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 linear matrix inequalities. Illustrative examples are exploited to show the effectiveness of the derived linear-matrix-inequalities- (LMIS-) based stability conditions.
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  presented a different equation model for GRNs with constant time delays and proposed necessary and sufficient conditions for such GRNs. Ren and Cao  derived delay-dependent robust asymptotic stability criteria for a class of genetic regulatory networks with time-varying delays and parameter uncertainties. Wang et al.  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.  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, and denote, respectively, -dimensional Euclidean space and the set of all real matrices. The superscript “” denotes the matrix transposition and the notation (resp., ) where and are symmetric matrices, which means that is a positive semidefinite (resp., positive definite) matrix, is the identity matrix, and (resp., ) represents the largest (resp., smallest) eigenvalue of matrix . For symmetric block matrices or long matrix expressions, an asterisk is used to represent a term that is induced by symmetry. Let , and denote the family of continuous functions from to with the norm , where is the Euclidean norm in ; stands for the mathematical expectation operator. Let be a complete probability space with a filtration satisfying the usual conditions (i.e., the filtration contains all -null sets and is right continuous). Denote by the family of all -measurable -valued random variables such that .
2. Model Description
In this paper, we will consider the following genetic regulatory networks : where , and and are the concentrations of mRNA and protein of the th node at time , respectively; and denote the degradation or dilution rates of mRNAs and proteins, represents the translation rate, and is defined as follows: denotes the feedback regulation of the protein on the transcription, which is the monotonic function in Hill form, , and is the Hill coefficient; and are the time delays; , is the base transcriptional rate of the repressor of gene . Assume and are the equilibrium points of (2.1), defining , , it is easy to get where , from the definition of , it is easy to get
Taking the Markovian jumping parameters and stochastic delays into account, a Markovian jumping genetic regulatory networks model with mode-dependent delays is considered as where is a continuous-time Markovian process with right continuous trajectories and taking values in a finite set with the following transition probabilities: where and . Here, is the transition rate from to if , while .
and are the time-varying delays when the mode is in and we assume that they satisfy the following conditions where , , , , , and are known real constants, for any , denote
Remark 2.1. In , and 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 ). For any vector and matrix , one has the following inequality:
Lemma 2.3 (see ). For any positive definite matrix , scalar and vector function such that the integrations concerned are well defined, then the following inequality holds:
Lemma 2.4 (see ) (Schur complement). Given constant matrices , , and where and . Then if and only if
Lemma 2.5 (see ). Assume , and are constant matrices with appropriate dimensions, , then is equivalent to
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 , matrices , any diagonal positive definite matrix , and any matrices , with appropriate dimensions such that the following LMIs hold: where
Proof. Choose a Lyapunov-Krasovskii functional candidate:
Let be the weak infinite generator. Then for each along the trajectory of (2.5) one has Similarly
Note that Similarly, Noting the sector condition (2.4), for any positive matrix we have
For any matrices and with appropriate dimensions, we have
By Lemma 2.3 we can get the following inequalities: From (3.3) to (3.11) we can get where 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.
Corollary 3.2. The genetic regulatory networks (2.5) is asymptotically stable, if there exist matrix sets , matrices , any diagonal positive definite matrix , and any matrices and with appropriate dimensions such that the following LMIs hold: where, and are defined in Theorem 3.1.
Remark 3.3. In the proof of Theorem 3.1, if we ignore the terms , , and , , 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 , and , , , 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 , , , and are the parametric uncertainties satisfying: , , , , and are the known real constant matrices with appropriate dimensions, satisfies
Theorem 3.4. The genetic regulatory networks (3.16) is robust asymptotically stable, if there exist , , , , , , , , , , , real number , any diagonal positive definite matrix , and any matrices and with appropriate dimensions such that the following LMIs hold: where
Proof. Consider the same Lyapunov-Krasovskii functional (3.3), do the differential along the trajectory (3.16), one can readily get
By Lemmas 2.4 and 2.5, we can get that 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 , , , and , we can get the following corollary.
Corollary 3.5. The genetic regulatory networks (3.16) is robust asymptotically stable, if there exist , , , , , , , , , , , real number , any diagonal positive definite matrix , and any matrices and with appropriate dimensions such that the following LMIs hold: where and 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.
Consider (2.5) where
The nonlinear regulation function is taken as , so we can easily get , the transmission probability is assumed to be , and time delays are chosen as Then we have
By using Matlab Toolbox, solving (3.1) we can obtain the feasible solutions Hence the Markovian jumping genetic regulatory networks with mode-dependent delays is asymptotically stable. Assume , ,, , , and . Then we can calculate the maximal allowable bounds of with different values of .
Consider (3.16) where
The uncertain parameters for every mode of the Markovian genetic regulatory networks are given by