# Robust Stability of Markovian Jumping Genetic Regulatory Networks with Mode-Dependent Delays

Authors: Guang He; Jian-An Fang; Xiaotai Wu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The robust stability analysis problem is investigated for a class of Markovian jumping genetic regulatory networks with parameter uncertainties and mode-dependent delays, which varies randomly according to the Markov state and exists in both translation and feedback regulation processes. The purpose of the addressed stability analysis problem is to establish some easily verifiable conditions under which the Markovian jumping genetic regulatory networks with parameter uncertainties and mode-dependent delays is asymptotically stable. By utilizing a new Lyapunov functional and a lemma, we derive delay-dependent sufficient conditions ensuring the robust stability of the gene regulatory networks in the form of linear matrix inequalities. Illustrative examples are exploited to show the effectiveness of the derived linear-matrix-inequalities- (LMIS-) based stability conditions.


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## Body

## 1. Introduction

In the past few years, genetic regulatory networks (GRNs) have been playing more and more important role in biological and biomedical sciences. With the study of genetic regulatory networks, scientists can gain insight into the underlying process of living systems at the molecular level; the dynamic behaviors of the GRNs in living organisms have received increasing attentions in the past decade [1–9].Generally, GRNs can be described by two types of models, the Boolean networks models [10–12] and differential equation models [13–17]. Recently, the differential models have received an increasing amount of research attention since it can be provide detailed understanding of the nonlinear behavior exhibited by biological systems. Hence, our present research further examines the differential GRN models with both mode-dependent time delays and Markovian jumping parameters.Time delays are inevitably occurred due to the slow processes of transcription, translation, and translocation or the finite switching speed of amplifiers. The theoretical models without consideration of time delays may provide wrong predictions [15, 18]. The stability problem of genetic regulatory network with time delays has been investigated by many researches [15, 19–24]. For instance, Chen and Aihara [15] presented a different equation model for GRNs with constant time delays and proposed necessary and sufficient conditions for such GRNs. Ren and Cao [22] derived delay-dependent robust asymptotic stability criteria for a class of genetic regulatory networks with time-varying delays and parameter uncertainties. Wang et al. [24] developed a model for genetic regulatory networks with polytopic parameter uncertainties and derived delay-dependent stability criteria for such network. Moreover, due to the modeling inaccuracies and changes in the environment of the model, parameter uncertainties can be often encountered in the genetic regulatory networks. Therefore, the problem of robust stability analysis for uncertain GRNs emerges as a research topic of primary importance.On the other hand, as shown in [25, 26], GRNs with Markovian jump parameters are a system with transitions among the states governed by a Markov chain taking values in a finite set. Therefore, it is of significance to model genetic regulatory networks with hybrid system. Recently, Hybrid system with time-varying delays has received increasing attention [27, 28]. Specially, the stability of Markovian genetic regulatory networks, which are subject to mode switching (or jumping), has been thoroughly investigated in [25, 26]. It should be pointed out that the delays in [25, 26] were a deterministic case. Ribeiro et al. [29] has pointed out that the transmission delay may occur randomly in GRNs and their probabilistic characteristics can often be obtained by statistical methods.However, most of the reported works focus on the effect of a deterministic time delay case for the Markovian jumping genetic regulatory networks; a very few studies on the effect of stochastic delays have been reported.In this paper, firstly, we deal with the stability problem of Markovian jumping genetic regulatory networks with mode-dependent delays, that is, the delay varies randomly according to the Markov state. Then, the results are extended to an uncertain case. By utilizing a new Lyapunov-Krasovskii function and a novel lemma, we derive new delay-dependent stability criteria in the form of linear matrix inequalities (LMIs), which can be easily checked by LMI Toolbox. Finally, two numerical examples are provided to show the effectiveness of the results.Notations 1.
Throughout this paper,ℝn and ℝn×m denote, respectively, n-dimensional Euclidean space and the set of all n×m real matrices. The superscript “T” denotes the matrix transposition and the notation X≥Y (resp., X>Y) where X and Y are symmetric matrices, which means that X-Y is a positive semidefinite (resp., positive definite) matrix, I is the n×n identity matrix, and λmax(A) (resp., λmin(A)) represents the largest (resp., smallest) eigenvalue of matrix A. For symmetric block matrices or long matrix expressions, an asterisk ⋆ is used to represent a term that is induced by symmetry. Let h>0, and C([-h,0;ℝn] denote the family of continuous functions ϕ from [-h,0] to ℝn with the norm ∥ϕ∥=sup-h≤θ≤0|ϕ(θ)|, where |·| is the Euclidean norm in ℝn; 𝔼{·} stands for the mathematical expectation operator. Let (Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). Denote by Lℱ0p([-h,0];ℝn) the family of all ℱ0-measurable C([-h,0];ℝn)-valued random variables ξ={ξ(θ):-h≤θ≤0} such that sup-h≤θ≤0𝔼|ξ(θ)|p<∞.


