn ,Inn ,Onð2mnþ6nÞ ,

M3 ¼ ½OnðmnþnÞ ,Inn ,Onð2mnþ5nÞ ,

M4 ¼ ½Onðmnþ2nÞ ,Inn ,Onð2mnþ4nÞ ,

M5 ¼ ½Onðmnþ3nÞ ,Inn ,Onð2mnþ3nÞ ,

M7 ¼ ½Onð2mnþ4nÞ ,Inn ,Onðmnþ2nÞ ,

M8 ¼ ½Onð2mnþ5nÞ ,Inn ,OnðmnþnÞ ,

M9 ¼ ½Onð2mnþ6nÞ ,Inn ,Onmn ,

M9 ¼ ½Onð3mnþ6nÞ ,Inn , L1 ðaÞ ¼

1 bs e R1 ðaÞ, s

L2 ðaÞ ¼ P1 ðaÞBðaÞAT ðaÞðsR1 ðaÞ þ ðssÞR3 ðaÞÞBðaÞ,

L3 ðaÞ ¼ P2 ðaÞDðaÞC T ðaÞðtR2 ðaÞ þ ðttÞR4 ðaÞÞDðaÞ, L5 ðaÞ ¼

1 bs e R3 ðaÞ, ss

L6 ðaÞ ¼

1 bt e R2 ðaÞ, t

L4 ðaÞ ¼ 12DT ðaÞGT ,

L7 ðaÞ ¼

1 bt e R4 ðaÞ: tt

Proof. To prove the theorem, choose a Lyapunov–Krasovskii functional candidate as V ðt,aÞ ¼ V1 ðt,aÞ þ V2 ðt,aÞ þ V3 ðt,aÞ þ V4 ðt,aÞ,

