Stochastic stability of Markovian jumping uncertain stochastic genetic regulatory networks with interval time-varying delays

Mathematical Biosciences 226 (2010) 97–108

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Stochastic stability of Markovian jumping uncertain stochastic genetic regulatory networks with interval time-varying delays q P. Balasubramaniam *, R. Rakkiyappan, R. Krishnasamy Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India

a r t i c l e

i n f o

Article history: Received 15 June 2009 Received in revised form 13 April 2010 Accepted 14 April 2010 Available online 27 April 2010 Keywords: Asymptotic stability Genetic regulatory networks Linear matrix inequality Lyapunov–Krasovskii functional Markovian jumping parameters Uncertain systems

a b s t r a c t This paper investigates the robust stability problem of stochastic genetic regulatory networks with interval time-varying delays and Markovian jumping parameters. The structure variations at discrete time instances during the process of gene regulations known as hybrid genetic regulatory networks based on Markov process is proposed. The jumping parameters considered here are generated from a continuous-time discrete-state homogeneous Markov process, which is governed by a Markov process with discrete and finite state space. The new type of Markovian jumping matrices Pi and Qi are introduced in this paper. The parameter uncertainties are assumed to be norm bounded and the discrete delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are unavoidable. Based on the Lyapunov–Krasovskii functional and stochastic stability theory, delay-interval dependent stability criteria are obtained in terms of linear matrix inequalities. Some numerical examples are given to illustrate the effectiveness of our theoretical results.  2010 Elsevier Inc. All rights reserved.

1. Introduction Regulatory networks have become an important new area of research in the biological and biomedical sciences [1–3]. Specifically, the DNA information controlling gene expression (regulation) is the key to understand the differences between species and evolution [4]. Gene expression is a process consisting of transcription and translation. During transcription process, messenger RNAs (mRNAs) are synthesized from genes by the regulation of transcription factors, which are proteins. During translational process, the sequence of nucleotides in the mRNA is used in the synthesis of a protein. Genetic regulatory networks (GRNs), structured by networks of regulatory interactions between DNA, RNA, proteins. GRNs are applied for gaining insight into the underlying processes of living systems at the molecular level. Considerable attention has been contributed to the theoretical analysis and experimental GRNs. A large amount of results have been reported on dynamical behaviors of GRNs for example [5–11]. Recently, by extracting functional information from observable data, significant advances on discovering the structure of the genetic network have been made. Further deeper insights have

q The work of the first author was supported by CSIR Project, New Delhi, India under the sanctioned No. 25(0161)/08/EMR-II. The work of the second author was supported by CSIR-SRF under Grant No. 09/715(0013)/2009-EMR-I. * Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail addresses: [email protected], [email protected] (P. Balasubramaniam).

0025-5564/$ - see front matter  2010 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2010.04.002

been gained on both the static and dynamic behaviors of genetic networks. Similar to other dynamic systems [12,13], stability is a natural requirement for GRNs with clear biological significance [14]. On the other hand, it is also observed that time delays are present during the slow reaction process, such as transcription, translation and translocation involving multistage reactions in genetic networks [15]. It has been shown in [16], by mathematical modelling observation data, that the oscillatory expression of three proteins is likely to be the consequence of transcriptional delays. In fact, delay is often the key factor to the instability of a given system and plays an important role in the analysis of gene regulation dynamics. The theoretical results obtained for gene networks with or without time delays are scattered in the literature. To mention a few, a simple gene circuit, a regulator and transcriptional repressor modules in Escherichia coli [5] has been designed and studied for testing the role of negative feedback in the stability analysis of gene networks. Considering the fact that a genetic network is composed of a number of molecules that interact and regulate the expression of other genes by proteins, the authors presented a GRN model. This was described by a delay differential equation to study the local stability, using the characteristic equation discussed in [6]. A non-linear model for GRNs with SUM regulatory functions was proposed in the form of the Lure system. Also sufficient conditions for ensuring the stability of the gene networks were derived in terms of linear matrix inequalities (LMIs) in [9]. Moreover, from the viewpoint of potential applications, the study of asymptotic stability is more important and meaningful since the dynamic process of a gene network provides a better

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understanding of the mechanism of the interactions between biochemical molecules. During the past decades quantitative and qualitative network models such as boolean networks model [17,18], linear differential equation model [19–21] and a single negative feedback loop network [5] have been developed for gene regulatory networks. It has been shown in [22], by mathematically modelling recent data, that the observed oscillatory expression and activity of three proteins is most likely to be driven by transcriptional delays and time delay is often inevitable when analyzing the dynamical behaviors of GRNs [6,23,24]. Due to small numbers of transcriptional factors and other key signaling proteins, there is a considerable experimental evidence that noise plays a very important role in gene regulation [25]. Genetically identical cells and organisms exhibit remarkable diversity even when they have identical histories of environmental exposure. Noise, in the process of gene expression, may contribute to this phenotypic variability [28]. It is suggested that this noise has multiple sources including the stochastic disturbances of the biochemical reactions of gene expression that affects the kinetics of the networks [29,33]. We know very little about the noise in the genetic network which is different from the method discussed in [31] and [32]. The external noise is used to control a single gene autoregulatory network in the concentration of protein, we assume here that the noise perturbations are unknown and additively perturb the network. Authors in [26] proposed a stochastic model for gene expression in prokaryotes to study the origins of noise in gene expressions. Further stochastic differential equations have been applied in stochastic simulations based chemical master equation. In general, the stochastic noise arises in gene expression in one of two ways. The intrinsic noise is inherent in the biochemical reactions. Its magnitude is proportional to the inverse of the system size and its origin is often thermal. The external noise originates in the random variation of one or more of the externally set control parameters [27]. It is noted that the hybrid system modelling involves some kind of natural switching. Furthermore, Markov chains have also been widely used as a generic framework for modelling gene networks. A finite state homogeneous Markov chain model has been constructed from microarray data in [30]. Markov chain models incorporating rule-based transitions between states are capable of mimicking biological phenomena. All the existing Markovian models are based on the qualitative approaches and they can be used to characterize the state of a gene network in terms of discrete logical variables. However, this might not be enough to describe the dynamics of gene networks accurately. In [9], a non-linear model for GRNs with SUM regulatory functions and some stability criteria are presented. In [33], sufficient stability conditions for the stochastic GRNs with disturbance attenuation are obtained. Note that, in [9] and [33], parametric uncertainties are not taken into account while the ranges of time-varying delays ranged from 0 to an upper bound. However, in practice, a time-varying interval delay is often encountered. In [11], the stability analysis of GRNs with interval time-varying delays and parametric uncertainties is addressed. However, in [11] the information of the derivative of time-varying delays have not been included to study stability results. In [34–37], the authors studied stochastic stability analysis for GRNs with time-varying delays. Delay-range dependent stability results for GRNs have been studied in [34] with both time-varying and parameter uncertainties without Markovian jumping parameters. To the best of authors’ knowledge, the problem of delay-range dependent stability results for Markovian jumping stochastic genetic regulatory networks with interval time-varying delays has not been fully investigated and it is very challenging. In this paper, we are concerned with the stochastic robust stability analysis for GRNs with interval time-varying delays and Markovian jumping parameters. Based on a hybrid stochastic

model for GRNs with Markovian uncertain switching probabilities discussed in [38], stochastic stability results have been derived at by constructing an appropriate new Lyapunov–Krasovskii functional, employing some free-weighting matrices and LMI technique. Less conservative delay-range dependent and ratedependent stability criteria are derived based on the consideration of the ranges for time-varying delays. Finally, five numerical examples are given to illustrate the effectiveness and conservativeness of the proposed method. 1.1. Notations Throughout this paper, Rn and Rnm denote, respectively, the ndimensional Euclidean space and the set of all n  m real matrices. For symmetric matrices X and Y, the notation X P Y (X > Y) means that X  Y is positive-semidefinite (positive-definite); MT denotes the transpose of the matrix M; I is the identity matrix with appropriate dimension; jj is the Euclidean norm in Rn . Moreover, let ðX; F; fFt gtP0 ; PÞ be a probability space with a filtration fFt gtP0 satisfying the usual conditions (that is, the filtration contains all P-null sets and is right continuous). L2F0 ðð1; 0; Rn Þ denotes the family of all F0 -measurable. EðÞ stands for the expectation operator with respect to the given probability measure P; and matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem description and preliminaries Generally, a GRN consists of a group of genes that interact and regulate the expression of other genes by proteins. The change in expression of a gene is controlled by the stimulation and inhibition of proteins in transcriptional, translational and post-translational processes [14]. From [6], GRNs with time delays containing of n mRNAs and n proteins can be described by the following equations:



_ i ðtÞ ¼ ai mi ðtÞ þ bi ðp1 ðt  rðtÞÞ; p2 ðt  rðtÞÞ; . . . ; pn ðt  rðtÞÞÞ; m _pi ðtÞ ¼ ci pi ðtÞ þ di mi ðt  sðtÞÞ; i ¼ 1; 2; . . . ; n; ð1Þ