## 2. Model Description

In this paper, we will consider the following genetic regulatory networks [25]:
(2.1)m˙(t)=-Am(t)+Bf(p(t-σ(t)))+L,p˙(t)=-Cp(t)+Dm(t-τ(t)),
where m(t)=[m1(t),m2(t),…,mn(t)]T,p(t)=[p1(t),p2(t),…,pn(t)]T, and mi(t) and pi(t) are the concentrations of mRNA and protein of the ith node at time t, respectively; A=diag(a1,a2,…,an) and C=diag(c1,c2,…,cn) denote the degradation or dilution rates of mRNAs and proteins, D=diag(d1,d2,…,dn) represents the translation rate, and B=(bij)∈ℝn×n is defined as follows:
(2.2)bij{>0iftranscriptionjisanactivatorofgenei;=0,ifthereisnolinkfromnodejtoi;<0iftranscriptionjisanrepressorofgenei;f(·) denotes the feedback regulation of the protein on the transcription, which is the monotonic function in Hill form, fi(x)=xhi/(1+xhi), and hi is the Hill coefficient; τ(t) and σ(t) are the time delays; L=[l1,l2,…,ln]T, li is the base transcriptional rate of the repressor of gene i. Assume m⋆ and p⋆ are the equilibrium points of (2.1), defining x(t)=m(t)-m⋆, y(t)=p(t)-p⋆, it is easy to get
(2.3)x˙(t)=-Ax(t)+Bg(y(t-σ(t))),y˙(t)=-Cy(t)+Dx(t-τ(t)),
where g(y(t))=f(y(t)+P⋆)-f(p⋆), from the definition of g, it is easy to get
(2.4)g(x)(g(x)-Kx)≤0.Taking the Markovian jumping parameters and stochastic delays into account, a Markovian jumping genetic regulatory networks model with mode-dependent delays is considered as(2.5)x˙(t)=-A(r(t))x(t)+B(r(t))g(y(t-σr(t)(t))),y˙(t)=-C(r(t))y(t)+D(r(t))x(t-τr(t)(t)),
where r(t) is a continuous-time Markovian process with right continuous trajectories and taking values in a finite set S={1,2,…,N} with the following transition probabilities:
(2.6)P{r(t+Δt)=j:r(t)=i}={γijΔt+O(Δt)ifj≠i,1+γijΔt+O(Δt)ifj=i,
where Δt>0 and limΔt→0O(Δt)/Δt=0. Here, γij≥0 is the transition rate from i to j if i≠j, while γii=-∑j=1,j≠iN‍γij.τ
r
(
t
)
(
t
) and σr(t)(t) are the time-varying delays when the mode is in r(t) and we assume that they satisfy the following conditions
(2.7)0≤d1i≤τi(t)≤d2i,τ˙i(t)≤hi,0≤e1i≤σi(t)≤e2i,σ˙i(t)≤μi,
where d1i, d2i, e1i, e2i, hi, and μi are known real constants, for any i∈S, denote
(2.8)d1=min{d1i,i∈S},d2=max{d2i,i∈S},e1=min{e1i,i∈S},e2=max{e2i,i∈S}.Remark 2.1.
In [25], hi and μi are assumed to be less than 1. But in practice, they are not always less than 1. In this paper, we develop the criteria without this restrict. In the following we will give some lemmas, which will play an indispensable role in deriving our criteria.Lemma 2.2 (see [24]).
For any vectorx,y∈ℝn and matrix Q>0, one has the following inequality:
(2.9)2xTy≤xTQx+yTQ-1y.Lemma 2.3 (see [30]).
For any positive definite matrixM>0, scalar γ>0 and vector function ω:[0,γ]→ℝn such that the integrations concerned are well defined, then the following inequality holds:
(2.10)(∫0γ‍ω(s)ds)TM(∫0γ‍ω(s)ds)≤γ(∫0γ‍ωT(s)Mω(s)ds).Lemma 2.4 (see [31]) (Schur complement).
Given constant matricesX, Y, and Z where X=XT and 0<Y=YT. Then X+ZTY-1Z<0 if and only if
(2.11)[XZTZ-Y]<0,or[-YZZTX]<0.Lemma 2.5 (see [32]).
AssumeΩ,X1, and X2 are constant matrices with appropriate dimensions, 0≤α(t)≤1, then
(2.12)Ω-X1<0,Ω-X2<0,
is equivalent to
(2.13)Ω-α(t)X1-(1-α(t))X2<0.


## 3. Main Results

In this section, we first deal with the asymptotical stability problem for the system (2.5). By employing a new Lyapunov-Krasovskii function, some less conservative sufficient criteria for the stability problem of Markovian jumping genetic regulatory networks with mode-dependent delays are derived in terms of LMIs. Then the results are extended to uncertain case.Theorem 3.1.
