Exponential Stability Analysis for Genetic Regulatory Networks with Both Time-Varying and Continuous Distributed Delays

Abstract

The global exponential stability is investigated for genetic regulatory networks with time-varying delays and continuous distributed delays. By choosing an appropriate Lyapunov-Krasovskii functional, new conditions of delay-dependent stability are obtained in the form of linear matrix inequality (LMI). The lower bound of derivatives of time-varying delay is first taken into account in genetic networks stability analysis, and the main results with less conservatism are established by interactive convex combination method to estimate the upper bound of derivative function of the Lyapunov-Krasovskii functional. In addition, two numerical examples are provided to illustrate the effectiveness of the theoretical results.

1. Introduction

From the late 20th century to the early 21st century, more than a decade’s time, life science, especially molecular biology science, had great surprising changes. The research of genetic regulatory networks has become an important area in the molecular biology science and received great attention. There are plenty of results [1, 2]. Genetic regulatory networks can be seen as biochemically dynamical systems, and it is natural to simulate them by using dynamical system model. There are a variety of models that have been proposed but mainly are the Boolean network model (discrete model) [3–6] and the differential equation model (continuous model) [7, 8]. In the Boolean models, each gene’s activity is expressed with ON or OFF, and each gene’s state is described by the Boolean function of other related genes’ states. In differential equation model, the variables delineate the concentrations of gene products, such as mRNAs and proteins, which are continuous values. According to a large number of biological experiments, we know that gene expression is usually continuously variable, so it is more reasonable to describe genetic regulation networks with differential equation model than with the Boolean network model.

Gene expression is a complex process regulation by the stimulation and inhibition of protein including transcription, translation, and posttranslation processes, and a large number of reactions and reacting species participate in this process. There are fast reaction and slow reaction in real genetic regulatory systems. The fast reaction includes dimerization, binding reaction, and phosphorylation, and the slow reaction contains transcription, translation, and translocation or the finite switching speed of amplifiers. Due to the slow reaction, time delays exist in genetic regulatory networks. In [9], there is a biochemistry experiment on mice which has proved that there exists a time lag of about 15 min in the peaks between the mRNA molecules and the proteins of the gene Hes1.

The emergence of the time delays will influence the genetic regulatory networks’ dynamic behaviors, which cause the researchers’ interest. The stability is one of the very important dynamic characteristics. Hence, it is necessary to consider the stability of genetic regulatory networks with time delay; see [10–16]. In [11], random time delays are taken into account, and some stability criteria for the uncertain delayed genetic networks with SUM regulatory logic where each transcription factor acts additively to regulate a gene were obtained; asymptotical stability criteria were proposed for genetic regulatory networks with interval time-varying delays and nonlinear disturbance in [12]. In [14], authors gave some stochastic asymptotic stability sufficient conditions for the uncertain stochastic genetic regulatory networks with both mixed time-varying delays by constructing the Lyapunov functional and employing stochastic analysis methods. In [15, 16], the authors studied genetic regulatory networks with constant delay.

Motivated by the above discussions, we analyze the exponential stability of genetic regulatory networks with time-varying delays and continuous distributed delays. It is worth mentioning that the asymptotical stability of genetic regulatory networks is studied in most literature; see [17, 18]. But the exponential stability of our research is of better stability than asymptotical stability. Literature [11] discusses the exponential stability of genetic regulatory networks with random time delays, which are essentially constant delays. Our paper considers the system with interval time-varying delays and continuous distributed delays, which is more reasonable than literature [11]. Literature [19] studies robust exponential stability for stochastic genetic regulatory networks with time-varying delays, whose derivatives’ upper bound is less than 1. In our models, the derivatives’ upper bound of interval time-varying delays has no limit of less than 1. And we study the lower bound of derivatives of time-varying delay to systems stability effect for the first time. In the theorem for evidence, convex combination and interactive convex combination method were adopted, which have less conservatism.

This paper is organized as follows. In Section 2, model description and some assumptions are given. In Section 3, several sufficient results are obtained to check the exponential stability for genetic regulatory networks with time-varying delays and continuous distributed delays. Some numerical examples are given to demonstrate the effectiveness of our analysis in Section 4. Finally, conclusions are drawn in Section 5.

