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

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 linearmatrix inequalities. Illustrative examples are exploited to show the effectiveness of the derived linear-matrix-inequalitiesLMISbased stability conditions.


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, R n and R n×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 λ max A resp., λ min 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; R n denote the family of continuous functions φ from −h, 0 to R n with the norm φ sup −h≤θ≤0 |φ θ |, where | • | is the Euclidean norm in R n ; E{•} stands for the mathematical expectation operator.Let Ω, F, {F t } t≥0 , P be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions i.e., the filtration contains all P -null sets and is right continuous .Denote by L p

Model Description
In this paper, we will consider the following genetic regulatory networks 25 : where Δt > 0 and lim Δt → 0 O Δt /Δt 0. Here, γ ij ≥ 0 is the transition rate from i to j if i / j, while γ ii − N j 1,j / i γ ij .τ r t t and σ r t t are the time-varying delays when the mode is in r t and we assume that they satisfy the following conditions where d 1i , d 2i , e 1i , e 2i , h i , and μ i are known real constants, for any i ∈ S, denote

2.8
Remark 2.1.In 25 , h i and μ i 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 vector x, y ∈ R n and matrix Q > 0, one has the following inequality: Lemma 2.3 see 30 .For any positive definite matrix M > 0, scalar γ > 0 and vector function ω : 0, γ → R n such that the integrations concerned are well defined, then the following inequality holds: Lemma 2.5 see 32 .Assume Ω, X 1 , and X 2 are constant matrices with appropriate dimensions,

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 modedependent 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 any diagonal positive definite matrix Λ, and any matrices U, V with appropriate dimensions such that the following LMIs hold: where Proof.Choose a Lyapunov-Krasovskii functional candidate: Let L be the weak infinite generator.Then for each r t i, i ∈ S along the trajectory of 2.5 one has x T s N 3 x s ds.

Mathematical Problems in Engineering
Similarly t−e 2 y T s M 3 y s ds,

Mathematical Problems in Engineering
Noting the sector condition 2.4 , for any positive matrix Λ we have For any matrices U and V with appropriate dimensions, we have

3.10
By Lemma 2.3 we can get the following inequalities:

3.11
From 3.3 to 3.11 we can get

3.13
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 Theorem 3.  that is, we do not use Lemma 2.5, then we will have the following corollary.
Remark 3.3.In the proof of Theorem 3.1, if we ignore the terms R 3 5 η 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 T  5 R 3 5 η 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: where ΔA i , ΔB i , ΔC i , and ΔD i are the parametric uncertainties satisfying: 3.17 , and H di are the known real constant matrices with appropriate dimensions, F i satisfies

3.18
Theorem 3.4.The genetic regulatory networks 3.16 is robust asymptotically stable, if there exist real number {ε i , i ∈ S}, any diagonal positive definite matrix Λ, and any matrices U and V with appropriate dimensions such that the following LMIs hold: where

3.20
Proof.Consider the same Lyapunov-Krasovskii functional 3.3 , do the differential along the trajectory 3.16 , one can readily get

3.21
By Lemmas 2.4 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 Theorem 3.1, if we ignore the terms number {ε i , i ∈ S}, any diagonal positive definite matrix Λ, and any matrices U and V with appropriate dimensions such that the following LMIs hold: where Ω 1i and Ω 2i are defined in Theorem 3.4.

Illustrative Examples
In this section, two numerical examples are given to illustrate the effectiveness of the derived results.

4.1
The nonlinear regulation function is taken as g x x 2 / 1 x 2 , so we can easily get k i 0.65, the transmission probability is assumed to be γ −0.

4.5
The uncertain parameters for every mode of the Markovian genetic regulatory networks are given by It can be seen from Tables 1 and 2 that using the lemma will yield less conservative results.

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 − τ i t /d 2 ξ T t × T 1 R 1 1 ξ t , − 1− τ i t /d 2 ξ T t T 2 R 1 2 ξ t , − σ i t /e 2 η T t T 4 R 3 4 η t , and − 1 − σ i t /e 2 η T t × T 5 R 3 5 η 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 LMIsbased stability conditions.
, and m i t and p i t are the concentrations of mRNA and protein of the ith node at time t, respectively; A diag a 1 , a 2 , . . ., a n and C diag c 1 , c 2 , . . ., c n denote the degradation or dilution rates of mRNAs and proteins, D diag d 1 , d 2 , . . ., d n represents the translation rate, and B b ij ∈ R n×n is defined as follows: e 1 y T t M 3 y t γ ii 1, if we deal with the terms −d 2 e 2 η T t T 4 R 3 4 η t , and − 1 − σ i t /e 2 η T t T 5 R 3 5 η t , we can get the following corollary.The genetic regulatory networks 3.16 is robust asymptotically stable, if there exist Hence the Markovian jumping genetic regulatory networks with mode-dependent delays is asymptotically stable.Assume h 1 0.2, μ 1 0.1, μ 2 0.2, e 1 0.1, e 2 0.5, and d 1 0.2.Then we can calculate the maximal allowable bounds of d 2 with different values of h 2 .

Table 1 :
The maximal allowable bounds of d 2 .