The genetic regulatory networks (2.5) is asymptotically stable, if there exist matrix sets {Pi>0,Qi>0,∀i∈S}, matrices Nj,Mj(j=1,2,3)>0Rj(j=1,2,3,4)>0, any diagonal positive definite matrix Λ, and any matrices U, V with appropriate dimensions such that the following LMIs hold:
(3.1)Ω1i-ϵ1TR1ϵ1-ϵ1TR2ϵ1<0,Ω1i-ϵ1TR1ϵ1-ϵ3TR2ϵ3<0,Ω1i-ϵ2TR1ϵ2-ϵ1TR2ϵ1<0,Ω1i-ϵ2TR1ϵ2-ϵ3TR2ϵ3<0,Ω2i-ϵ4TR3ϵ4-ϵ4TR4ϵ4<0,Ω2i-ϵ4TR3ϵ4-ϵ6TR4ϵ6<0,Ω2i-ϵ5TR3ϵ5-ϵ4TR4ϵ4<0,Ω2i-ϵ5TR3ϵ5-ϵ6TR4ϵ6<0,
where
(3.2)ϵ1=[0I0-I0],ϵ2=[I-I000],ϵ3=[0-II00],ϵ4=[0I0-I00],ϵ5=[I-I0000],ϵ6=[0-II000],Ω1i=[Ξ1R100-UAi⋆Ξ2R2R2+R1-d1R2d20⋆⋆-R2-N100⋆⋆⋆-R1-R2-N2+d1R2d20⋆⋆⋆⋆Ξ3],Ω2i=[Π1R300-VCi0⋆Π2R4R3+R4-e1R4e20Λ⋆⋆-R4-M1000⋆⋆⋆-R3-R4-M2+e1R4e200⋆⋆⋆⋆Π30⋆⋆⋆⋆⋆Π4],Ξ1=-PiAi-AiTPi+∑j=1N‍γijPj+∑j=13‍Nj-γii(d2-d1)N3-R1+Pi,Ξ2=-(1-hi)N3-2R1-2R2+DiTVDi+DiTQiDi+d1R2d2,Ξ3=-U+d22R1+(d2-d1)2R2,Π1=-QiCi-CiTQi+∑j=1N‍γijQj+∑j=13‍Mj-γii(e2-e1)M3-R3+Qi,Π2=-(1-μi)M3-2R3-2R4+e1R4e2,Π3=-V+e22R3+(e2-e1)2R4,Π4=-2ΛK-1+BiTUBi+BiTPiBi.Proof.
Choose a Lyapunov-Krasovskii functional candidate:(3.3)V(i,t,x(t),y(t))=V1(i,t,x(t),y(t))+V2(i,t,x(t),y(t))+V3(i,t,x(t),y(t))+V4(i,t,x(t),y(t)),
where
(3.4)V1(i,t,x(t),y(t))=xT(t)Pix(t)+yT(t)Qiy(t),V2(i,t,x(t),y(t))=∑j=12‍∫t-djt‍xT(s)Njx(s)ds+∑j=12‍∫t-ejt‍yT(s)Mjy(s)ds+∫t-τi(t)t‍xT(s)N3x(s)ds+∫t-σi(t)t‍yT(s)M3y(s)ds,V3(i,t,x(t),y(t))=-γii∫-d2-d1‍∫t+θt‍xT(s)N3x(s)dsdθ-γii∫-e2-e1‍∫t+θt‍yT(s)M3y(s)dsdθ,V4(i,t,x(t),y(t))=d2∫-d20‍∫t+θt‍x˙T(s)R1x˙(s)dsdθ+(d2-d1)∫-d2-d1‍∫t+θt‍x˙T(s)R2x˙(s)dsdθ+e2∫-e20‍∫t+θt‍y˙T(s)R3y˙(s)dsdθ+(e2-e1)∫-e2-e1‍∫t+θt‍y˙T(s)R4y˙(s)dsdθ.
Letℒ be the weak infinite generator. Then for each r(t)=i,i∈S along the trajectory of (2.5) one has
(3.5)ℒV1(i,t,x(t),y(t))=2xT(t)Pi(-Aix(t)+Big(y(t-σi(t))))+∑j=1N‍γijxT(t)Pjx(t)+2yT(t)Qi(-Ciy(t)+Dix(t-τi(t)))+∑j=1N‍γijyT(t)Qjy(t),ℒV2(i,t,x(t),y(t))≤∑j=13‍xT(t)Njx(t)-(1-hi)xT(t-τi(t))N3x(t-τi(t))+∑j=13‍yT(t)Mjy(t)-(1-μi)yT(t-σi(t))M3y(t-σi(t))-∑j=12‍xT(t-dj)Njx(t-dj)-∑j=12‍yT(t-ej)Mjy(t-ej)+∑j=1N‍γij∫t-τj(t)t‍xT(s)N3x(s)ds+∑j=1N‍γij∫t-σj(t)t‍yT(s)M3y(s)ds,∑j=1N‍γij∫t-τj(t)t‍xT(s)N3x(s)ds=γii∫t-τi(t)t‍xT(s)N3x(s)ds+∑j≠i‍γij∫t-τj(t)t‍xT(s)N3x(s)ds≤γii∫t-d1t‍xT(s)N3x(s)ds+∑j≠i‍γij∫t-d2t‍xT(s)N3x(s)ds=γii∫t-d1t‍xT(s)N3x(s)ds-γii∫t-d2t‍xT(s)N3x(s)ds=-γii∫t-d2t-d1‍xT(s)N3x(s)ds.