tZ0,

where V1 ðt,aÞ ¼ xT ðtÞP1 ðaÞxðtÞ þ yT ðtÞP2 ðaÞyðtÞ, Z

t

V2 ðt,aÞ ¼ tsðtÞ=m

ebðstÞ jT ðsÞQ1 ðaÞjðsÞ ds þ

Z

t

ttðtÞ=m

ebðstÞ fT ðsÞQ2 ðaÞfðsÞ ds

ð23Þ

Y. Yao et al. / Journal of the Franklin Institute 348 (2011) 2718–2733

Z

t

ebðstÞ cT ðsÞQ3 ðaÞcðsÞ ds þ

þ þ

Z

ttðtÞ=m Z tt bðstÞ T

e

2727

ts

ebðstÞ xT ðsÞQ4 ðaÞxðsÞ ds ts

y ðsÞQ5 ðaÞyðsÞ ds,

tt

Z

s

Z

t

_ ds dy þ ebðstÞ x_ T ðsÞR1 ðaÞxðsÞ

V3 ðt,aÞ ¼ ty

0

sZ

Z

n X

bðstÞ T

t

_ ds dy ebðstÞ y_ T ðsÞR2 ðaÞyðsÞ

ty t

Z

t

_ ds dy, ebðstÞ y_ T ðsÞR4 ðaÞyðsÞ

_ ds dy þ x_ ðsÞR3 ðaÞxðsÞ

ty

Z

Z

Z

t

e s

t 0

þ

V4 ðt,aÞ ¼

Z

ty

t

yi ðtÞ

gi ðsÞ ds,

gi 0

i¼1

   T   sðtÞ m1 sðtÞ jðtÞ ¼ xðtÞ,x t , , . . . ,x t m m    T   tðtÞ m1 tðtÞ fðtÞ ¼ yðtÞ,y t , , . . . ,y t m m     T    tðtÞ m1 tðtÞ cðtÞ ¼ gðyðtÞÞ,g y t : , . . . ,g y t m m For a fixed a, the derivatives of Vi ðt,aÞ, i ¼ 1,2,3,4 are given by _ þ 2yT ðtÞP2 ðaÞyðtÞ, _ V_ 1 ðt,aÞ ¼ 2xT ðtÞP1 ðaÞxðtÞ Z t Z t bðstÞ T _ V 2 ðt,aÞrb e j ðsÞQ1 ðaÞjðsÞ dsb ebðstÞ fT ðsÞQ2 ðaÞfðsÞ ds tsðtÞ=m Z ts ttðtÞ=m Z t ebðstÞ cT ðsÞQ3 ðaÞcðsÞ dsb ebðstÞ xT ðsÞQ4 ðaÞxðsÞ ds b ts ZttðtÞ=m tt ebðstÞ yT ðsÞQ5 ðaÞyðsÞ ds þ jT ðtÞQ1 ðaÞjðtÞ b      tt  d2 T sðtÞ sðtÞ bs=m e 1 Q1 ðaÞj t þ fT ðtÞQ2 ðaÞfðtÞ j t   m m m  d1 T tðtÞ tðtÞ ebt=m 1 Q2 ðaÞf t þ cT ðtÞQ3 ðaÞcðtÞ f t   m m m  d1 T tðtÞ tðtÞ ebt=m 1 Q3 ðaÞc t þ ebs xT ðtsÞQ4 ðaÞxðtsÞ c t m m m ebs xT ðtsÞQ4 ðaÞxðtsÞ þ ebt yT ðttÞQ5 ðaÞyðttÞebt yT ðttÞQ5 ðaÞyðttÞ, Z sRt Z t R t ebðstÞ y_ T ty ty _ ds dyb _ ds dy V3 ðt,aÞ ¼ b ebðstÞ x_ T ðsÞR1 ðaÞxðsÞ ðsÞR2 ðaÞyðsÞ Z0 t Z t Z 0s Z t _ ds dyb _ ds dy ebðstÞ x_ T ðsÞR3 ðaÞxðsÞ ebðstÞ y_ T ðsÞR4 ðaÞyðsÞ b s

ty

_ þs x_ T ðtÞR1 ðaÞxðtÞ

Z

t

ty

t

_ ds þ t y_ T ðtÞR2 ðaÞyðtÞ _ ebs x_ T ðsÞR1 ðaÞxðsÞ ts

Y. Yao et al. / Journal of the Franklin Institute 348 (2011) 2718–2733

2728

Z

t

_ ds þ ðssÞx_ T ðtÞR3 ðaÞxðtÞ _ ebt y_ T ðsÞR2 ðaÞyðsÞ

 tt

_ þðttÞy_ T ðtÞR4 ðaÞyðtÞ

Z

Z

ts

_ ds ebs x_ T ðsÞR3 ðaÞxðsÞ ts

t

_ ds, ebt y_ T ðsÞR4 ðaÞyðsÞ

tt

V_ 4 ðt,aÞ ¼

n X

_ gi gi ðyi ðtÞÞy_ i ðtÞ ¼ gT ðyðtÞÞGyðtÞ:

ð24Þ

i¼1

From Lemma 1, it is obvious that Z t 1 _ dsr ½xðtÞxðtsÞT R1 ðaÞ½xðtÞxðts,  x_ T ðsÞR1 ðaÞxðsÞ s ts 

Z

t

Z

ts

1 _ dsr ½yðtÞyðttÞT R2 ðaÞ½yðtÞyðtt, y_ T ðsÞR2 ðaÞyðsÞ t tt _ dsr x_ T ðsÞR3 ðaÞxðsÞ

 ts

Z

tt

_ dsr y_ T ðsÞR4 ðaÞyðsÞ

 tt

1 ½xðtsÞxðtsÞT R3 ðaÞ½xðtsÞxðtsÞ, ss

1 ½yðttÞyðttÞT R4 ðaÞ½yðttÞyðts, tt

ð25Þ

The sector condition (3) ensures that for the positive diagonal matrix K, the following inequality holds: gT ðyðtÞÞGCðaÞyðtÞrgT ðyðtÞÞGCðaÞK 1 gðyðtÞÞ:

ð26Þ

Then from Eqs. (23) to (26) we have V_ ðt,aÞ þ bV ðt,aÞr2xT ðtÞP1 ðaÞ½AðaÞxðtÞ þ BðaÞgðyðttðtÞÞÞÞ þ2yT ðtÞP2 ðaÞ½CðaÞyðtÞ þ DðaÞxðtsðtÞÞÞ þ bxT ðtÞP1 ðaÞxðtÞ       d2 T sðtÞ sðtÞ þbyT ðtÞP2 ðaÞyðtÞ þ jT ðtÞQ1 ðaÞjðtÞebs=m 1 Q1 ðaÞj t þ fT ðtÞQ2 ðaÞfðtÞ j t m m m

     d1 T tðtÞ tðtÞ f t e 1 Q2 ðaÞf t þ cT ðtÞQ3 ðaÞcðtÞ m m m       d1 T tðtÞ tðtÞ bt=m c t e 1 Q3 ðaÞc t þ ebs xT ðtsÞQ4 ðaÞxðtsÞ m m m bt=m