Notations. Throughout this paper, 𝐑, 𝐑^𝑛, and 𝐑^𝑛×𝑚 denote, respectively, the set of all real numbers, real 𝑛-dimensional space, and real 𝑛×𝑚-dimensional space. 𝐙₊ denote the set of all positive integers. ‖⋅‖ denote the Euclidean norms in 𝐑^𝑛. 𝐼 and 0 denote, respectively, the identity matrix and the zero matrix with appropriate dimension. For a vector or matrix 𝐴, 𝐴^T denotes its transpose. For a square matrix 𝐴, 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the maximum eigenvalue and minimum eigenvalue of matrix 𝐴, respectively, and sym(𝐴) is used to represent 𝐴+𝐴^T. For simplicity, in symmetric block matrices, we often use ∗ to represent the term that is induced by symmetry.

2. Problem Formulation and Some Preliminaries

In this paper, we are devoted to studying the stability to an autoregulatory genetic network with time delays described by the following delay differential equations: 𝑚̇_𝑖(𝑡)=−𝑎_𝑖𝑚_𝑖(𝑡)+𝜔_𝑖𝑝₁(𝑡−𝜎(𝑡))𝑝_𝑛((𝑡,𝑝̇𝑡−𝜎))𝑖=1,…,𝑛,_𝑖(𝑡)=−𝑐_𝑖𝑝_𝑖(𝑡)+𝑑_𝑖𝑚_𝑖(𝑡−𝜏(𝑡)),𝑖=1,…,𝑛, (1) where 𝑚_𝑖(𝑡)’s and 𝑝_𝑖(𝑡)’s are the concentrations of mRNAs and proteins, respectively; 𝑎_𝑖’s and 𝑐_𝑖’s are the degradation rates of mRNAs and proteins, respectively; 𝑑_𝑖’s are the translation rates of proteins; 𝜔_𝑖(⋅)’s are the regulatory functions of mRNAs, being generally monotonic to each argument; and 𝜎(𝑡) and 𝜏(𝑡) are time-varying delays.

Assumption 1. 𝜎(𝑡) and 𝜏(𝑡) are the time-varying delay satisfying 0≤𝜎₁≤𝜎(𝑡)≤𝜎₂,𝜎₃≤𝜎̇(𝑡)≤𝜎₄<∞,0≤𝜏₁≤𝜏(𝑡)≤𝜏₂,𝜏₃≤𝜏̇(𝑡)≤𝜏₄<∞, (2) where 𝜎₁,𝜎₂,𝜎₃,𝜎₄,𝜏₁,𝜏₂,𝜏₃,𝜏₄ are some constants.

Remark 2. In this paper, the lower bound of derivatives of time-varying delay is considered for the first time in the research of genetic regulatory networks. When information on lower bound of time-varying delay’s derivatives can be measured, our results are better than the previous works.

In genetic regulatory networks, some genes can be activated by one of a few different transcription factors (“OR” logic), and others can be activated by two or more transcription factors which must be bounded at the same time (“AND” logic). In this paper, we take a model of genetic regulatory networks where each transcription factor acts additively to regulate the 𝑖th gene (“SUM” logic) [20]. The regulatory function takes the form 𝜔_𝑖(𝑝₁(𝑡),…,𝑝_𝑛(𝑡))=∑^𝑛_𝑗=1𝜔_𝑖𝑗(𝑝_𝑗(𝑡)), and 𝜔_𝑖𝑗(𝑝_𝑗(𝑡)) is a monotonic function with the following Hill form [21]: 𝜔_𝑖𝑗𝑝_𝑗(𝑡)=𝛼_𝑖𝑗𝑝_𝑗(𝑡)/𝛽_𝑗^𝐻𝑗1+𝑝_𝑗(𝑡)/𝛽_𝑗^𝐻𝑗𝛼iftranscriptionfactor𝑗isanactivatorofgene𝑖,_𝑖𝑗11+𝑝_𝑗(𝑡)/𝛽_𝑗^𝐻𝑗iftranscriptionfactor𝑗isarepressorofgene𝑖, (3) where 𝐻_𝑗 is the Hill coefficient, 𝛽_𝑗 is a positive constant, and 𝛼_𝑖𝑗 is the constant transcriptional rate of 𝑗th transcriptional factor to 𝑖th gene.