_ i ðtÞ and p_ i ðtÞ are the concentrations of mRNA and protein where m of the ith node at time t, respectively. In this network, there is one output but multiple inputs for a single node or gene. In Eq. (1), ai and ci are the degradation rates of the mRNA and protein, respectively. di is the translation rate and bi() is the regulatory function of the ith gene, which is generally a non-linear function of the variables (p1(t), p2(t), . . ., pn(t)), but has a form of monotonicity with each variable [9,39,40]. s(t) and r(t) are the time-varying delays satisfying 0 6 s1 6 sðtÞ 6 s2 ; s_ ðtÞ 6 l and 0 6 r1 6 rðtÞ 6 r2 ; r_ ðtÞ 6 g. The gene activity is tightly controlled in a cell and gene regulation function bi() plays an important role in the dynamics. Some genes can be activated by one of a few different possible transcription factors (‘OR’ logic). Other genes require that two or more transcription factors must all be bound for activation (‘AND’ logic). Here, we focus on a model of genetic networks where each transcription factor acts additively to regulate the ith gene. The regulatory function is of P the form bi ðp1 ðtÞ; p2 ðtÞ; . . . ; pn ðtÞÞ ¼ nj¼1 bij ðpj ðtÞÞ, which is also called SUM logic [41,42]. The function bij(pj(t)) is a monotonic function of the Hill form [8,43]. If transcription factor j is an activator of gene i, then

bij ðpj ðtÞÞ ¼

8 > < aij

1þðpj ðtÞ=bj Þ

> : aij

1 H 1þðpj ðtÞ=bj Þ j

ðpj ðtÞ=bj Þ

Hj Hj

; if transcription factor j is an activator of gene i; ; if transcription factor j is a repressor of gene i;

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where H is the Hill coefficient, bj is a positive constant and aij is the dimensionless transcriptional rate of transcription factor j to gene i, which is a bounded constant. Hence, Eq. (1) can be rewritten into the following form [9,11]:

8 n P >
j

j¼1

> :_ pi ðtÞ ¼ ci pi ðtÞ þ di mi ðt  sðtÞÞ;

ð2Þ

i ¼ 1; 2; . . . ; n;

  where g j ðxÞ ¼ ðx=bj ÞHj = 1 þ ðx=bj ÞHj , ui is defined as a basal rate, P ui ¼ j2V i1 aij and Vi1 is the set of all the j which is a repressor of gene i. The coupling matrix W ¼ ðwij Þ 2 Rnn of the genetic network is defined as follows: if transcription factor j is an activator of gene i, wij = aij; if there is no link from node j to node i, wij = 0; if transcription factor j is a repressor of gene i, wij = aij. In other words, the matrix defines the coupling topology, direction and the transcriptional rate of the genetic network. In compact matrix form, Eq. (2) can be written as



_ mðtÞ ¼ AmðtÞ þ Wgðpðt  rðtÞÞÞ þ u; _ pðtÞ ¼ CpðtÞ þ Dmðt  sðtÞÞ; T

_ xðtÞ ¼ AxðtÞ þ Wf ðyðt  rðtÞÞÞ; _ yðtÞ ¼ CyðtÞ þ Dxðt  sðtÞÞ;

Pfrðt þ dtÞ ¼ jjrðtÞ ¼ ig ¼



ð4Þ

i ðyÞ 6 ki . tion, it satisfies, for all x; y 2 R with x–y; 0 6 gi ðxÞg xy

From the relationship of f() and g(), we know that f() satisfies the sector condition

ðcij þ Dcij Þdt þ oðdtÞ;

if i–j

1 þ ðcii þ Dcii Þdt þ oðdtÞ; if i ¼ j

T

where x(t) = [x 1(t), x 2(t), . . ., x n(t)]T, y(t) = [y1 (t), y2(t), . . ., yn (t)]T, f(y(t)) = [f1(y1(t)), f2(y2(t)), . . ., fn(yn(t))]T with f(y(t)) = g(y(t) + p*)  g(p*). Since gi is a monotonically increasing function with satura-

fi ðxÞ 6 ki : 06 x

Let r(t), t P 0, be a right-continuous Markov chain on the probability space taking values in a finite space S = {1, 2, . . ., N} with generator P = (cij)NN given by

ð3Þ

where m(t) = [m1(t), m2(t), . . ., mn(t)] , p(t) = [p1(t), p2(t), . . ., pn(t)] , m(t  s(t)) = [m1(t  s(t)),m2(t  s(t)),. . ., mn(t  s(t))]T, g(p(t  r(t))) = [g1 (p 1(t  r (t))), g 2(p 2(t  r(t))), . . ., gn (p n(t  r (t)))]T, A = diag{a 1, a2, . . ., an}, C = diag{c1, c2, . . ., cn}, D = diag{d1, d2, . . ., dn} and u = [u1, u2, . . ., un]T. In the following, we will always shift an intended equilibrium point (m*, p*) of the system (3) to the origin by letting x(t) = m(t)  m*, y(t) = p(t)  p*. Hence, system (3) can be transformed into the following form:



In fact, the system matrices of gene networks may change randomly at discrete time instances governed by Markov process and very often, the switching probabilities are not always known precisely in prior. Therefore, the Markov jump system [44–46] might be the suitable way to model the process of gene regulation. Taking into account the parameter fluctuations and the intracellular noise perturbations [11,28,47], an uncertain stochastic Markovian gene network model is considered as follows: 8 dxðtÞ ¼ ½ðAðrðtÞÞ þ DAðrðtÞÞÞxðtÞ þ ðWðrðtÞÞ > > > > > þ DWðrðtÞÞÞf ðyðt  rðtÞÞÞdt þ ½G0 ðrðtÞÞxðtÞ > > > > < þ G1 ðrðtÞÞxðt  sðtÞÞ ð9Þ > þ G2 ðrðtÞÞyðtÞ þ G3 ðrðtÞÞyðt  rðtÞÞdxðtÞ; > > > > > dyðtÞ ¼ ½ðCðrðtÞÞ þ DCðrðtÞÞÞyðtÞ þ ðDðrðtÞÞ > > > : þ DDðrðtÞÞÞxðt  sðtÞÞdt;

where dt > 0,cij P 0 is the known transition rate from i to j if i – j P where cii ¼  j–i cij and the uncertain transition rate Dcij satisfies P jDcijj 6 dij and Dcii ¼  j–i Dcij ; i; j 2 S. Here, the Markov chain r() is assumed to be independent of the Wiener process x(t). Then, one can rewrite Markovian gene network (9) as

8 dxðtÞ ¼ ½ðAi þ DAi ÞxðtÞ þ ðW i þ DW i Þf ðyðt  rðtÞÞÞdt > > > < þ½G0i xðtÞ þ G1i xðt  sðtÞÞ þ G2i yðtÞ > þG3i yðt  rðtÞÞdxðtÞ; > > : dyðtÞ ¼ ½ðC i þ DC i ÞyðtÞ þ ðDi þ DDi Þxðt  sðtÞÞdt;

ð10Þ

where DAi, DWi, DCi and DDi are the parametric uncertainties satisfying:

½DAi ; DW i ; DC i ; DDi  ¼ Ei F i ½Hai ; Hbi ; Hci ; Hdi ;

ð11Þ

Ei, Hai, Hbi, Hci, Hdi are the known real constant matrices with appropriate dimensions and Fi satisfies

F Ti F i 6 I; i 2 S:

ð12Þ

ð5Þ

The following lemmas will be used in the proof of the main results.

In [33], the genetic networks with noise perturbations is modeled as follows:

Lemma 2.1. Let D and N be a real constant matrices of appropriate dimensions, matrix F(t) satisfies FT(t)F(t) 6 I. Then



_ xðtÞ ¼ AxðtÞ þ Wf ðyðt  rðtÞÞÞ þ rðyðtÞÞnðtÞ; _ yðtÞ ¼ CyðtÞ þ Dxðt  sðtÞÞ;

where n(t) = [n1(t), n2(t), . . ., nl(t)]T, with ni(t) as a scalar zero mean Gaussian white noise process and ni(t) is independent of nj(t) for all i – j. rðyðtÞÞ 2 Rnl is called the noise intensity matrix. Recall that the time derivative of a Wiener process is a white noise process. We have dx(t) = n(t)dt, where x(t) is an l-dimensional Wiener process. Hence, system (6) can be rewritten as the following stochastic differential equations [33]:



dxðtÞ ¼ ½AxðtÞ þ Wf ðyðt  rðtÞÞÞdt þ rðyðtÞÞdxðtÞ; dyðtÞ ¼ ½CyðtÞ þ Dxðt  sðtÞÞdt:

Lemma 2.2. For any constant matrix M > 0, any scalars a and b with a < b, and a vector function xðtÞ : ½a; b ! Rn such that the integrals concerned is well defined, then the following holds:

"Z

b

#T xðsÞ ds

M

"Z

a

#

b

xðsÞ ds 6 ðb  aÞ a

Z

b

xT ðsÞMxðsÞ ds: a

ð7Þ

In this paper, we first consider the following GRNs with both interval time-varying delays and stochastic noise:

8 > < dxðtÞ ¼ ½AxðtÞ þ Wf ðyðt  rðtÞÞÞdt þ ½G0 xðtÞ þG1 xðt  sðtÞÞ þ G2 yðtÞ þ G3 yðt  rðtÞÞdxðtÞ; > : dyðtÞ ¼ ½CyðtÞ þ Dxðt  sðtÞÞdt;

(i) For any scalar  > 0, DF(t)N + NTFT(t)DT 6 1DDT + NTN. (ii) For any P > 0,2aTb 6 aTP1a + bTPb.