ebs xT ðtsÞQ4 ðaÞxðtsÞ þ ebt yT ðttÞQ5 ðaÞyðttÞ ebt yT ðttÞQ5 ðaÞyðttÞ þ ½AðaÞxðtÞ þ BðaÞgðyðttðtÞÞÞT ðsR1 ðaÞ þ ðssÞR3 ðaÞÞ½AðaÞxðtÞ þ BðaÞgðyðttðtÞÞÞ þ½CðaÞyðtÞ þ DðaÞxðtsðtÞÞT ðtR2 ðaÞ þ ðttÞR4 ðaÞÞ 1 ½CðaÞyðtÞ þ DðaÞxðtsðtÞÞ ebs ½xðtÞxðtsÞT R1 ðaÞ½xðtÞxðtsÞ s

Y. Yao et al. / Journal of the Franklin Institute 348 (2011) 2718–2733

2729

1  ebt ½yðtÞyðttÞT R2 ðaÞ½yðtÞyðttÞ t 

1 bs s ½xðtsÞxðtsÞT R3 ðaÞ½xðts ÞxðtsÞ ss



1 bt e ½yðttÞyðttÞT R4 ðaÞ½yðttÞyðttÞ tt

þgT ðyðtÞÞG½CðaÞK 1 gðyðtÞÞ þ DðaÞxðtsÞÞ þ bgT ðyðtÞÞGyðtÞ ¼ tT ðtÞFðaÞtðtÞ: ð27Þ where nðtÞ ¼ ½jðtÞ,xðtsðtÞÞ,xðtsÞ,xðtsÞ,fðtÞ,yðttðtÞÞ,yðttÞ,yðttÞ,cðtÞ,gðyðttðtÞÞÞT and FðaÞ is defined in (21). Therefore we have V_ ðt,aÞ þ bV ðt,aÞrtT ðtÞFðaÞtðtÞo0 for all nonzero tðtÞ. Using a similar method from Theorem 1, we can complete the proof.

&

4. Illustrative example In this section, we present two examples to illustrate the usefulness of our theoretical results. Example 1. Consider a genetic regulatory network as follows: ( _ ¼ AðiÞxðtÞ þ BðiÞgðyðttÞÞ, xðtÞ _ ¼ CðiÞyðtÞ þ DðiÞxðtsÞ, i ¼ 1,2 yðtÞ with  Að1Þ ¼

0 1 

Að2Þ ¼

1 0

3 0 0 3

,

 Bð1Þ ¼

,

Bð2Þ ¼



1

2

0:8

0

1 0 1

2



,

Cð1Þ ¼



 ,

Cð2Þ ¼

2

0

0

2

2

0

0

2

,

 Dð1Þ ¼



 ,

Dð2Þ ¼

1

0

0

1

1

0

0

1

,

,

In addition, nonlinear regulatory functions fi ðaÞ in (4) are taken as the usually used Hill form, that is, fi ðaÞ ¼ a2 =ð1 þ a2 Þ, i.e., the Hill coefficient is 2. We can know that ki ¼ 0:65 is the maximal value of the derivative of fi ðxi Þ. The time delays are assumed to t ¼ 0:6,s ¼ 0:3. We choose b ¼ 0:575, m ¼ 1:2, m ¼ 3, thus Ta 4Tan ¼ ln m=b ¼ 0:3171. By using the Matlab LMI toolbox, we can solve the LMIs (7), (8) and obtain feasible solutions. For the space consideration, here we only list the matrix variables P1 ð1Þ,P2 ð1Þ,

Y. Yao et al. / Journal of the Franklin Institute 348 (2011) 2718–2733

2730

R1 ð1Þ,R2 ð1Þ:



P1 ð1Þ ¼  R1 ð1Þ ¼

6:5243 3:5851

3:5851 , 15:5646

2:6046

0:6281

0:6281

2:8575

 P2 ð1Þ ¼



 ,

R2 ð1Þ ¼

1:8290 1:2261 , 1:2261 4:6950

2:1401

1:4005

1:4005

5:4508

:

Therefore from Theorem 1 we can know that the network is exponentially stable under the above conditions. Under the same conditions, if we choose m ¼ 2 then we could not get feasible solutions of the LMIs. The value of m becomes larger, the easier it is to get the feasible solutions of the LMIs. And by choosing different partitioning numbers, we might obtain a smaller average dwell time Tan . Moreover, all the conditions obtained depend not only upon the time delays, but also upon the partitioning number. Therefore we get a less conservative results. To confirm the theoretical results via simulation, we let the initial conditions to be xð0Þ ¼ ½0:2,0:8T and yð0Þ ¼ ½0:3,0:9T . Then it can be seen from Fig. 1 that the concentrations of mRNA and protein convergent exponentially. Example 2. The dynamics of repressilator has been theoretically predicted and experimentally investigated in Escherichia coli [1]. The repressilator is a cyclic negative feedback loop comprising three repressor genes ðlacl,tetR, and clÞ and their promoters. The kinetics of the system are described as follows: 8 " # > > < dxi ¼ xi þ R1 dt, 1 þ ynj > > : dy ¼ ½R ðyi xi Þ dt, i

2

where i ¼ lacl,tetR,cl; j ¼ cl,lacl,tetR. mi and pi are the concentrations of the three mRNA and repressor-protein, and R2 40 denotes the ratio of the protein decay rate to mRNA mRNA concentrations

0.8 0.7

0.8

0.6

0.7 0.6

0.4

y (t)

x (t)

0.5 0.3 0.2

0.5 0.4

0.1

0.3

0

0.2

−0.1

0.1

−0.2

Protein concentrations

0.9

0

1

2 3 Times (sec)

4

5

0

0

1

2 3 Times (sec)

Fig. 1. mRNA and protein concentrations xj ,yj ,j ¼ 1,2.

4

5

Y. Yao et al. / Journal of the Franklin Institute 348 (2011) 2718–2733 mRNA concentrations

1

0.8 0.7

0.6

0.6

0.4

0.5 p (t)

m (t)

Protein concentrations

0.9

0.8

0.2

0.4 0.3 0.2

0

0.1

−0.2 −0.4

2731

0 0

0.5

1

1.5 2 2.5 Times (sec)

3

3.5

4

−0.1

0

0.5

1

1.5 2 2.5 Times (sec)

3

3.5

4

Fig. 2. mRNA and Protein concentrations mj, pj, j¼ 1,2,3.

decay rate. Taking into account the transcriptional time delay, mode switching, we rewrite the above equations into vector form with adjusting some parameters and shift the equilibrium point to the origin: ( dmðtÞ ¼ ½AðiÞmðtÞ þ BðiÞf ðpðttðtÞÞÞ dt, dpðtÞ ¼ ½CðiÞpðtÞ þ DðiÞmðtsðtÞÞ dt, i ¼ 1,2, where Að1Þ ¼ diagf6,6,6g Cð1Þ ¼ diagf5,5,5g

Dð1Þ ¼ diagf1,0:8,1:2g

Að2Þ ¼ diagf7,7,7g Cð2Þ ¼ diagf6,6,6g

Dð2Þ ¼ diagf1,1,1g

2

0

6 Bð1Þ ¼ 4 5 0

0 0 5

5

3

7 0 5, 0

2

0

0

6 0 Bð2Þ ¼ 4 5:5 0 5:5

5:5

3

7 0 5, 0

fi ðpi Þ ¼ p2i =ð1 þ p2i Þ, that is R1 ¼ 2:5 in the above equations, and sðtÞ ¼ 0:3 þ 0:2 sin t,tðtÞ ¼ 0:4 þ 0:1 cos t. It is easy to know that the maximal value of the derivative of fi ðpi Þ is less than ki ¼ 0.65 and s ¼ 0:0999, s ¼ 0:5001, d2 ¼ 0.2, t ¼ 0:2999, t ¼ 0:5001, d1 ¼ 0.1. The initial conditions is taken as x(0) ¼ [0.5,0.7,0.8]T and y(0) ¼ [0.4,0.9,0.6]T, m ¼ 2. The simulation result of the trajectories are shown in Fig. 2. Therefore from Theorem 2 we can know that the network is exponential stable. 5. Conclusions In this paper, we have proposed and dealt with the exponential stability problem of a class of delayed genetic network model with switching parameters. Some criteria have been established to the exponential stability by employing the piecewise Lyapunov functional method combined with the average dwell time approach. All the obtained conditions which are dependent on the delays and the partitioning are in terms of linear matrix

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Y. Yao et al. / Journal of the Franklin Institute 348 (2011) 2718–2733

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