Similarly
(3.6)∑j=1N‍γij∫t-σj(t)t‍yT(s)M3y(s)ds≤-γii∫t-e2t-e1‍yT(s)M3y(s)ds,ℒV3(i,t,x(t),y(t))=-γii(d2-d1)xT(t)N3x(t)+γii∫t-d2t-d1‍xT(s)N3x(s)ds-γii(e2-e1)yT(t)M3y(t)+γii∫t-e2t-e1‍yT(s)M3y(s)ds,ℒV4(i,t,x(t),y(t))=d22x˙T(t)R1x˙(t)-d2∫t-d2t‍x˙T(s)R1x˙(s)ds+(d2-d1)2x˙T(t)R2x˙(t)+(e2-e1)2y˙T(t)R4y˙(t)ds-e2∫t-e2t‍y˙T(s)R3y˙(s)ds+e22y˙T(t)R3y˙(t)-(d2-d1)∫t-d2t-d1‍x˙T(s)R2x˙(s)ds-(e2-e1)∫t-e2t-e1‍y˙T(s)R4y˙(s)ds.
Note that(3.7)-d2∫t-d2t‍x˙T(s)R1x˙(s)ds=-d2∫t-d2t-τi(t)‍x˙T(s)R1x˙(s)ds-d2∫t-τi(t)t‍x˙T(s)R1x˙(s)ds=-(d2-τi(t))∫t-d2t-τi(t)‍x˙T(s)R1x˙(s)ds-τi(t)∫t-d2t-τi(t)‍x˙T(s)R1x˙(s)ds-τi(t)∫t-τi(t)t‍x˙T(s)R1x˙(s)ds-(d2-τi(t))∫t-τi(t)t‍x˙T(s)R1x˙(s)ds≤-xT(t-τi(t))R1x(t-τi(t))+2xT(t-τi(t))R1x(t-d2)-xT(t-d2)R1x(t-d2)-xT(t)R1x(t)+2xT(t)R1x(t-τi(t))-xT(t-τi(t))R1x(t-τi(t))-[τi(t)d2-τi(t)][xT(t-τi(t))-xT(t-d2)]R1[x(t-τi(t))-x(t-d2)]-[d2-τi(t)τi(t)][xT(t)-xT(t-τi(t))]R1[x(t)-x(t-τi(t))]≤-xT(t-τi(t))R1x(t-τi(t))+2xT(t-τi(t))R1x(t-d2)-xT(t-d2)R1x(t-d2)-xT(t)R1x(t)+2xT(t)R1x(t-τi(t))-xT(t-τi(t))R1x(t-τi(t))-(τi(t)d2)[xT(t-τi(t))-xT(t-d2)]R1[x(t-τi(t))-x(t-d2)]-[1-(τi(t)d2)][xT(t)-xT(t-τi(t))]R1[x(t)-x(t-τi(t))].
Similarly,
(3.8)-(d2-d1)∫t-d2t-d1‍x˙T(s)R2x˙(s)ds≤-xT(t-τi(t))R2x(t-τi(t))+2xT(t-τi(t))R2x(t-d2)-xT(t-d2)R2x(t-d2)-xT(t-d1)R2x(t-d1)+2xT(t-d1)R2x(t-τi(t))-xT(t-τi(t))R2x(t-τi(t))+(d1d2)[xT(t-τi(t))-xT(t-d2)]R2[x(t-τi(t))-x(t-d2)]-(τi(t)d2)[xT(t-τi(t))-xT(t-d2)]R2[x(t-τi(t))-x(t-d2)]-[1-(τi(t)d2)][xT(t-d1)-xT(t-τi(t))]R2[x(t-d1)-x(t-τi(t))],-e2∫t-e2t‍y˙T(s)R3y˙(s)ds≤-yT(t-σi(t))R3y(t-σi(t))+2yT(t-σi(t))R3y(t-e2)-yT(t-e2)R3y(t-e2)-yT(t)R3y(t)+2yT(t)R3y(t-σi(t))-yT(t-σi(t))R3y(t-σi(t))-[σi(t)e2][yT(t-σi(t))-yT(t-e2)]R3[y(t-σi(t))-y(t-e2)]-[1-(σi(t)e2)][yT(t)-yT(t-σi(t))]R3[y(t)-y(t-σi(t))],-(e2-e1)∫t-e2t-e1‍y˙T(s)R4y˙(s)ds≤-yT(t-σi(t))R4y(t-σi(t))+2yT(t-σi(t))R4y(t-e2)-yT(t-e2)R4y(t-e2)-yT(t-e1)R4y(t-e1)+2yT(t-e1)R4y(t-σi(t))-yT(t-σi(t))R4y(t-σi(t))+(e1e2)[yT(t-σi(t))-yT(t-e2)]R4[y(t-σi(t))-y(t-e2)]-(σi(t)e2)[yT(t-σi(t))-yT(t-e2)]R4[y(t-σi(t))-y(t-e2)]-[1-(σi(t)e2)][yT(t-e1)-yT(t-σi(t))]R4[y(t-e1)-y(t-σi(t))].