ð6Þ

Lemma 2.3. (Schur Complement) Given constant matrices X1, X2 and X3 with appropriate dimensions, where XT1 ¼ X1 and XT2 ¼ X2 > 0, then

X1 þ XT3 X1 2 X3 < 0 ð8Þ

where A, W, C and D are as same as the ones in system (4). The noise intensity matrix and x(t) are defined as in system (6).

if and only if

"

X1 

XT3 X2

# < 0;

or



X2 



X3 < 0: X1

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P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

3. Stability results for stochastic GRNs In this section, we derive the main results which ensures the stability analysis of stochastic GRNs with Markovian jumping parameters and time-varying delays. Theorem 3.1. The Markovian jumping stochastic genetic regulatory networks (20) is asymptotically stable in the mean square if there exist matrices Ql, Rj, Zn, for l = 1, 2, 3, j = 1, 2, . . .,4, n = 1, 2, . . .,6 such that the following LMIs are feasible:

2

P1i 6  Pi ¼ 6 4   2 Q 1i 6  Qi ¼ 6 4   2

P2i P5i   Q 2i Q 5i  

U1 U2 U ¼ 4  U4 

3 P3i P4i P6i P7i 7 7 > 0; P8i P9i 5  P 10i 3 Q 3i Q 4i Q 6i Q 7i 7 7 > 0; Q 8i Q 9i 5  Q 10i 3

ð13Þ

U3

0 5 < 0; U5



The detailed proof of this result is given in Appendix A. It is now certain that the intracellular and extracellular noise perturbations are unavoidable during the modeling of genetic network. Therefore, it would be interesting to consider the dynamics for the genetic networks with both parameter fluctuations. Stochastic disturbances related to stability analysis is obviously one of the most important problems. In this theorem, by means of stochastic analysis theory, the global asymptotic stability conditions in the mean-square sense are obtained for the addressed uncertain stochastic GRNs with Markovian jumping parameters. To ensure the negativity of E½LVðxðtÞ; yðtÞ; t; iÞ for any possible state, it requires U < 0. This implies that the equilibrium solution of the network (10) is globally asymptotically stable in the mean-square.

4. Robust stability results for stochastic GRNs

P1i 6  6 Pi ¼ 6 4   2 Q 1i 6  6 Qi ¼ 6 4 

3 P3i P4i P6i P7i 7 7 7 > 0; P8i P9i 5  P 10i 3 Q 3i Q 4i Q 5i Q 6i Q 7i 7 7 7 > 0;  Q 8i Q 9i 5

P2i P5i   Q 2i

 2 6 6 6 6 6 6 6 6 P¼6 6 6 6 6 6 6 6 6 4





3





 r2

0

Z6

Theorem 4.4. The uncertain stochastic Markovian GRNs (10) with uncertain switching probabilities is robustly stochastically stable in the mean square, if there exist matrices P1i > 0, Q1i > 0, Qh, Rk, Zl, for h = 1, 2, 3, k = 1, 2, . . ., 4, l = 1, 2, . . ., 6, scalars 1i, e2i and {aij > 0, bij > 0, " i, j 2 S, i – j} such that the following LMI conditions hold: 6 6 6 6 6 6 6 6 6 6 6 6 P¼6 6 6 6 6 6 6 6 6 6 6 4

P1 U2 U3 P2 Z 5 P2 Z 6 P3 Ei 1i P4 P5 Ei 2i P6 7  U4 0 0 0 0 0 0 0 7 7 0 0 0 0 0 0 7   U5 7 

Remark 4.3. Compared with the existing Lyapunov functional [34], the proposed one contains the augmented vectors. It can be seen that the augmented vectors f1(t) and f2(t) in the proposed Lyapunov functional (23) contains the integral terms R T R T R T R T t tsðtÞ ts1 t xðsÞ ds , xðsÞ ds , xðsÞ ds , yðsÞ ds , tsðtÞ tsðtÞ ts2 trðtÞ R T R T trðtÞ tr1 yðsÞ ds , and trðtÞ yðsÞ ds . These integral terms play an tr2

2

Q 10i

Z5

Remark 4.2. Theorem 4.1 provides delay-dependent stability criterion for stochastic uncertain GRNs with interval time-varying delays and Markovian jumping parameters. Such stability criterion is derived based on the assumption that the time-varying delays are differentiable and the values of l, g are known. Especially in [34], stochastic uncertain GRNs have been taken into account and estimating the derivative of Lyapunov–Krasovskii functional for stochastic uncertain GRNs with interval time-varying delays. In this paper, we have proposed an improved delay-dependent stability criterion for stochastic uncertain GRNs with Markovian jumping parameters than the results proposed in [34]. Moreover, a new type of Markovian jumping matrices Pi and Qi are taken into account for deriving the stability criteria. This type of augmented matrices has not been used in any of the existing literature for the stability criteria of stochastic uncertain GRNs with Markovian jumping parameters.

important role in the reduction of conservativeness.

Theorem 4.1. The uncertain stochastic Markovian GRNs (10) is robustly stochastically stable in the mean square, if there exist matrices Qj, Rk, Zl, for j = 1, 2, 3, k = 1, 2, . . ., 4, l = 1, 2, . . ., 6 and scalars 1i, e2i such that the following LMIs are feasible:

2

The detailed proof of this result is given in Appendix B. Moreover, parameter uncertainties when modeling the GRNs are fully neglected in [9]. In contrast, the result in this Theorem provides the mean square asymptotic stability condition of Markovian jumping GRNs, where both parameter uncertainties and random noise are taken into account. It is shown in the subsequent example that the GRNs, under certain conditions, can withstand a certain level of uncertainties and noises, which provides a further step for understanding signal fidelity in gene networks and designing robust noise-tolerant gene circuits.

0

0

Z 5 Ei

0









 r12

0

0

Z 6 Ei

0

   

   

   

   

   

1i I   

0

0 0

0 0 0

1i I  

2i I 

2i I

7 7 7 7 7 < 0; 7 7 7 7 7 7 7 5

ð14Þ

b 5 Ei 2i P6 P7 b 1 U2 U3 P2 Z 5 P2 Z 6 P3 Ei 1i P4 P P   U4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   U5 





 rZ 52

0









 rZ126















































0

P8 0 0

0

Z 5 Ei

0

0

0

0

0

Z 6 Ei

0

0

0

1i I

0

0

0

0

0



1i I

0

0

0

0







2i I

0

0

0











2i I

0

0













P9

0















P10

3 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

ð15Þ The detailed proof of this result is given in Appendix B. Moreover, the parameters of the genetic networks may change randomly at discrete time instances, that is, the GRN may have finite modes, and the modes may switch (or jump) from one to another at different times governed by a Markov chain. Furthermore, the switching probabilities are not always precisely known a priori. Therefore, a GRN with mode switching could be modelled as a hybrid one; in other words, the state space of the GRN contains

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P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

both discrete and continuous states, which can be suitably described by the Markovian jumping systems. In this theorem, the global asymptotic stability conditions in the mean-square sense are obtained for the addressed uncertain stochastic GRNs with uncertain switching probabilities by means of stochastic analysis theory. To ensure the negativity of E½LVðxðtÞ; yðtÞ; t; iÞ for any possible state, it requires P < 0. This implies that the equilibrium solution of the network (10) is globally asymptotically stable in the mean-square. Remark 4.5. The above theorem gives the delay-dependent robust stochastic stability result for the uncertain stochastic Markovian GRNs with both uncertain switching probabilities and interval time-varying delays. Moreover, the same type of Markovian jumping matrices Pi and Qi as proposed in [38] are taken into account and less conservative delay-dependent results are derived than those reported in [38]. To reduce the number of decision variables, here we take the above Markovian jumping matrices Pi and Qi as in [38].