Therefore, (1) can be rewritten as 𝑚̇_𝑖(𝑡)=−𝑎_𝑖𝑚_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑝_𝑗(𝑡−𝜎(𝑡))+𝑒_𝑖,𝑝̇𝑖=1,…,𝑛,_𝑖(𝑡)=−𝑐_𝑖𝑝_𝑖(𝑡)+𝑑_𝑖𝑚_𝑖(𝑡−𝜏(𝑡)),𝑖=1,…,𝑛, (4) where 𝑓_𝑗(𝑥)=(𝑥/𝛽_𝑗)^𝐻𝑗/(1+(𝑥/𝛽_𝑗))^𝐻𝑗 is monotonically increasing function; 𝑒_𝑖’s are basal rate defined by 𝑒_𝑖=∑_{𝑗∈𝑈𝑘}𝛼_𝑖𝑗 with 𝑈_𝑘={𝑗∣the𝑗thtranscription factor being a repressor of the kth gene, 𝑗=1,…,𝑛}; and matrix 𝐵=(𝑏_𝑖𝑗)∈𝐑^𝑛×𝑛 is defined as 𝑏_𝑖𝑗=𝛼_𝑖𝑗,iftranscriptionfactor𝑗isanactivatorofgene𝑖,0,ifthereisnolinkfromnode𝑗tonode𝑖,−𝛼_𝑖𝑗,iftranscriptionfactor𝑗isarepressorofgene𝑖. (5)

In the compact matrix form, (4) can be rewritten as 𝑚̇(𝑡)(𝑡)𝑝((𝑡𝑝̇(𝑡)(𝑡)((𝑡,=−𝐴𝑚+𝐵𝑓𝑡−𝜎))+𝐸,=−𝐶𝑃+𝐷𝑚𝑡−𝜏)) (6) where 𝑚(𝑡)=𝑚₁(𝑡),…,𝑚_𝑛(𝑡)^𝑇,𝑝(𝑡)=𝑝₁(𝑡),…,𝑝_𝑛(𝑡)^𝑇,𝑓𝑝(𝑡)=𝑓₁𝑝₁(𝑡),…,𝑓_𝑛𝑝_𝑛(𝑡)^𝑇,𝐴=diag𝑎₁,…,𝑎_𝑛,𝐼=𝑒₁,…,𝑒_𝑛^𝑇,𝐶=diag𝑐₁,…,𝑐_𝑛,𝐷=diag𝑑₁,…,𝑑_𝑛. (7)

In order to get the stability results, the following assumption is necessarily imposed on (6).

Assumption 3. 𝑓_𝑖∶𝐑→𝐑, 𝑖=1,…,𝑛, are monotonically increasing functions with saturation and moreover satisfy 𝑚⁻_𝑖≤𝑓_𝑖(𝑎)−𝑓_𝑖(𝑏)𝑎−𝑏≤𝑚⁺_𝑖,∀𝑎,𝑏∈𝐑,𝑖=1,…,𝑛, (8) where 𝑚⁻_𝑖 and 𝑚⁺_𝑖 are constants.

Remark 4. In the most existing literature, (8) was strengthened to 0≤((𝑓_𝑖(𝑎)−𝑓_𝑖(𝑏))/(𝑎−𝑏))≤𝑙_𝑖,forall𝑎,𝑏∈𝐑, where 𝑙_𝑖’s are positive constant. Therefore, Assumption 1 is somewhat general and, in fact, similar to that of [12]; see the discussion below.

The vectors 𝑚^∗,𝑝^∗ are said to be an equilibrium point of system (6), if they satisfy 0=−𝐴𝑚^∗+𝐵𝑓𝑝^∗(𝑡−𝜎(𝑡))+𝐸,0=−𝐶𝑃^∗+𝐷𝑚^∗(𝑡−𝜏(𝑡)). (9)

Let 𝑥(𝑡)=𝑚(𝑡)−𝑚^∗ and let 𝑦(𝑡)=𝑝(𝑡)−𝑝^∗; we get 𝑥̇(𝑡)(𝑡)𝑦((𝑡,𝑦̇(𝑡)(𝑡)((𝑡,=−𝐴𝑥+𝐵𝑔𝑡−𝜎))=−𝐶𝑦+𝐷𝑥𝑡−𝜏)) (10) where 𝑥(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]^𝑇, 𝑦(𝑡)=[𝑦₁(𝑡),…,𝑦_𝑛(𝑡)]^𝑇, 𝑔(𝑦(𝑡))=[𝑔(𝑦₁(𝑡)),…,𝑔(𝑦_𝑛(𝑡))]^𝑇, and 𝑔_𝑖(𝑦_𝑖(𝑡))=𝑓_𝑖(𝑦_𝑖(𝑡)+𝑝^∗_𝑖)−𝑓_𝑖(𝑝^∗_𝑖).