Noting the sector condition (2.4), for any positive matrix Λ we have
(3.9)2yT(t-σi(t))Λg(y(t-σi(t)))-2gT(y(t-σi(t)))ΛK-1g(y(t-σi(t)))≥0.
For any matricesU and V with appropriate dimensions, we have
(3.10)-2x˙T(t)U[x˙(t)+Aix(t)-Big(y(t-σi(t)))]=0,-2y˙T(t)V[y˙(t)+Ciy(t)-Dix(t-τi(t))]=0.
By Lemma2.3 we can get the following inequalities:
(3.11)2xT(t)PiBig(y(t-σi(t)))≤xT(t)Pix(t)+gT(y(t-σi(t)))BiTPiBig(y(t-σi(t))),2yT(t)QiDix(t-τi(t))≤yT(t)Qiy(t)+xT(t-τi(t))DiTQiDix(t-τi(t)),2x˙T(t)UBig(y(t-σi(t)))≤x˙T(t)Ux˙(t)+gT(y(t-σi(t)))BiTUBig(y(t-σi(t))),2y˙T(t)VDix(t-τi(t))≤y˙T(t)Vy˙(t)+xT(t-τi(t))DiTVDix(t-τi(t)).
From (3.3) to (3.11) we can get
(3.12)ℒV(i,t,x(t),y(t))≤ξTΩ1iξ(t)-(τi(t)d2)ξT(t)ϵ1TR1ϵ1ξ(t)-[1-(τi(t)d2)]ξT(t)ϵ2TR1ϵ2ξ(t)-(τi(t)d2)ξT(t)ϵ1TR2ϵ1ξ(t)-[1-(τi(t)d2)]ξT(t)ϵ3TR2ϵ3ξ(t)+ηTΩ2iη(t)-(σi(t)e2)ηT(t)ϵ4TR3ϵ4η(t)-[1-(σi(t)e2)]ηT(t)ϵ5TR3ϵ5η(t)-(σi(t)e2)ηT(t)ϵ4TR4ϵ4η(t)-[1-(σi(t)e2)]ηT(t)ϵ6TR4ϵ6η(t),
where
(3.13)ξ(t)=[xT(t),xT(t-τi(t)),xT(t-d1),xT(t-d2),x˙T(t)]T,η(t)=[yT(t),yT(t-σi(t)),yT(t-e1),yT(t-e2),y˙T(t),gT(y(t-σi(t)))]T.
By Lemma 2.5, (3.12) < 0 is equivalent to (3.1). Then by the Lyapunov-Krasovskii stability theorem that the genetic regulatory networks (2.5) is asymptotically stable in the mean square. Hence, this completes the proof.In the proof of Theorem3.1, if we deal with the terms -d2∫t-d2t‍x˙T(s)R1x˙(s)ds,  -(d2-d1)∫t-d2t-d1‍x˙T(s)R2x˙(s)ds, -e2∫t-e2t‍y˙T(s)R3y˙(s)ds, -(e2-e1)∫t-e2t-e1‍y˙T(s)R4y˙(s)ds as follows:
(3.14)-d2∫t-d2t‍x˙T(s)R1x˙(s)ds≤-xT(t-τi(t))R1x(t-τi(t))+2xT(t-τi(t))R1x(t-d2)-xT(t-d2)R1x(t-d2)-xT(t)R1x(t)+2xT(t)R1x(t-τi(t))-xT(t-τi(t))R1x(t-τi(t)),-(d2-d1)∫t-d2t-d1‍x˙T(s)R2x˙(s)ds≤-xT(t-τi(t))R2x(t-τi(t))+2xT(t-τi(t))R2x(t-d2)-xT(t-d2)R2x(t-d2)-xT(t-d1)R2x(t-d1)+2xT(t-d1)R2x(t-τi(t))-xT(t-τi(t))R2x(t-τi(t)),-e2∫t-e2t‍y˙T(s)R3y˙(s)ds≤-yT(t-σi(t))R3y(t-σi(t))+2yT(t-σi(t))R3y(t-e2)-yT(t-e2)R3y(t-e2)-yT(t)R3y(t)+2yT(t)R3y(t-σi(t))-yT(t-σi(t))R3y(t-σi(t)),-(e2-e1)∫t-e2t-e1‍y˙T(s)R4y˙(s)ds≤-yT(t-σi(t))R4y(t-σi(t))+2yT(t-σi(t))R4y(t-e2)-yT(t-e2)R4y(t-e2)-yT(t-e1)R4y(t-e1)+2yT(t-e1)R4y(t-σi(t))-yT(t-σi(t))R4y(t-σi(t))
that is, we do not use Lemma 2.5, then we will have the following corollary.Corollary 3.2.