5. Numerical examples In this section, we give five examples showing the effectiveness and less conservativeness of our theoretical results. The aim is to examine the robust stability of a given stochastic gene networks under proper conditions by applying Theorem 3.1 and Theorem 3.2. Elwitz and Leiber [43] studied the dynamics of a repressilator which is acyclic negative-feedback loop comprising five repressor genes (lacl, tetR, cl) and their promotors. The Kinetics of the system are determined by the following differential equations:

(

_ i ðtÞ ¼ mi ðtÞ þ l a þ a0 ; m 1þp ðts Þ j

1

ð16Þ

p_ i ðtÞ ¼ bpi ðtÞ þ bmi ðt  s2 Þ;

where i = lacl, tetR, cl and j = cl, lacl, tetR, respectively; mi and pi are the concentrations of the five mRNAs and repressor-proteins; a0 is the growth rate of protein in a cell in the presence of saturating amounts of repressor, while a0 + a is the growth rate in its absence; b denotes the ratio of the protein decay rate to the mRNA decay rate; l is a Hill coefficient. Shifting to the equilibrium point of (16) to the origin and taking stochastic perturbation into account, one has the following example.

to system (18) and by choosing l = g = 0 and s1 = r1 = 0, it has been found that the maximum upper bound of s2 = r2 = 1.1366. Example 2. To show the effectiveness of the theoretical results we have employed a synthetic oscillatory network of transcriptional regulators in E. coli has been presented as the mathematical model of the repressilator as experimentally investigated in [43]. In this example, we generalize the model proposed in [43] by introducing time delays and random noises. The appropriate GRN is shown in Fig. 1, which describes the gene, mRNA and protein interactions. These genes are regulated by other genes; then they are expressed through transcription to obtain mRNA (which is not shown for simplicity) and again through translation to gave out their products, i.e., proteins. These proteins could subsequently act as the transcription factors of other genes to regulate the expressions of others. Consider the genetic networks with both time-varying delays and stochastic noise, but without parameter uncertainties as in Example 1, where

A ¼ diagf3; 3; 3g;

0

2:5

> > > > :

þG3 yðt  rðtÞÞdxðtÞ;

2:5

3

7 0 5: 0

Let G0 = G1 = G2 = G3 = 0.4I. The maximum allowable upper bounds of s2 = r2 for different values of l = g and s1 = r1 obtained by Theorem 3.1 are calculated and listed in Table 1. Example 3. Consider the following uncertain genetic network with stochastic perturbation:

8 > < dxðtÞ ¼ ½ðA þ EFðtÞHa ÞxðtÞ þ ðW þ EFðtÞHb Þf ðyðt  rðtÞÞÞdt þ½G0 xðtÞ þ G1 xðt  sðtÞÞ þ G2 yðtÞ þ G3 yðt  rðtÞÞdxðtÞ; > : dyðtÞ ¼ ½ðC þ EFðtÞHc ÞyðtÞ þ ðD þ EFðtÞHd Þxðt  sðtÞÞdt; ð18Þ

I Transcription Translation Repression

Example 1. Consider the following genetic networks with both time-varying delays and stochastic noise, but without parameter uncertainties:

8 dxðtÞ ¼ ½AxðtÞ þ Wf ðyðt  rðtÞÞÞdt > > > > < þ½G0 xðtÞ þ G1 xðt  sðtÞÞ þ G2 yðtÞ

C ¼ diagf2:5; 2:5; 2:5g;

D ¼ diagf0:8; 0:8; 0:8g 2 0 0 6 W ¼ 0:54 2:5 0

Activation

II

III Gene

ð17Þ

Protein

dyðtÞ ¼ ½CyðtÞ þ Dxðt  sðtÞÞdt;

where

Fig. 1. Gene regulation network comprising three genes.

A ¼ C ¼ diagf0:9; 0:9; 0:9; 0:9; 0:9g; 2

0

1 1 0

0

D ¼ diagf1; 1; 1; 1; 1g;

3

7 6 6 1 0 0 1 1 7 7 6 6 1 0 0 07 W ¼ 0:56 0 7: 7 6 4 1 1 0 0 0 5 0 0 0 1 0 Let G0 = G1 = 0, G2 = G3 = 0.1I and f(x) = x2/1 + x2, which means that k1 = 0.65, K = diag{0.65, 0.65, 0.65, 0.65, 0.65}. Applying Theorem 3.1

Table 1 The Maximal allowable upper bounds of s2 = r2 with different values of l = g and s1 = r1.

l=g

0

0.5

0.9

1

1.5

s1 = r1 = 0 s1 = r1 = 1 s1 = r1 = 10 s1 = r1 = 100

1.1869 2.1724 11.1671 101.1661

0.5756 1.5055 10.5053 100.5046

0.3883 1.3469 10.3036 100.2863

0.4036 1.3752 10.2916 100.2894

0.4787 1.3783 10.3086 100.3077

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P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

Table 2 The Maximal allowable upper bounds of s2 = r2 with different values of l = g and s1 = r1.

15 x1(t)

10

l=g

0

0.5

0.9

1

r1 = s1 = 0 r1 = s1 = 1 r1 = s1 = 10 r1 = s1 = 100

0.6284 1.6244 10.6243 100.6243

0.4542 1.3084 10.2744 100.2756

0.4080 1.1866 10.1763 100.1736

0.4114 1.1831 10.1761 100.1745

x2(t)

5 0

where

A ¼ diagf4; 2; 5; 2:5; 3:5g;

−5

C ¼ diagf1; 1; 1; 1; 1g;

D ¼ diagf0:7; 0:3; 0:6; 0:4; 0:4g; 2

3

−10

3

2

0:4 0:1 0:2 0:1 0:1 1 0 0 1 7 7 6 0:4 0:1 0:1 0:2 7 0 1 1 07 6 0:1 7 7 6 6 0 0 0 07 0:1 0 7 7; E ¼ 6 0:2 0:1 0:3 7: 7 7 6 0 1 0 05 0:1 0:4 0:1 5 4 0:1 0:1 0 1 0 0 0 0:1 0:2 0 0:1 0:4

0 6 60 6 W ¼ 0:86 61 6 40

Let Ha = 0.01I, Hb = 0.1I, Hc = 0.05I, Hd = 0.2I. The maximum allowable upper bounds of s2 = r2 for different values of l = g and s1 = r1 obtained by Theorem 4.1 are calculated and listed in Table 2.

−15

0

5

10

15

20

25

30

35

40

Fig. 2. State trajectory of the system for Example 1 with i = 1.

4 3

y1(t) y2(t)

2

Example 4. Consider the system (20) with



 ;



1

   2:5 0 A1 ¼ W1 ¼ ; C1 ¼ ; 0 5 2:5 0 0 2:5       0:8 0 5 0 0 0 D1 ¼ ; A2 ¼ ; W2 ¼ ; 0 0:8 0 5 3:5 0       3:5 0 0:6 0 1 0 ; D2 ¼ ; K¼ ; C2 ¼ 0 3:5 0 0:6 0 1   0:1 0 G01 ¼ ; 0 0:1   0:6 0 ; G02 ¼ G11 ¼ G12 ¼ G21 ¼ G22 ¼ G31 ¼ G32 ¼ 0 0:6   6 6 C¼ ; 3 3 5 0

0

0

0 −1 −2 −3 −4 0

P11 ¼ P12 ¼

10:5730



0:0073 8:5957  10:5742 0:0050 0:0050

0:0073

8:7527

 ;

Q 11

;

Q 12



 ¼ ; 0:3436 24:1952   26:4339 0:3693 ¼ : 0:3693 23:5567 26:6481

A1 ¼

0 1

;

W1 ¼

1

2

0:8

0

 ;

C1 ¼



2 0 0 2

30

35

40

x1(t) x2(t)

4 2 0

−4

0

 ;

D1 ¼

5



1 0 0 1

 ;

10

15

20

25

Fig. 4. State trajectory of the system for Example 1 with i = 2.

ð19Þ A2 ¼



25

−2

with



20

6

−6

8 > < dxðtÞ ¼ ½ðAi þ DAi ÞxðtÞ þ ðW i þ DW i Þf ðyðt  rðtÞÞÞdt þ½G0i xðtÞ þ G2i yðtÞdxðtÞ; > : dyðtÞ ¼ ½ðC i þ DC i ÞyðtÞ þ ðDi þ DDi Þxðt  sðtÞÞdt;

1 0

15

0:3436

Example 5. Consider the following system:



10

Fig. 3. State trajectory of the system for Example 1 with i = 1.

by applying Theorem 3.1 to system (20) and by choosing g = 0, l = 0, s1 = r1 = 0, s2 = r2 = 1. It has resulted in feasible solutions of the LMI (10). It shows that the genetic networks with stochastic perturbation is asymptotically stable in the mean square. Figs. 2– 5 represent the state trajectories of the systems given by Example 4 for the values of i = 1, 2. Limited to the length of the paper, we have given only a few feasible solution matrices



5



3 0 0 3

 ;

W2 ¼



1 0 1

2

 ;

C2 ¼



2 0 0 2

 ;

D2 ¼



1 0 0

1

 :

The uncertain parameters for every mode of the Markovian gene networks (19) are given by Ha1 = [0.2, 0.1], Hb1 = [0.1, 0.2],

103

P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

2 1.5

the robust asymptotic stability (in the mean square sense) for Markovian jumping stochastic GRNs with interval time-varying delays have been proposed. Delay-dependent robust stability condition for the stochastic genetic regulatory networks with Markovian jumping parameters and interval time-varying delays have been obtained in terms of LMIs, which can be easily solved by standard software. Finally, five numerical examples are given to illustrate the usefulness of the obtained results. To show the effectiveness of the obtained theoretical results, we have employed a synthetic oscillatory network of transcriptional regulators in E. coli, which has been presented as the mathematical model of the repressilator, experimentally investigated in [43]. We have generalized the model proposed in [43] by introducing time delays, random noises and Markovian jumping parameters.

y1(t) y2(t)

1 0.5 0 −0.5 −1 −1.5 −2

0

5

10

15

20

25

Fig. 5. State trajectory of the system for Example 1 with i = 2.