By the definition of 𝑔_𝑖(⋅), it satisfies sector condition: 𝑚⁻_𝑖≤𝑔_𝑖(𝑎)𝑎≤𝑚⁺_𝑖,𝑖=1,…,𝑛, (11) which implies that 𝑔_𝑖(𝑎)−𝑚⁻_𝑖𝑎𝑎𝑚≥0,^+𝑖𝑎−𝑔_𝑖(𝑎)𝑎≥0. (12) Let 𝑀₀=diag(𝑚⁻₁,…,𝑚⁻_𝑛), 𝑀₁=diag(𝑚⁺₁,…,𝑚⁺_𝑛), and 𝑚=max{|𝑚⁻₁|,…,|𝑚⁻_𝑛|,|𝑚⁺₁|,…,|𝑚⁺_𝑛|}.

The initial condition of system (10) is assumed to be 𝑥(𝑡)=𝜑₁(𝑡),𝑦(𝑡)=𝜓₁(𝑡)𝜎,−𝜌≤𝑡≤0,𝜌=max₂,𝜏₂. (13)

Based on system (10), we also consider the genetic regulatory networks with continuous distributed delays: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝐵𝑔𝑦(𝑡−𝜎(𝑡))+𝑊^𝑡_{𝑡−𝜗(𝑡)}𝑔𝑦(𝑠)𝑦̇(𝑡)(𝑡)((𝑡,𝑑𝑠,=−𝐶𝑦+𝐷𝑥𝑡−𝜏)) (14) where 0≤𝜗(𝑡)≤𝜗₁.

For completeness, we recall the following definition and lemmas.

Definition 5. System (10) or (14) is said to be globally exponentially stable, if there exist constants 𝜆>0 and 𝑀≥1, such that, for any initial value 𝑧_𝑡0, ‖𝑧(𝑡)‖≤𝑀𝑧_𝑡0𝐶1𝑒^{−𝜆(𝑡−𝑡0)} (15) hold, for all 𝑡≥0, where 𝑧(𝑡)=[𝑥(𝑡),𝑦(𝑡)]^𝑇 and ‖𝑧(𝑡)‖_𝐶¹=sup_{−𝜌≤𝜃≤0}{‖𝑧(𝑡+𝜃)‖,‖𝑧(̇𝑡+𝜃)‖}.

Lemma 6 (see [22]). For any positive definite matrix 𝑀∈𝐑^𝑛×𝑛, there exists a scalar 𝑞>0 and a vector-valued function 𝜔∶[0,𝑞]→𝐑^𝑛 such that ^𝑞₀𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝑞₀𝜔(𝑠)𝑑𝑠≤𝑞^𝑞₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (16)

Lemma 7 (see [23]). Let ℎ₁,…,ℎ_𝑁∶𝐑^𝑚→𝐑 take positive values in an open subset 𝐃 of 𝐑^𝑚. Then, the reciprocally convex combination of ℎ_𝑖 over 𝐃 satisfies min_{𝛼𝑖∣𝛼𝑖>0,∑𝑖𝛼𝑖=1𝑖}1𝛼_𝑖ℎ_𝑖𝜂=_𝑖ℎ_𝑖𝜂+max_{𝑘𝑖,𝑗(𝜂)𝑖≠𝑗}𝑘_𝑖,𝑗𝜂 (17) subject to 𝑘_𝑖,𝑗:𝐑^𝑚⟼𝐑,𝑘_𝑗,𝑖𝜂=𝑘_𝑖,𝑗𝜂,ℎ_𝑖𝜂𝑘_𝑖,𝑗𝜂𝑘_𝑖,𝑗𝜂ℎ_𝑗𝜂≥0. (18)

3. Main Results

In this section, several theorems are presented of genetic regulatory networks with both time-varying delays and continuous distributed delays. Firstly, a globally exponential stability result is developed for the genetic regulatory network with time-varying delays.