The genetic regulatory networks (2.5) is asymptotically stable, if there exist matrix sets {Pi>0,Qi>0,∀i∈S}, matrices Nj,Mj(j=1,2,3)>0Rj(j=1,2,3,4)>0, any diagonal positive definite matrix Λ, and any matrices U and V with appropriate dimensions such that the following LMIs hold:
(3.15)Ω1i<0,Ω2i<0,
where, Ω1i and Ω2i are defined in Theorem 3.1.Remark 3.3.
In the proof of Theorem3.1, if we ignore the terms -(τi(t)/d2)ξT(t)ϵ1TR1ϵ1ξ(t), -[1-(τi(t)/d2)]ξT(t)ϵ2TR1ϵ2ξ(t), and -(σi(t)/e2)ηT(t)ϵ4TR3ϵ4η(t), -[1-(σi(t)/e2)]ηT(t)ϵ5TR3ϵ5η(t), we can also get sufficient conditions ensuring the robust stability of the genetic regulatory networks. But the conditions are conservative to some extent. By considering the terms -(τi(t)/d2)ξT(t)ϵ1TR1ϵ1ξ(t), and -[1-(τi(t)/d2)]ξT(t)ϵ2TR1ϵ2ξ(t), -(σi(t)/e2)ηT(t)ϵ4TR3ϵ4η(t), -[1-(σi(t)/e2)]ηT(t)ϵ5TR3ϵ5η(t), we can get a less conservative criterion. The illustrate examples will show this in Section 4.In the following, we will extend our results to uncertain case. We consider the following Markovian jumping genetic regulatory networks with mode-dependent delays and parameter uncertainties:(3.16)x˙(t)=-(Ai+ΔAi)x(t)+(Bi+ΔBi)g(y(t-σi(t))),y˙(t)=-(Ci+ΔCi)y(t)+(Di+ΔDi)x(t-τi(t)),
where ΔAi, ΔBi,  ΔCi, and ΔDi are the parametric uncertainties satisfying:
(3.17)[ΔAi,ΔBi,ΔCi,ΔDi]=EiFi[Hai,Hbi,Hci,Hdi].Ei, Hai, Hbi, Hci, and Hdi are the known real constant matrices with appropriate dimensions, Fi satisfies
(3.18)FiTFi≤I,i∈S.Theorem 3.4.
The genetic regulatory networks (3.16) is robust asymptotically stable, if there exist P1>0, P2>0, …, Pn>0, Q1>0, Q2>0, …, Qn>0, Nj, Mj(j=1,2,3)>0, Rj(j=1,2,3,4)>0, real number {ɛi,i∈S}, any diagonal positive definite matrix Λ, and any matrices U and V with appropriate dimensions such that the following LMIs hold:
(3.19)Ω¯1i-ϵ1TR1ϵ1-ϵ1TR2ϵ1<0,Ω¯1i-ϵ1TR1ϵ1-ϵ3TR2ϵ3<0,Ω¯1i-ϵ2TR1ϵ2-ϵ1TR2ϵ1<0,Ω¯1i-ϵ2TR1ϵ2-ϵ3TR2ϵ3<0,Ω¯2i-ϵ4TR3ϵ4-ϵ4TR4ϵ4<0,Ω¯2i-ϵ4TR3ϵ4-ϵ6TR4ϵ6<0,Ω¯2i-ϵ5TR3ϵ5-ϵ4TR4ϵ4<0,Ω¯2i-ϵ5TR3ϵ5-ϵ6TR4ϵ6<0,
where
(3.20)Ω1i¯=[Γ¯1iϕ1i⋆ψ1i],Ω¯2i=[Γ¯2iϕ2i⋆φ2i],Γ¯1i=[Ξ¯1R100-UAi⋆Ξ¯2¯R2R2+R1-d1R2d20⋆⋆-R2-N100⋆⋆⋆-R1-R2-N2+d1R2d20⋆⋆⋆⋆Ξ¯3,],Ξ¯1=-PiAi-AiTPi+∑j=1N‍γijPj+∑j=13‍Nj-γii(d2-d1)N3-R1+Pi+2ɛiHaiTHai,Ξ¯2=-(1-hi)N3-2R1-2R2+DiTVDi+DiTQiDi+2ɛiHdiTHdi+d1R2d2,ψ1i=diag(-ɛiI,-I,-I,-I,-ɛiI),ϕ1i=[2PiEi00000000000000000000000000002UEi],Γ¯2i=[Π¯1R300-VCi0⋆Π2R4R3+R4-e1R4e20Λ⋆⋆-R4-M1000⋆⋆⋆-R3-R4-M2+e1R4e200⋆⋆⋆⋆Π30⋆⋆⋆⋆⋆Π¯4],Π¯1=-QiCi-CiTQi+∑j=1N‍γijQj+∑j=13‍Mj-γii(e2-e1)M3-R3+Qi+2ɛiHciTHci,Π¯4=-2ΛK-1+BiTUBi+BiTPiBi+2ɛiHbiTHbi,ϕ2i=[2QiEi0000000000000000000000000000000002VEi0000000],ψ2i=diag(-ɛiI,-I,-I,-I,-I,-ɛiI).Proof.