Table 3 The Maximal allowable upper bounds of s2 = r2 with different values of l = g and s1 = r1.

l=g

0.5

0.9

1

r1 = s1 = 0 r1 = s1 = 1 r1 = s1 = 10 r1 = s1 = 100

1.1501 1.8749 10.8439 100.8439

0.5180 1.4487 10.4485 100.4485

0.5084 1.4432 10.4430 100.4430

Hc1 = [0.1, 0.1], Hd1 = [0.2, 0.3], E1 = [0.3, 0.4]T and F1(t) = sin(t); Ha2 = [0.3, 0.2], Hb2 = [0.1, 0.1], Hc2 = [0.1, 0.3], Hd2 = [0.2, 0.2], E2 = [0.2, 0.1]T and F2(t) = cos(t). x2 The regulation function in this example is taken as f ðxÞ ¼ 1þx 2 and one can easily get ki = 0.65, K = diag{0.65, 0.65}. The stochastic perturbation matrices in this example are taken as

G01 G21



 ¼ ; G02 ¼ ; 0 0:1 0 0:1     0:2 0 0:2 0 ¼ ; G22 ¼ : 0 0:3 0 0:15 0:2

0





0:2

0

Acknowledgments The authors sincerely thank the Associate Editor and two anonymous reviewers for their careful reading, constructive comments and fruitful suggestions to improve the quality of the manuscript. Appendix A In this appendix, we will derive mean square asymptotic stability results for Markovian jumping stochastic GRNs without parameter uncertainties:

8 dxðtÞ ¼ ½Ai xðtÞ þ W i f ðyðt  rðtÞÞÞdt > > > < þ½G0i xðtÞ þ G1i xðt  sðtÞÞ þ G2i yðtÞ > þG 3i yðt  rðtÞÞdxðtÞ; > > : dyðtÞ ¼ ½C i yðtÞ þ Di xðt  sðtÞÞdt:

Theorem 3.1. The Markovian jumping stochastic GRNs (20) is asymptotically stable in the mean square if there exist matrices Ql, Rj, Zn, for l = 1, 2, 3, j = 1, 2, . . ., 4, n = 1, 2, . . ., 6 such that the following LMIs are feasible:

2

P1i

P2i

P3i

P5i

P6i



P8i







Q 1i

Q 2i

Q 3i

Q 5i

Q 6i



Q 8i





6 6  Pi ¼ 6 6  4

 3 3 The transmission probability is assumed to be C ¼ and 1 1   0:2 0:2 the uncertain probabilities is DC ¼ . In order to com0:3 0:3



pare results obtained in this paper with those in [38], let l = g = 0, s1 = r1 = 0. It was reported in [38] that the above system is robustly asymptotically stable in the mean square when 0 < s(t) 6 0.4 and 0 < r(t) 6 0.2. However, by our Theorem 4.4 and using Matlab LMI Toolbox, for l = 0, s1 = r1 = 0 it is found that the above GRNs (19) is robustly asymptotically stable in the mean square for any finite s2 and r2. More computational results are shown in Table 3. Therefore, the results given in this paper are less conservative than those in [38].

2

6 6  Qi ¼ 6 6  4 2



U1

U¼6 4  

U2

In this paper, we have made an effort to show the possibility of applying control theory to investigate the stability of stochastic GRNs. It is proved that gene regulation is an intrinsically noisy process due to intracellular and extracellular noise perturbations and environmental fluctuations. Utilizing stochastic differential equation theory, the obtained results are extended to the case with Markovian jumping parameters and uncertain switching probabilities. In this paper, several new sufficient conditions guaranteeing

P4i

3

7 P7i 7 7 > 0; P9i 7 5 P10i

U3

U4

Q 4i

3

7 Q 7i 7 7 > 0; Q 9i 7 5

ð21Þ

Q 10i 3

7 0 5 < 0; U5



with

U1 ¼ ðuj;k;i Þ1818i ; 2

6. Conclusion

ð20Þ

3T

s2 NT1 0 6 U2 ¼ 4 s12 MT1 0

s2 NT2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 ; s12 MT2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 s12 ST2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s12 ST1 2

3T NT1 0 N T2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 T U3 ¼ 4 M1 0 MT2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 5 ; ST1

0

ST2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

U4 ¼ diagfs2 Z 1 ; s12 Z 2 ; s12 ðZ 1 þ Z 2 Þg;

U5 ¼ diagfZ 3 ; Z 4 ; ðZ 3 þ Z 4 Þg;

104

P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

u1;1;i ¼ P 2i þ PT2i þ

N X

cij P1j þ Q 1 þ Q 2 þ Q 3 þ s2 Y 1 þ s12 Y 2 þ N1

u9;9;i ¼ R1 

j¼1

u1;3;i ¼ ð1  lÞP2i þ ð1  lÞP3i  ð1  lÞP4i þ S1  N1 þ NT2  M1 ; u1;6;i ¼ P 1i W i ; u1;7;i ¼ P 4i þ M1 ; u1;8;i ¼ P3i  S1 ; u1;11;i ¼ GT0i V T1  ATi U T1 ; u1;12;i ¼ GT0i V T2  ATi U T2 ; N X

cij P 2j þ PT5i ; u1;14;i ¼

j¼1

u1;15;i ¼

N X

cij P3j þ P 6i ;

cij P 4j þ P7i ;

u2;2;i ¼ Q 2i þ

Q T2i

þ

u13;15;i ¼ cij Q 1j þ R1 þ R2 þ R3 þ r

T 2Ci Z5Ci

u14;15;i ¼

j¼1

þr

T 12 C i Z 6 C i

þ r2 Y 3 þ r12 Y 4 

1

r2

Z 5  Q 1i C i 

C Ti Q T1i ;

u16;16;i ¼

u2;3;i ¼ Q 1i Di  r2 C Ti Z 5 Di  r12 C Ti Z 6 Di ; 1

u16;18;i ¼

Z5 ;

r2

u17;18;i ¼ N X

u2;16;i ¼ Q T5i þ

u2;12;i ¼

GT2i V T2 ;

N X

N X

j¼1

cij Q 4j  C Ti Q 4i ;

j¼1

 N2 

NT2

 M2 

MT2

þr

u4;9;i ¼

Z6 ;

N X

cij P10j 

j¼1

cij Q 5j 

1

r2

u16;17;i ¼

Y 3;

ðY 1 þ Y 2 Þ;

N X

1

s12

Y2;

cij Q 6j ;

j¼1

cij Q 7j ; u17;17;i ¼

N X

cij Q 8j 

j¼1

cij Q 9j ; u18;18;i ¼

N X

1

r12

cij Q 10j 

j¼1

ðY 3 þ Y 4 Þ;

1

r12

Y4;

þ V 3 ðxðtÞ; yðtÞ; t; iÞ þ V 4 ðxðtÞ; yðtÞ; t; iÞ

V 1 ðxðtÞ; yðtÞ; t; iÞ ¼ fT1 ðtÞPi f1 ðtÞ þ fT2 ðtÞQ i f2 ðtÞ; Z t Z t V 2 ðxðtÞ; yðtÞ; t; iÞ ¼ xT ðsÞQ 1 xðsÞ ds þ xT ðsÞQ 2 xðsÞ ds ts1

V 3 ðxðtÞ; yðtÞ; t; iÞ ¼

Z

Z

ts2

t

yT ðsÞR1 yðsÞ ds þ

tr1

r12

ðZ 5 þ Z 6 Þ 

u4;10;i ¼

1

1

r12

Z6 

1

r2

ðZ þ Z Þ;

r12 5 6 T u4;16;i ¼ ð1  gÞQ 5i þ ð1  gÞQ T6i  ð1  gÞQ T7i ; ÞQ T8i

þ

Z 5 ; u4;7;i ¼ KT 2 ;

ÞQ T9i ;

u4;17;i ¼ ð1  gÞQ 6i þ ð1  g  ð1  g u4;18;i ¼ ð1  gÞQ 7i þ ð1  gÞQ 9i  ð1  gÞQ T10i ; u5;5;i ¼ R4  T 1  T T1 ; u5;11;i ¼ GT3i V T1 ; u5;12;i ¼ GT3i V T2 ; u6;6;i ¼ ð1  gÞR4  T 2  T T2 ; u6;11;i ¼ W Ti U T1 ; u6;12;i ¼ W Ti U T2 ; u7;7;i ¼ Q 1 ; u7;13;i ¼ PT7i ; u7;14;i ¼ P T9i ; u7;15;i ¼ PT10i ;

Z

t

yT ðsÞR2 yðsÞ ds;

tr2

t

tsðtÞ

1

ð22Þ

where

þ

DTi Q 4i ;

r12

cij P9j ; u15;15;i ¼

1

s12

þ V 5 ðxðtÞ; yðtÞ; t; iÞ;

u3;15;i ¼ ð1  lÞP7i þ ð1  lÞP9i  ð1  lÞPT10i ; u3;16;i ¼ DTi Q 2i ; u3;17;i ¼ DTi Q 3i ;