Theorem 8. For system (10) with Assumptions 1 and 3, the equilibrium point is globally exponentially stable (that is, there are two positive constants 𝛼 and 𝜆 such that ‖𝑧(𝑡)‖≤𝛼𝑒^−𝜆𝑡‖𝑧(𝑡₀)‖_𝐶1, for all 𝑡≥𝑡₀) if there exist positive definite matrices 𝐻₂,𝐻₄,𝑃₁,𝑃₂,𝑄_𝑖,𝑖=1,…,7,𝑊_𝑖, 𝑖=1,…,4, Γ₁=diag(𝛾₁₁,…,𝛾_1𝑛), and Γ₂=diag(𝛾₂₁,𝛾₂₂,...,𝛾_2𝑛), such that, for any appropriate dimensions constant matrices 𝐻₁,𝐻₃,𝑋,𝑌, the following LMIs hold: 𝑊₂𝑋𝑋𝑊₂𝑊≥0,⁴𝑌𝑌𝑊₄≥0,Ω₁+Ω₂+𝑂^𝑇₃𝑒^−𝜆𝜏2𝑄₂+𝑒^−𝜆𝜏1𝑄₃𝑂₃+𝑂^𝑇₈𝑒^−𝜆𝜎2𝑄₅+𝑒^−𝜆𝜎1𝑄₆𝑂₈<0, (19)(20) where Ω₁=diag(𝜆𝑃₁+𝑄₁−𝑒^−𝜆𝜏1𝑊₁−𝐻^𝑇₁𝐴−𝐴^𝑇𝐻₁, 𝑒^−𝜆𝜏1(𝑄₁−𝑄₂)−𝑒^−𝜆𝜏1𝑊₁−𝑒^−𝜆𝜏2𝑊₂, 𝑒^−𝜆𝜏1(1−𝜏₄)𝑄₃− 𝑒^−𝜆𝜏2[(1−𝜏₃)𝑄₂+2𝑊₂−𝑋^𝑇−𝑋], −𝑒^−𝜆𝜏2𝑊₂,𝜏²₁𝑊₁+(𝜏₂−𝜏₁)²𝑊₂−𝐻^𝑇₂−𝐻₂, 𝜆𝑃₂+𝑄₄−𝑒^−𝜆𝜎1𝑊₃− 2𝑀^𝑇₁Γ₁𝑀₀−𝐻^𝑇₃𝐶−𝐶^𝑇𝐻₃, −𝑒^−𝜆𝜎1(𝑄₄−𝑄₅+𝑊₃)−𝑒^−𝜆𝜎2𝑊₄, 𝑒^−𝜆𝜎1(1−𝜎₄)𝑄₆−𝑒^−𝜆𝜎2[(1−𝜎₃)𝑄₅+2𝑊₄− 𝑌^𝑇−𝑌]−𝑀^𝑇₁Γ₂𝑀₀, −𝑒^−𝜆𝜎2(𝑄₄+𝑊₄), 𝜎²₁𝑊₃+(𝜎₂−𝜎₁)²𝑊₄−𝐻^𝑇₄−𝐻₄, 𝑄₇−2Γ₁, −𝑒^−𝜆𝜎2𝑄₇(1−𝜎_𝑑)−2Γ₂), Ω₂=sym(𝑂^𝑇₁𝑒^−𝜆𝜏1𝑊₁𝑂₂+𝑂^𝑇₁[𝑃₁−𝐻^𝑇₁−𝐻^𝑇₂𝐴]𝑂₅+𝑂^𝑇₁𝐻^𝑇₁𝐵𝑂₁₂+𝑂^𝑇₂𝑒^−𝜆𝜏2(𝑊₂−𝑋)𝑂₃+𝑂^𝑇₂𝑒^−𝜆𝜏2𝑋𝑂₄+𝑂^𝑇₃𝑒^−𝜆𝜏2(𝑊₂−𝑋)𝑂₄+𝑂^𝑇₃𝐻^𝑇₃𝐷𝑂₆+𝑂^𝑇₃𝐻^𝑇₄𝐷𝑂₁₀+𝑂^𝑇₅𝐻^𝑇₂𝐵𝑂₁₂+𝑂^𝑇₆𝑒^−𝜆𝜎1𝑊₃𝑂₇+𝑂^𝑇₆[𝑃₂−𝐻^𝑇₃−𝐻^𝑇₄𝐶]𝑂₁₀+𝑂^𝑇₆[Γ₁𝑀₀+𝑀^𝑇₁Γ₁]𝑂₁₁+𝑂^𝑇₇𝑒^−𝜆𝜎2(𝑊₄−𝑌)𝑂₈+𝑂^𝑇₇𝑒^−𝜆𝜎2𝑌𝑂₉+𝑂^𝑇₈𝑒^−𝜆𝜎2(𝑊₄−𝑌)𝑂₉+𝑂^𝑇₈[Γ₂𝑀₀+𝑀^𝑇₁Γ₂]𝑂₁₂), and 𝑂_𝑖=[0_{𝑛×(𝑖−1)𝑛},𝐼_𝑛×𝑛,0_{𝑛×(13−𝑖)𝑛}], 𝑖=1,…,12.

