Abstract

The problem of finite-time stability of switched genetic regulatory networks (GRNs) with time-varying delays via Wirtinger’s integral inequality is addressed in this study. A novel Lyapunov–Krasovskii functional is proposed to capture the dynamical characteristic of GRNs. Using Wirtinger’s integral inequality, reciprocally convex combination technique and the average dwell time method conditions in the form of linear matrix inequalities (LMIs) are established for finite-time stability of switched GRNs. The applicability of the developed finite-time stability conditions is validated by numerical results.

1. Introduction

In recent years, GRNs have received much research attention, and many interesting results have been reported [1–8]. Generally, there are two types of gene network models, the Boolean model [9] and differential equation model [10]. In the Boolean model, the state converges to a terminal state via a series of state transitions that is determined by the Boolean rules. In this model, the activity of each gene is expressed in one of two states, ON or OFF, which is determined by a Boolean function by its own and by other related states. Whereas in the differential equation model, the variables describe the concentrations of gene products, such as mRNA and proteins as continuous values of the gene regulation system, and also, this model talks about the concentrations of gene products such as mRNA and proteins as variables in GRNs [11–14]. There are many research results on the stability analysis for GRNs with time delay (e.g., [15–18]).

Time delays are ubiquitous in many fields because of finite propagation speeds of signals, finite processing times, finite reaction times, and finite switching speed of amplifiers. Since the biological system especially GRNs is a slow process of transcription, translation, and translocation [19–23], the time delay cannot be avoided. From the long term investigations, time delay will bring instability of the system, sustained oscillations such as bifurcation [24–26]. So, it is of great importance to deal delayed GRNs. For instance, in [27], authors presented a different equation model for GRNs with constant time delays and proposed stability analysis for GRNs with time delays. In [28], authors developed delay dependent criteria for stability of GRNs with delay and free-weighting matrices.

On the other hand, Markovian switching and switched systems have been studied extensively over the past decades, for its capacity in modeling practical systems and its potential applications. Including a variety of subsystems as constituent parts, switched systems are governed by a switching rule to coordinate the switching. Recently, the stability problem of Markovian jump GRNs and switched GRNs has been investigated in [29–34]. As we all know, most of gene networks contain some kinds of switching mechanisms. For instance, by increasing stimulation or by changing some regulatory mechanisms, a bistable system can switch from one steady state to the other. In [33, 35–39], authors investigated the stability for switched systems with time delays by utilizing an average dwell time approach.

Recently, many kinds of finite-time issues have attracted particular research interests, and there have been some results on finite-time stabilization and synchronization [40–47]. However, to the best of the authors knowledge, there have been very few results on the finite-time stability problem for delayed GRNs with time delays [48, 49], and the purpose of this study is therefore to shorten such a gap.

Motivated by the above discussion, in this study, we are concerned with the finite-time stability of switched GRNs, where the parameter values switch from one mode to another. By utilizing the average dwell time approach and by using a novel Lyapunov–Krasovskii functional, it is shown that the finite-time stability problem is solvable if a set of linear matrix inequalities (LMIs) is feasible. Finally, three examples are provided in the end of the study to show the effectiveness of the proposed criteria.

The rest of this study is organized as follows: in Section 2, preliminaries and problem formulation are given. In Section 3, some conditions are established to ensure the finite-time stability of the considered system. In Section 4, three examples are illustrated to show the effectiveness of the obtained theoretical results. And finally, conclusions are given in Section 5. Notations: throughout this study, , , and denote, respectively, the set of all real numbers, real n-dimensional space, and real -dimensional space. denote the Euclidean norms in . For a vector or matrix , denotes its transpose. For a square matrix , and denote the maximum eigenvalue and minimum eigenvalue of matrix , respectively, and sym is used to represent . For simplicity, in symmetric block matrices, we often use to represent the term that is induced by symmetry.

2. Problem Description and Preliminaries

Consider the following nonlinear GRNs with time-varying delays described bywhere , :, are the concentrations of mRNA and protein, respectively; is the regulatory functions of mRNAs, and are the constant matrices, and they are the rates of degradation; represents the translation rate; is the regulative matrix, and and are the time-varying delays.

For obtaining our conclusions, we make the following assumptions.

Assumption 1. , s = 1, 2, …, n are monotonically increasing functions with saturation and satisfywhere , s = 1, 2, …, n are the nonnegative constants.

Assumption 2. and are the time-varying delays satisfyingwhere are the constants. The initial condition of system (1) is assumed to beUse the following transformation:where and constitute an equilibrium point of system (1) and then shift the intended equilibrium point to the origin. In this way, the system equation turns to bewhere , and .
According to Assumption 1 and the definition of , we know that is bounded, that is, , such that , s = 1, 2, …, n and satisfies the following sector condition:Let .
Sometimes, GRNs were described by the continuous time switched system, as in [33]; so system (6) can be described as the switching system with switching signal:where is the switching signal, which is a piece constant function depending on time t. For each , the matrices are constant matrices of appropriate dimensions.
For the switching signal , we have the following switching sequence: ; in other words, when , subsystem is activated. To assume , .
For proving the theorem, we recall the following definition and lemmas.

Definition 1 (See [50]). The system (8) is said to be finite-time stable with respect to positive real numbers , ifwhere

Definition 2 (See [51]). For any , let denote the switching number of on an interval . Ifholds for given , , then the constant is called the average dwell time and is the chatter bound. Without loss of generality, we choose throughout this study.

Lemma 1 (See [52]). For any constant matrix , , scalars , and vector function such that the integrations concerned are well defined, and the following inequality holds

Lemma 2 (See [53]). Let have positive values in an open subset D of . Then, the reciprocally convex combination of over D satisfiessubjected to

Lemma 3 (See [54]). For a positive definite matrix , the following inequality holds for all continuously differentiable function in :where , and .

3. Main Results

In this section, we present a finite-time stability theorem for switching genetic regulatory networks with interval time-varying delays (8).

Theorem 1. The switched genetic networks (8) is finite-time stable with respect to positive real numbers and constants , and ; if there exist symmetric positive definite matrices for all , the diagonal matrix , and positive scalars and such that the following LMIs holdand the average dwell time of the switching signal p (t) satisfieswhere