Consider the same Lyapunov-Krasovskii functional (3.3), do the differential along the trajectory (3.16), one can readily get
(3.21)ℒV(i,t,x(t),y(t))≤ξTΓ1i¯ξ(t)-(τi(t)d2)ξT(t)ϵ1TR1ϵ1ξ(t)-[1-(τi(t)d2)]ξT(t)ϵ2TR1ϵ2ξ(t)-(τi(t)d2)ξT(t)ϵ1TR2ϵ1ξ(t)-[1-(τi(t)d2)]ξT(t)ϵ3TR2ϵ3ξ(t)+ηTΓ2i¯η(t)-(σi(t)e2)ηT(t)ϵ4TR3ϵ4η(t)-[1-(σi(t)e2)]ηT(t)ϵ5TR3ϵ5η(t)-(σi(t)e2)ηT(t)ϵ4TR4ϵ4η(t)-[1-(σi(t)e2)]ηT(t)ϵ6TR4ϵ6η(t)+2ɛi-1xT(t)PiEiEiTPix(t)+2ɛi-1yT(t)QiEiEiTQiy(t)+2ɛi-1x˙T(t)UEiEiTUx(t)+2ɛi-1y˙T(t)VEiEiTVy˙(t).
By Lemmas2.4 and 2.5, we can get that ℒV(i,t,x(t),y(t))<0 is equivalent to (3.19). Hence the Markovian jumping genetic regulatory network with mode-dependent delays and parameter uncertainties is robust asymptotically stable. This completes the proof.As mentioned in Theorem3.1, if we ignore the terms -(τi(t)/d2)ξT(t)ϵ1TR1ϵ1ξ(t),  -[1-(τi(t)/d2)]ξT(t)ϵ2TR1ϵ2ξ(t),  -(σi(t)/e2)ηT(t)ϵ4TR3ϵ4η(t), and -[1-(σi(t)/e2)]ηT(t)ϵ5TR3ϵ5η(t), we can get the following corollary.Corollary 3.5.
The genetic regulatory networks (3.16) is robust asymptotically stable, if there exist P1>0, P2>0, …, Pn>0, Q1>0, Q2>0, …, Qn>0, Nj, Mj(j=1,2,3)>0, Rj(j=1,2,3,4)>0, real number {ɛi,i∈S}, any diagonal positive definite matrix Λ, and any matrices U and V with appropriate dimensions such that the following LMIs hold:
(3.22)Ω1i¯<0,Ω2i¯<0,
where Ω1i¯ and Ω2i¯ are defined in Theorem 3.4.


## 4. Illustrative Examples

In this section, two numerical examples are given to illustrate the effectiveness of the derived results.Example 4.1.
Consider (2.5) where
(4.1)A1=[3003],B1=[1-20.80],C1=[2002],D1=[1001],A2=[3003],B2=[-1012],C2=[2002],D2=[-1001].
The nonlinear regulation function is taken asg(x)=x2/(1+x2), so we can easily get ki=0.65, the transmission probability is assumed to be γ=[-0.40.40.6-0.6], and time delays are chosen as
(4.2)τ1(t)=0.4+0.2cos(t),τ2(t)=0.4+0.1sin(t),σ1(t)=0.2+0.1sin(t),σ2(t)=0.2+0.1cos(t).
Then we have
(4.3)h1=0.2,h2=0.1,d1=0.2,d2=0.6,μ1=0.1,μ2=0.1,e1=0.1,e2=0.3.