1

j¼1 N X

cij P8j 

VðxðtÞ; yðtÞ; t; iÞ ¼ V 1 ðxðtÞ; yðtÞ; t; iÞ þ V 2 ðxðtÞ; yðtÞ; t; iÞ T 2 Di Z 5 Di

u3;7;i ¼ M2 ; u3;8;i ¼ S2 ; u3;11;i ¼ GT1i V T1 ; u3;12;i ¼ GT1i V T2 ; u3;13;i ¼ ð1  lÞPT5i þ ð1  lÞPT6i  ð1  lÞPT7i ; u3;14;i ¼ ð1  lÞP6i þ ð1  lÞPT8i  ð1  lÞPT9i ;

u4;4;i ¼ ð1  gÞR3 

N X

cij P7j ; u14;14;i ¼

Proof. Consider the following Lyapunov–Krasovskii functional:

þ r12 DTi Z 6 Di ;

u3;18;i ¼

N X

cij P6j ;

j¼1

the remaining terms are zero.

cij Q 3j  C Ti Q 3i ; u2;18;i ¼ Q 7i þ

u3;3;i ¼ ð1  lÞQ 3 þ S2 þ

j¼1 N X

N X

s12 ¼ s1  s2 ; r12 ¼ r1  r2 ; K ¼ diagfk1 ; k2 ; . . . ; kn g;

cij Q 2j  C Ti Q 2i ;

ST2

j¼1 N X

s2

u13;14;i ¼

Y1;

j¼1

j¼1

u2;17;i ¼ Q 6i þ

j¼1 N X

1

cij P5j 

j¼1

u2;6;i ¼ KT 1 ; u2;9;i ¼ Q 4i ; u2;10;i ¼ Q 3i ; u2;11;i ¼

N X

j¼1

u2;4;i ¼ ð1  gÞQ 2i þ ð1  gÞQ 3i  ð1  gÞQ 4i þ

GT2i V T1 ;

1

ðZ þ Z Þ; u10;16;i ¼ Q T6i ; r12 5 6 u10;17;i ¼ Q T8i ; u10;18;i ¼ Q 9i ; u11;11;i ¼ s2 Z 1 þ s12 Z 2  U 1  U T1 ; u11;12;i ¼ U T2  V 1 ;

u10;10;i ¼ R2 

u13;13;i ¼

j¼1 N X

Z ;

u11;13;i ¼ P2i ; u11;14;i ¼ P3i ; u11;15;i ¼ P 4i ; u12;12;i ¼ P1i þ s2 Z 3 þ s12 Z 4  V 2  V T2 ;

j¼1

N X

1

r12 6 u9;16;i ¼ Q T7i ; u9;17;i ¼ Q T9i ; u9;18;i ¼ Q T10i ;

þ NT1  P1i Ai  ATi PT1i ;

u1;13;i ¼

u8;8;i ¼ Q 2 ; u8;13;i ¼ PT6i ; u8;14;i ¼ PT8i ; u8;15;i ¼ P9i ;

Z

xT ðsÞQ 3 xðsÞ ds

t



 yT ðsÞR3 yðsÞ þ f T ðyðsÞÞR4 f ðyðsÞÞ ds;

trðtÞ

V 4 ðxðtÞ; yðtÞ; t; iÞ ¼

Z

Z

0

 s2

þ þ þ þ þ

Z

t

tþh 0

Z

g T1 ðsÞZ 1 g 1 ðsÞ ds dh t

s tþh Z 2s1 Z t s Z 2s1  s2 Z 0

g T2 ðsÞZ 3 g 2 ðsÞ ds dh

tþh Z t

Z

tþh t

g T2 ðsÞZ 4 g 2 ðsÞ ds dh

_ ds dh y_ T ðsÞZ 5 yðsÞ

r tþh Z 2r1 Z t r2

g T1 ðsÞZ 2 g 1 ðsÞ ds dh

tþh

_ ds dh; y_ T ðsÞZ 6 yðsÞ

105

P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

V 5 ðxðtÞ; yðtÞ; t; iÞ ¼

Z

Z

0

s2

þ þ

Z s1 Z

Z

By Lemma 2.1(ii), for any matrices Zi P 0, i = 1, 2, . . ., 4, the following inequalities hold:

xT ðsÞY 1 xðsÞ ds dh

tþh

s2 0 r2

þ

t

Z

t

xT ðsÞY 2 xðsÞ ds dh

Z

tþh

t

tsðtÞ

T g 1 ðsÞ ds 6 s2 nT ðtÞNZ 1 1 N nðtÞ

t

yT ðsÞY 3 yðsÞ ds dh

þ

tþh t

Z r1 Z r2

2nT ðtÞN

yT ðsÞY 4 yðsÞ ds dh;

2nT ðtÞM

tþh

Z

Z

ts1

2nT ðtÞS

tsðtÞ

ts2

 R T R T R T  t trðtÞ tr1 fT2 ðtÞ ¼ yT ðtÞ ; yðsÞds yðsÞds yðsÞds trðtÞ tr2 trðtÞ

g 1 ðtÞ ¼ Ai xðtÞ þ W i f ðyðt  rðtÞÞÞ;

0 ¼ 2nT ðtÞN xðtÞ  xðt  sðtÞÞ 

Z

t

tsðtÞ

g 1 ðsÞds 

Z

t

tsðtÞ

T

0 ¼ 2n ðtÞM xðt  s1 Þ  xðt  sðtÞÞ  Z

ts1

tsðtÞ

Z

#

þ

Z

ts1

g 1 ðsÞ ds

Z

g 1 ðsÞ ds 2nT ðtÞS

Z

tsðtÞ

ts2

!T

Z

þ

Z

g 2 ðsÞdxðsÞ ;

tsðtÞ

ts2

0 ¼ 2nT ðtÞV½G0i xðtÞ þ G1i xðt  sðtÞÞ þ G2i yðtÞ þ G3i yðt  rðtÞÞ  g 2 ðtÞ;



Z

tsðtÞ

ts2

ð27Þ

 T N ¼ NT1 0 NT2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;  T T M ¼ M1 0 M T2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  T S ¼ ST1 0 ST2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;  T U ¼ 0 0 0 0 0 0 0 0 0 0 U T1 U T2 0 0 0 0 0 0 ;  T V ¼ 0 0 0 0 0 0 0 0 0 0 V T1 V T2 0 0 0 0 0 0 :

!

ts1

g 2 ðsÞdxðsÞ

T ðZ 1 þ Z 2 Þ

g 2 ðsÞdxðsÞ :

ð34Þ

By Itô’s formula [48], we can obtain the following stochastic differential:

dVðxðtÞ; yðtÞ; t; iÞ ¼ LVðxðtÞ; yðtÞ; t; iÞdt þ 2xT ðtÞP 1i g 2 ðtÞ dwðtÞ; where L is the diffusion operator and

LVðxðtÞ; yðtÞ; t; iÞ ¼ LV 1 ðxðtÞ; yðtÞ; t; iÞ þ LV 2 ðxðtÞ; yðtÞ; t; iÞ þ LV 3 ðxðtÞ; yðtÞ; t; iÞ þ LV 4 ðxðtÞ; yðtÞ; t; iÞ þ LV 5 ðxðtÞ; yðtÞ; t; iÞ; LV 1 ðxðtÞ; yðtÞ; t; iÞ ¼ 2fT1 ðtÞPi E1 nðtÞ þ trace½g T2 ðtÞP1i g 2 ðtÞ

In addition, from Eq. (5), we have

i ¼ 1; 2; . . . ; n;

þ fT1 ðtÞ

fi ðyi ðt  rðtÞÞÞ½fi ðyi ðt  rðtÞÞÞ  ki yi ðt  rðtÞÞ 6 0;

n X

N X

cij Pj f1 ðtÞ þ 2fT2 ðtÞQ i E2 nðtÞ

j¼1

i ¼ 1; 2; . . . ; n: þ fT2 ðtÞ

Thus, for any Tj = diag{t1j, t2j, . . ., tnj} P 0, j = 1, 2, it follows that

N X

cij Q j f2 ðtÞ;

ð35Þ

j¼1

t i2 fi ðyi ðt  rðtÞÞÞ

LV 2 ðxðtÞ; yðtÞ; t; iÞ ¼ xT ðtÞðQ 1 þ Q 2 ÞxðtÞ  xT ðt  s1 ÞQ 1 xðt

i¼1 T

 ½fi ðyi ðt  rðtÞÞÞ  ki yi ðt  rðtÞÞ ¼ 2f ðyðtÞÞT 1 f ðyðtÞÞ

 s1 Þ  xT ðt  s2 ÞQ 2 xðt  s2 Þ þ yT ðtÞ

þ 2yT ðtÞKT 1 f ðyðtÞÞ  2f T ðyðt  rðtÞÞÞT 2 f ðyðt  rðtÞÞÞ

 ðR1 þ R2 ÞyðtÞ  yT ðt  r1 ÞR1 yðt  r1 Þ

þ 2yT ðt  rðtÞÞKT 2 f ðyðt  rðtÞÞÞ:

ð33Þ

g 2 ðsÞdxðsÞ 6 s12 nT ðtÞSðZ 1 þ Z 4 Þ1 ST nðtÞ

ð26Þ

where

ð32Þ

g 2 ðsÞdxðsÞ

tsðtÞ

0 ¼ 2n ðtÞU½Ai xðtÞ þ W i f ðyðt  rðtÞÞÞ  g 1 ðtÞ;

i¼1

g 2 ðsÞdxðsÞ ;

ts1

tsðtÞ

 Z4

ð25Þ

t i1 fi ðyi ðtÞÞ½fi ðyi ðtÞÞ  ki yi ðtÞ  2

!