Proof. Based on system (10), we construct the following Lyapunov-Krasovskii functional: 𝑉𝑥(𝑡),𝑦(𝑡)=𝑉₁𝑥(𝑡),𝑦(𝑡)+𝑉₂𝑥(𝑡),𝑦(𝑡)+𝑉₃𝑥(𝑡),𝑦(𝑡), (21) where 𝑉₁𝑥(𝑡),𝑦(𝑡)=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+𝑦^𝑇(𝑡)𝑃₂𝑦(𝑡),𝑉₂𝑥(𝑡),𝑦(𝑡)=^𝑡_𝑡−𝜏1𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑑𝑠_{𝑡−𝜏1𝑡−𝜏(𝑡)}𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠+^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝜎1}𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₄𝑦(𝑠)+𝑑𝑠_{𝑡−𝜎1(𝑡)𝑡−𝜎}𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₅𝑦(𝑠)+𝑑𝑠_{𝑡−𝜎(𝑡)𝑡−𝜎2}𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₆𝑦(𝑠)+𝑑𝑠_{𝑡𝑡−𝜎(𝑡)}𝑒^{𝜆(𝑠−𝑡)}𝑔^𝑇𝑦(𝑠)𝑄₇𝑔𝑦(𝑠𝑉)𝑑𝑠,₃𝑥(𝑡)(𝑡)=,𝑦_{0−𝜏1𝑡𝑡+𝜃}𝜏₁𝑒^{𝜆(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑊₁𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝜏1−𝜏2𝑡𝑡+𝜃}𝜏₂−𝜏₁𝑒^{𝜆(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑊₂𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜎1𝑡𝑡+𝜃}𝜎₁𝑒^{𝜆(𝑠−𝑡)}𝑦̇^𝑇(𝑠)𝑊₃𝑦̇(𝑠)𝑑𝑠𝑑𝜃+^−𝜎1_−𝜎2^𝑡_𝑡+𝜃𝜎₂−𝜎₁𝑒^{𝜆(𝑠−𝑡)}𝑦̇^𝑇(𝑠)𝑊₄𝑦̇(𝑠)𝑑𝑠𝑑𝜃. (22) Taking the derivatives of 𝑉_𝑖, 𝑖=1,2,3, we have 𝑉̇₁𝑥(𝑡),𝑦(𝑡)=2𝑥^𝑇(𝑡)𝑃₁𝑥̇(𝑡)+2𝑦^𝑇(𝑡)𝑃₂𝑦̇(𝑡),𝑉̇₂𝑥(𝑡),𝑦(𝑡)=−𝜆^𝑡_𝑡−𝜏1𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠−𝜆^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠−𝜆^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑𝑠−𝜆^𝑡_𝑡−𝜎1𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₄𝑦(𝑠)𝑑𝑠−𝜆^𝑡−𝜎1_{𝑡−𝜎(𝑡)}𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₅𝑦(𝑠)𝑑𝑠−𝜆^{𝑡−𝜎(𝑡)}_𝑡−𝜎2𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₆𝑦(𝑠)𝑑𝑠−𝜆^𝑡_{𝑡−𝜎(𝑡)}𝑒^{𝜆(𝑠−𝑡)}𝑔