By using Matlab Toolbox, solving (3.1) we can obtain the feasible solutions
(4.4)P1=[0.67200.00620.00620.7652],P2=[0.77430.05630.05630.7245],Q1=[1.05290.00210.00211.2663],Q2=[1.09260.00360.00361.3185],N1=[0.36020.02780.02780.3244],N2=[0.35210.02440.02440.3215],N3=[1.62600.02840.02841.8337],M1=[0.00700.00160.00160.0052],M2=[0.0338-0.0002-0.00020.0353],M3=[0.0891-0.0029-0.00290.0920],R1=[0.15340.02530.02530.1688],R2=[0.4415-0.0022-0.00220.3993],R3=[0.41920.00530.00530.3845],R4=[1.2767-0.0102-0.01022.0128],U=[0.04100.00320.00320.0373],V=[0.11460.00240.00240.1054],Λ=[1.3501002.0371].
Hence the Markovian jumping genetic regulatory networks with mode-dependent delays is asymptotically stable. Assume h1=0.2, μ1=0.1,μ2=0.2, e1=0.1, e2=0.5, and d1=0.2. Then we can calculate the maximal allowable bounds of d2 with different values of h2.Example 4.2.
Consider (3.16) where
(4.5)A1=[5005],B1=[0010],C1=[2.5002.5],D1=[1001],A2=[5005],B2=[0020],C2=[3003],D2=[0.8000.8].
The uncertain parameters for every mode of the Markovian genetic regulatory networks are given by(4.6)Ha1=[-0.200-0.2],Hb1=[-0.10.20.080],Hc1=[-0.100-0.1],Hd1=[0.15000.15]E1=[0.3000.3],Ha2=[-0.300-0.3],Hb2=[0.1000.1],Hc2=[-0.100-0.1],Hd2=[0.1000.1],E2=[0.1000.1].
The nonlinear regulation function is taken as g(x)=x2/(1+x2), so we can easily get ki=0.65, the transmission probability is assumed to be γ=[-0.40.40.6-0.6].
Choosingh1=0.1, h2=0.1, d1=0, d2=0.2, μ1=0.1, μ2=0.1, e1=0.2, and e2=0.4. Applying Theorem 3.4 to system (3.16), we can get the following feasible solutions:
(4.7)P1=[3.6297-0.0003-0.00032.7195],P2=[3.6878-0.0003-0.00032.5334],Q1=[6.3467-0.0029-0.00295.0050],Q2=[5.6235-0.0024-0.00244.5655],N1=[4.6944003.5184],N2=[4.4643003.2752],N3=[10.6313-0.0028-0.00287.8495],M1=[0.2268000.2555],M2=[0.2297000.2511],M3=[0.2155000.2095],R1=[0.4985000.4617],R2=[4.01940.00020.00023.8311],R3=[3.84860.00020.00023.6607],R4=[7.0716-0.0110-0.01103.3522],U=[0.2339000.1500],V=[0.6615000.6188],Λ=[5.1026001.6577],ɛ1=5.4301,ɛ2=4.9533.
Assume h1=0.1,  μ1=0.1, μ2=0.1, e1=0.2, e2=0.4, and d1=0. Then we can calculate the maximal allowable bounds of d2 with different values of h2.It can be seen from Tables1 and 2 that using the lemma will yield less conservative results.Table 1
The maximal allowable bounds ofd2.
Methods
h
2
=
0
h
2
=
0.5
h
2
=
1
h
2
=
1.5
Theorem3.1
d
2
=
5.84
d
2
=
2.65
d
2
=
0.46
d
2
=
0.46
Corollary3.2
d
2
=
5.8
d
2
=
2.635
d
2
=
0.407
d
2
=
0.406Table 2
The maximal allowable bounds ofd2.
Methods
h
2
=
0.5
h
2
=
1
h
2
=
1.5
h
2
=
2
Theorem3.4
d
2
=
323
d
2
=
3.475
d
2
=
2.376
d
2
=
2.14
Corollary3.5
d
2
=
309
d
2
=
3.11
d
2
=
1.97
d
2
=
1.76


## 5. Conclusions

In this paper, we have delt with the robust stability analysis problem for the Markovian jumping genetic regulatory networks with parameter uncertainties and mode-dependent delays. By employing a new Lyapunov-Krasovskii function and a lemma to deal with the terms-(τi(t)/d2)ξT(t)×ϵ1TR1ϵ1ξ(t), -[1-(τi(t)/d2)]ξT(t)ϵ2TR1ϵ2ξ(t), -(σi(t)/e2)ηT(t)ϵ4TR3ϵ4η(t), and -[1-(σi(t)/e2)]ηT(t)×ϵ5TR3ϵ5η(t), some less conservative sufficient conditions in the terms of LMIs to ensure the robust stability of the addressed Markovian jumping genetic networks are derived. Finally, two examples are given to illustrate the usefulness of the derived LMIs-based stability conditions.



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*Source: 504378-2012-12-05.xml*