t

T g 2 ðsÞdxðsÞ 6 s12 nT ðtÞMZ 1 4 M nðtÞ

tsðtÞ

ts2

g 2 ðsÞdxðsÞ

tsðtÞ

tsðtÞ

T

n X

Z

 Z3 2nT ðtÞM

ð31Þ

!T

t

tsðtÞ

ts2

0 6 2

Z

ð24Þ

 tsðtÞ g 2 ðsÞ dwðsÞ ;

fi ðyi ðtÞÞ½fi ðyi ðtÞÞ  ki yi ðtÞ 6 0;

g T1 ðsÞðZ 1 þ Z 2 Þg 2 ðsÞ ds;

T g 2 ðsÞdxðsÞ 6 s2 nT ðtÞNZ 1 3 N nðtÞ

g 2 ðsÞdwðsÞ ;

g 2 ðsÞ dwðsÞ ;

 Z 0 ¼ 2n ðtÞS xðt  sðtÞÞ  xðt  s2 Þ  

tsðtÞ

ð30Þ

tsðtÞ

þ

T

Z

t

ts1

tsðtÞ

Z

#

ð23Þ

"



Z

g T1 ðsÞZ 2 g 1 ðsÞ ds;

ts2

From the Newton–Leibnitz formula, the following equalities are true for any matrices N, M, S, U, V with appropriate dimensions:

"

ts1

g 1 ðsÞ ds 6 s12 nT ðtÞSðZ 1 þ Z 2 Þ1 ST nðtÞ þ

2nT ðtÞN

g 2 ðtÞ ¼ G0i xðtÞ þ G1i xðt  sðtÞÞ þ G2i yðtÞ þ G3i yðt  rðtÞÞ:

Z

tsðtÞ

Z

ð29Þ

T g 1 ðsÞ ds 6 s12 nT ðtÞMZ 1 2 M nðtÞ

þ

 R T R T R T  t tsðtÞ ts1 ; fT1 ðtÞ ¼ xT ðtÞ xðsÞds xðsÞds xðsÞds tsðtÞ ts2 tsðtÞ

g T1 ðsÞZ 1 g 1 ðsÞ ds;

tsðtÞ

tsðtÞ

where

t

ð28Þ

 yT ðt  r2 ÞR2 yðt  r2 Þ;

ð36Þ

106

P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

LV 3 ðxðtÞ; yðtÞ; t; iÞ ¼ xT ðtÞQ 3 xðtÞ  ð1  lÞxT ðt  sðtÞÞQ 3 xðt  sðtÞÞ T

where

"

T

þ y ðtÞR3 yðtÞ  ð1  gÞy ðt  rðtÞÞR3 yðt  rðtÞÞ þ f T ðyðtÞÞR4 f ðyðtÞÞ  ð1  gÞf T ðyðt  rðtÞÞÞ  R4 f ðyðt  rðtÞÞÞ;

nðtÞ ¼ xT ðtÞ yT ðtÞ xT ðt  sðtÞÞ yT ðt  rðtÞÞ  f T yðtÞ f T yðt  rðtÞÞ xT ðt  s1 Þ xT ðt  s2 Þ yT ðt  r1 Þ

ð37Þ

 yT ðt  r2 Þ g T1 ðtÞ g T2 ðtÞ !T Z

T Z t tsðtÞ xðsÞ ds xðsÞ ds 

LV 4 ðxðtÞ; yðtÞ; t; iÞ ¼ g T1 ðtÞðs2 Z 1 þ s12 Z 2 Þg 1 ðtÞ þ g T2 ðtÞðs2 Z 3 þ s12 Z 4 Þg 2 ðtÞ _T

_ þ y ðtÞðr2 Z 5 þ r12 Z 6 ÞyðtÞ Z Z t g T1 ðsÞZ 1 g 1 ðsÞ ds   ts2

Z



ts2



g T2 ðsÞZ 3 g 2 ðsÞd

ts2

ts1

Z

tsðtÞ t

g T1 ðsÞZ 2 g 1 ðsÞds 

t

_ y_ T ðsÞZ 5 yðsÞds 

Z

tr2

tr1



_ ds; y_ T ðsÞZ 6 yðsÞ

LV 5 ðxðtÞ; yðtÞ; t; iÞ ¼ x ðtÞðs2 Y 1 þ s12 Y 2 ÞxðtÞ þ y ðtÞðr2 Y 3 þ r12 Y 4 ÞyðtÞ Z ts1 Z t xT ðsÞY 1 xðsÞ ds  xT ðsÞY 2 xðsÞ ds  Z

ts2

t

yT ðsÞY 3 yðsÞds 

Z

tr2

tr1

yT ðsÞY 4 yðsÞds;

0 0

0

0 0 0 0 0 0 0 I 0 0 0 0 0 0 0

3

7 6 6 I 0 ð1  lÞI 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 7 E1 ¼ 6 6 0 0 ð1  lÞI 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 7; 5 4 0 0 ð1  lÞI 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 3

2

0 C i Di 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 60 I 0 ð1  gÞI 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 7 E2 ¼ 6 6 0 0 0 ð1  gÞI 0 0 0 0 0 I 0 0 0 0 0 0 0 0 7: 5 4 0 0 0 ð1  gÞI 0 0 0 0 I 0 0 0 0 0 0 0 0 0

T 1 T þ s12 SðZ 1 þ Z 2 Þ1 ST þ NZ 1 3 N þ MZ 4 M t

tsðtÞ

 Z3

Z

!

t

tsðtÞ

 Z4

Z

ts1

tsðtÞ

 ðZ 3 þ Z 4 Þ

g 2 ðsÞdxðsÞ þ ! g 2 ðsÞdxðsÞ þ

Z

Z

¼ nT ðtÞUnðtÞ þ

 Z3

 Z4

Z

ts1

tsðtÞ

 ðZ 3 þ Z 4 Þ

tsðtÞ

Z

! g 2 ðsÞdxðsÞ þ tsðtÞ

ts2

tsðtÞ

g 2 ðsÞdxðsÞ

ts2

Z

tsðtÞ

g 2 ðsÞdxðsÞ

ts2

Z ¼E

tsðtÞ

T

ðZ 3 þ Z 4 Þ

Z

tsðtÞ

ts2

g 2 ðsÞdxðsÞ

)

g T2 ðsÞðZ 3 Z 4 Þg 2 ðsÞ ds ;

8 !T !9 Z t < Z t = E g 2 ðsÞdxðsÞ Z 3 g 2 ðsÞdxðsÞ : tsðtÞ ; tsðtÞ ! Z t

¼E

tsðtÞ

g T2 ðsÞZ 3 g 2 ðsÞ ds ;

8 !T !9 Z ts1 < Z ts1 = g 2 ðsÞdxðsÞ Z 4 g 2 ðsÞdxðsÞ E : tsðtÞ ; tsðtÞ ! Z ts1

¼E

tsðtÞ

g T2 ðsÞZ 4 g 2 ðsÞ ds :

for all x(t), y(t) except for x(t) = y(t) = 0, where E is the mathematical expectation operator. By applying Schur complements, it is easy to see that (21) is equivalent to U < 0, except for x(t) = y(t) = 0, hence, from Lyapunov stability theorem, we can conclude that the stochastic Markovian GRNs (20) with both interval time-varying delays and stochastic noise is globally stochastically asymptotically stable in the mean square. This completes the proof. h

Appendix B



ts1

tsðtÞ

Z

( Z E

T

In this appendix, the problem of delay-dependent robust stability analysis for model (10) will be investigated. The corresponding result is summarized in the following theorem.

!T

g 2 ðsÞdxðsÞ þ

Z

!T g 2 ðsÞdxðsÞ

g 2 ðsÞdxðsÞ !

tsðtÞ

!T

g 2 ðsÞdxðsÞ

t

t

trðtÞ

!T 3 yðsÞ ds 5

EdVðxðtÞ; yðtÞ; t; iÞ ¼ E½LVðxðtÞ; yðtÞ; t; iÞ < 0;

ts1

g 2 ðsÞdxðsÞ

tsðtÞ

Z

Z

tsðtÞ

ts2

tr1

It follows from U < 0 that

Combining (23)–(39) and using Lemma 2.2, we have h T 1 T LVðxðtÞ; yðtÞ; t; iÞ 6 nT ðtÞ U11 þ s2 NZ 1 1 N þ s12 MZ 2 M

Z i þ SðZ 3 þ Z 4 Þ1 ST nðtÞ þ

tr2

Z

Note that [49]

ts2

tr2

ð39Þ where 2

tsðtÞ

T trðtÞ yðsÞ ds

1 T T þ MZ 1 4 M þ SðZ 3 þ Z 4 Þ S :

ð38Þ



yðsÞ ds

xðsÞ ds

1 T T 1 T 1 T U ¼ U11 þ s2 NZ 1 1 N þ s12 MZ 2 M þ s12 SðZ 1 þ Z 2 Þ S þ NZ 3 N

T

ts2

t

Z

!T

ts1

g T2 ðsÞZ 4 g 2 ðsÞdxðsÞ

tr2

T

ts2

!T

trðtÞ

ts1

ts2

Z

xðsÞ

Z

Z

!T g 2 ðsÞdxðsÞ

tsðtÞ

ts2

g 2 ðsÞdxðsÞ

T



g 2 ðsÞdxðsÞ ;

ð40Þ

Theorem 4.1. The uncertain stochastic Markovian genetic regulatory networks (10) is robustly stochastically stable in the mean square, if there exist matrices Qj, Rk, Zl, for j = 1, 2, 3, k = 1, 2, . . ., 4, l = 1, 2, . . ., 6 and scalars 1i, 2i such that the following LMIs are feasible:

2

3

P1i 6  6 Pi ¼ 6 4 

P2i

P 3i

P 4i

P5i

P 6i



P 8i

P 7i 7 7 7 > 0; P 9i 5







P10i

107

P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108

2

Q 2i

Q 1i 6 6  6 Qi ¼ 6 6  4

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 P¼6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

P1

3

Q 4i

7 7 7 7 > 0; 7 5

Q 5i

Q 6i

Q 7i



Q 8i

Q 9i





Q 10i

 2

Q 3i

U2

U3

2

P2 Z 5 P2 Z 6 P3 Ei 1i P4 P5 Ei 2i P6



U4

0

0

0

0

0

0

0





U5

0

0

0

0

0

0







 rZ52

0

0

0

Z 5 Ei

0









 rZ126

0

0

Z 6 Ei

0











1i I

0

0

0













1i I

0

0















2i I

0

















2i I

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 7 7 7 7 7 7 7 5

6 6 6 6 6 6 6 6 6 6 6 P¼6 6 6 6 6 6 6 6 6 6 6 4

^ 5 Ei 2i P6 P7 b 1 U2 U3 P2 Z 5 P2 Z 6 P3 Ei 1i P4 P P  U 4 0 0 0 0 0 0 0 0   U 5 0 0 0 0 0 0 0 





 rZ52

0









 rZ126























 

 

 

 

 

 

0

Z 5 Ei

P1 ¼ Pj;k;i

1818i

;

0

0

0

Z 6 Ei

0

0

0

1i I

0

0

0

0

0

1i I

0

0

0

0

 

2i I 

0 2i I

0 0

0 0



















P 9

0



















P10

with



b1 ¼ P b j;k;i P b 1;1;i ¼ P

N X

 1818i

;

cij P1j þ Q 1 þ Q 2 þ Q 3 þ s2 Y 1 þ s12 Y 2 þ N1 þ NT1  P1i Ai

j¼1

 ATi PT1i þ

b 2;2;i ¼ P

N X aij 2 d ; 4 ij j¼1;j–i

b 1;6;i ¼ P1i W i ; P

N X

b 1;11;i ¼ GT V T  AT U T ; P i 0i 1 1

b 1;7;i ¼ M1 ; P

b 1;12;i ¼ GT V T  AT U T ; P i 0i 2 2

cij Q 1j þ R1 þ R2 þ R3 þ r2 Y 3 þ r12 Y 4

j¼1

P2 ¼ ½ 0 C i Di 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T ;  

T

P3 ¼ PT1i 0 0 0 0 0 0 0 0 0 U T1 U T2 0 0 0 0 0 0 ; P4 ¼ ½ Hai 0 0 0 0 Hbi 0 0 0 0 0 0 0 0 0 0 0 0 T ; 

T

P5 ¼ 0 Q T1i 0 0 0 0 0 0 0 0 0 0 0 0 0 Q 2i Q 3i Q 4i ; T

P6 ¼ ½ 0 Hci Hdi 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  ; 



Pj;k;i –ðuj;k;i Þ when ððj; kÞ–ð2; 2Þ; ð2; 3Þ; ð3; 3ÞÞ;

P2;2;i ¼ Q 2i þ Q T2i þ

N X

cij Q 1j þ R1 þ R2 þ R3 þ r2 Y 3 þ r12 Y 4

j¼1



1

r2

P2;3;i ¼ Q 1i Di ;

1

r2

Z 5  Q 1i C i  C Ti Q T1i þ

b 2;3;i ¼ Q 1i Di ; P

b 2;4;i ¼ P

b 2;11;i ¼ GT V T ; P 2i 1

P3;3;i

¼ ð1  lÞQ 3 þ S2 þ

 N2 

b 2;6;i ¼ KT 1 ; P

Z5 ;

b 2;12;i ¼ GT V T ; P 2i 2

b 3;3;i ¼ ð1  lÞQ 3 þ S2 þ ST  N2  NT  M 2  M T ; P 2 2 2 b 3;8;i ¼ S2 ; P

b 3;7;i ¼ M2 ; P

b 4;4;i ¼ ð1  gÞR3  P b 4;9;i ¼ P

1

r12

1

r12

b 5;5;i ¼ R4  T 1  T T ; P 1

b 3;11;i ¼ GT V T ; P 1i 1

ðZ 5 þ 2Z 6 Þ 

b 4;10;i ¼ P

Z6 ;

1

r12

1

r2

Z5 ;

b 3;12;i ¼ GT V T ; P 1i 2 b 4;7;i ¼ KT 2 ; P

ðZ 5 þ Z 6 Þ;

b 5;11;i ¼ GT V T ; P 3i 1

b 6;12;i ¼ W T U T ; P i 2

b 6;11;i ¼ W T U T ; P i 1 ST2

1

r2

N X aij 2 d ; 4 ij j¼1;j–i

b 5;12;i ¼ GT V T ; P 3i 2

b 6;6;i ¼ ð1  gÞR4  T 2  T T ; P 2

Z 5  Q 1i C i  C Ti Q T1i ;

NT2

 M2 

M T2 ;

b 9;9;i ¼ R1  P

b 8;8;i ¼ Q 2 ; P

b 10;10;i ¼ R2  P Proof. The proof of this theorem is immediately follows from Theorem 3.1. Next, we will derive a robust stochastic stability result for the uncertain stochastic Markovian genetic regulatory networks (10) with uncertain switching probabilities. h Theorem 4.4. The uncertain stochastic Markovian GRNs (10) with uncertain switching probabilities is robustly stochastically stable in the mean square, if there exist matrices P1i > 0, Q1i > 0, Qh, Rk, Zl, for h = 1, 2, 3, k = 1, 2, . . ., 4, l = 1, 2, . . ., 6, scalars 1i, e2i and {aij > 0,bij > 0, " i, j 2 S, i – j} such that the following LMI conditions hold:

7 7 7 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 7 7 7 7 5

ð42Þ

b 1;8;i ¼ S1 ; P

0 0

0

b 1;3;i ¼ S1  N1 þ NT  M 1 ; P 2 

0



ð41Þ

with

0

P8

3

1

1

r12

b 7;7;i ¼ Q 1 ; P Z6 ;

ðZ þ Z Þ;

r12 5 6 b 11;11;i ¼ s2 Z 1 þ s12 Z 2  U 1  U T ; P b 11;12;i ¼ U T  V 1 ; P 1 2 b 12;12;i ¼ P1i þ s2 Z 3 þ s12 Z 4  V 2  V T ; P 2 b 13;13;i ¼  P b 15;15;i ¼  P b 17;17;i ¼  P

1

s2

Y1;

1

s12 1

r12

Y 2;

b 14;14;i ¼  P

1

s12

b 16;16;i ¼  P

ðY 3 þ Y 4 Þ;

1

r2

ðY 1 þ Y 2 Þ;

Y3;

b 18;18;i ¼  P

1

r12

Y 4;

108

P. Balasubramaniam et al. / Mathematical Biosciences 226 (2010) 97–108



T

b 5 ¼ 0 QT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; P 1i

P7 ¼ ½ P11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T ; P8 ¼ ½ 0 P12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T ; 



P9 ¼ diag ai1 I; . . . ; aiði1Þ I; aiðiþ1Þ I; . . . ; aiN I ; h

i

P10 ¼ diag bi1 I; . . . ; biði1Þ I; biðiþ1Þ I; . . . ; biN I ; 



P11 ¼ P1i  P11 ; . . . ; P 1i  P1ði1Þ ; P 1i  P1ðiþ1Þ ; . . . ; P1i  PiN ; 



P12 ¼ Q 1i  Q 11 ; . . . ; Q 1i  Q 1ði1Þ ; Q 1i  Q 1ðiþ1Þ ; . . . ; Q 1i  Q iN ; the remaining terms are zero. Proof. Based on the discussions in [38], it is easy to note that

Dcii ¼ 

X

Dcij ;

j–i

we have N X

Dcij P1j ¼

j¼1

N X

Dcij ðP 1j  P1i Þ;

j¼1;j–i

  N X aij 2 1 dij þ ½P1j  P1i 2 ; 4 aij j¼1;j–i " # N N X X bij 2 1 Dcij Q 1j 6 dij þ ½Q 1j  Q 1i 2 : bij 4 j¼1 j¼1;j–i 6

ð43Þ ð44Þ

Consider the Lyapunov–Krasovskii functional as defined in (22) except V1(x(t), y(t), t, i). Now we choose V1(x(t), y(t), t, i) as

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