^𝑇𝑦(𝑠)𝑄₇𝑔𝑦(𝑠)𝑑𝑠+𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)−𝑒^−𝜆𝜏1𝑥^𝑇𝑡−𝜏₁𝑄₁𝑥𝑡−𝜏₁+𝑒^−𝜆𝜏1𝑥^𝑇𝑡−𝜏₁𝑄₂𝑥𝑡−𝜏₁−𝑒^−𝜆𝜏2𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑥(𝑡−𝜏(𝑡))(1−𝜏̇(𝑡))−𝑒^−𝜆𝜏2𝑥^𝑇𝑡−𝜏₂𝑄₃𝑥𝑡−𝜏₂+𝑒^−𝜆𝜏1𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₃𝑥(𝑡−𝜏(𝑡))(1−𝜏̇(𝑡))+𝑦^𝑇(𝑡)𝑄₄𝑦(𝑡)−𝑒^−𝜆𝜎1𝑦^𝑇𝑡−𝜎₁𝑄₄𝑦𝑡−𝜎₁+𝑒^−𝜆𝜎1𝑦^𝑇𝑡−𝜎₁𝑄₅𝑦𝑡−𝜎₁−𝑒^−𝜆𝜎2𝑦^𝑇(𝑡−𝜎(𝑡))𝑄₅𝑦(𝑡−𝜎(𝑡))(1−𝜎̇(𝑡))−𝑒^−𝜆𝜎2𝑦^𝑇𝑡−𝜎₂𝑄₆𝑦𝑡−𝜎₂+𝑒^−𝜆𝜎1𝑦^𝑇(𝑡−𝜎(𝑡))𝑄₆𝑦(𝑡−𝜎(𝑡))(1−𝜎̇(𝑡))+𝑔^𝑇𝑦(𝑡)𝑄₇𝑔𝑦(𝑡)−𝑒^−𝜆𝜎2𝑔^𝑇×𝑦(𝑡−𝜎(𝑡))𝑄₇𝑔𝑦((𝑡𝜎̇(𝑡,𝑉̇𝑡−𝜎))(1−))₃𝑥(𝑡),𝑦(𝑡)=−𝜆⁰_−𝜏1^𝑡_𝑡+𝜃𝜏₁𝑒^{𝜆(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑊₁𝑥̇(𝑠)𝑑𝑠𝑑𝜃−𝜆^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝜏₂−𝜏₁𝑒^{𝜆(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑊₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃−𝜆⁰_−𝜎1^𝑡_𝑡+𝜃𝜎₁𝑒^{𝜆(𝑠−𝑡)}𝑦̇^𝑇(𝑠)𝑊₃𝑦̇(𝑠)𝑑𝑠𝑑𝜃−𝜆^−𝜎1_−𝜎2^𝑡_𝑡+𝜃𝜎₂−𝜎₁𝑒^{𝜆(𝑠−𝑡)}𝑦̇^𝑇(𝑠)𝑊₄𝑦̇(𝑠)𝑑𝑠𝑑𝜃+𝜏²₁𝑥̇^𝑇(𝑡)𝑊₁𝑥̇(𝑡)−^𝑡_𝑡−𝜏1𝜏₁𝑒^{𝜆(𝜃−𝑡)}𝑥̇^𝑇(𝜃)𝑊₁𝑥̇(𝜃)+𝜏𝑑𝜃₂−𝜏₁²𝑥̇^𝑇(𝑡)𝑊₂𝑥̇(𝑡)−^𝑡−𝜏1_𝑡−𝜏2𝜏₂−𝜏₁𝑒^{𝜆(𝜃−𝑡)}𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃+𝜎²₁𝑦̇^𝑇(𝑡)𝑊₃𝑦̇(𝑡)−^𝑡_𝑡−𝜎1𝜎₁𝑒^{𝜆(𝜃−𝑡)}𝑦̇^𝑇(𝜃)𝑊₃𝑦̇(𝜃)+𝜎𝑑𝜃₂−𝜎₁²𝑦̇^𝑇(𝑡)𝑊₄𝑦̇(𝑡)−^𝑡−𝜎1_𝑡−𝜎2𝜎₂−𝜎₁𝑒^{𝜆(𝜃−𝑡)}𝑦̇^𝑇(𝜃)𝑊₄𝑦̇(𝜃)𝑑𝜃. (23) From Lemma 6, we have −^𝑡_𝑡−𝜏1𝜏₁𝑒^{𝜆(𝜃−𝑡)}𝑥̇^𝑇(𝜃)𝑊₁𝑥̇(𝜃)𝑑𝜃≤−𝑒^−𝜆𝜏1𝑥(𝑡)−𝑥𝑡−𝜏₁^𝑇𝑊₁𝑥(𝑡)−𝑥𝑡−𝜏₁,−^𝑡_𝑡−𝜎1𝜎₁𝑒^{𝜆(𝜃−𝑡)}𝑦̇^𝑇(𝜃)𝑊₃𝑦̇(𝜃)𝑑𝜃≤−𝑒^−𝜆𝜎1𝑦(𝑡)−𝑦𝑡−𝜎₁^𝑇𝑊₃𝑦(𝑡)−𝑦𝑡−𝜎₁. (24) Meanwhile, −^𝑡−𝜏1_𝑡−𝜏2𝜏₂−𝜏₁𝑒^{𝜆(𝜃−𝑡)}𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃≤−𝑒^{−𝜆𝜏2𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂−𝜏₁𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝑑𝜃_{𝑡−𝜏1𝑡−𝜏(𝑡)}𝜏₂−𝜏₁𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃=−𝑒^{−𝜆𝜏2𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝑑𝜃_{𝑡−𝜏1𝑡−𝜏(𝑡)}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝑑𝜃_{𝑡−𝜏(𝑡)𝑡−𝜏2}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝑑𝜃_{𝑡−𝜏1𝑡−𝜏(𝑡)}𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃=−𝑒^{−𝜆𝜏2𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝑑𝜃_{𝑡−𝜏1𝑡−𝜏(𝑡)}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝜏(𝑡)𝑑𝜃−𝜏₁𝜏₂(𝑡)×−𝜏_{𝑡−𝜏(𝑡)𝑡−𝜏2}𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)+𝜏𝑑𝜃₂−𝜏(𝑡)𝜏(𝑡)−𝜏₁×^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃. (25) By Lemma 6, we obtain that −^𝑡−𝜏1_𝑡−𝜏2𝜏₂−𝜏₁𝑒^{𝜆(𝜃−𝑡)}𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃≤−𝑒^−𝜆𝜏2𝜏₂−𝜏₁𝜏₂(𝑡)𝑥((𝑡−𝜏𝑡−𝜏))−𝑥𝑡−𝜏₂^𝑇×𝑊₂𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏₂+𝜏₂−𝜏₁𝜏(𝑡)−𝜏₁𝑥𝑡−𝜏₁−𝑥(𝑡−𝜏(𝑡))^𝑇×𝑊₂𝑥𝑡−𝜏₁−𝑥(𝑡−𝜏(𝑡)). (26) And by Lemma 7, we get that −^𝑡−𝜏1_𝑡−𝜏2𝜏₂−𝜏₁𝑒^{𝜆(𝜃−𝑡)}𝑥̇^𝑇(𝜃)𝑊₂𝑥̇(𝜃)𝑑𝜃≤−𝑒^−𝜆𝜏2𝑥𝑡−𝜏₁−𝑥(𝑡−𝜏(𝑡))^𝑇×𝑊₂𝑥𝑡−𝜏₁((𝑡+𝑥((𝑡−𝑥𝑡−𝜏))𝑡−𝜏))−𝑥𝑡−𝜏₂^𝑇×𝑊₂𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏₂+2𝑥𝑡−𝜏₁−𝑥(𝑡−𝜏(𝑡))^𝑇×𝑋𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏₂. (27) Similar to (27), −^𝑡−𝜎1_𝑡−𝜎2𝑒^{𝜆(𝜃−𝑡)}𝑦̇^𝑇(𝜃)𝑊₄𝑦̇(𝜃)𝑑𝜃≤−𝑒^−𝜆𝜎2𝑦𝑡−𝜎₁−𝑦(𝑡−𝜎(𝑡))^𝑇×𝑊₄𝑦𝑡−𝜎₁((𝑡+𝑦((𝑡−𝑦𝑡−𝜎))𝑡−𝜎))−𝑦𝑡−𝜎₂^𝑇×𝑊₄𝑦(𝑡−𝜎(𝑡))−𝑦𝑡−𝜎₂+2𝑦𝑡−𝜎₁−𝑦(𝑡−𝜎(𝑡))^𝑇×𝑌𝑦(𝑡−𝜎(𝑡))−𝑦𝑡−𝜎₂. (28) By Assumption 3, for any Γ_𝑖=diag(𝛾_𝑖1,…,𝛾_𝑖𝑛)≥0, 𝑖=1,2, the following inequality is true: −2^𝑛_𝑖=1𝛾_𝑖1𝑔_𝑖𝑦_𝑖(𝑡)−𝑚⁺_𝑖𝑦_𝑖(𝑡)𝑔_𝑖𝑦_𝑖(𝑡)−𝑚⁻_𝑖𝑦_𝑖(𝑡)−2^𝑛_𝑖=1𝛾_𝑖2𝑔_𝑖𝑦_𝑖(𝑡−𝜏(𝑡))−𝑚⁺_𝑖𝑦_𝑖((𝑡×𝑔𝑡−𝜏))_𝑖𝑦_𝑖(𝑡−𝜏(𝑡))−𝑚⁻_𝑖𝑦_𝑖(𝑡−𝜏(𝑡))≥0. (29) It can be rewritten as −2𝑔𝑦(𝑡)−𝑀₁𝑦(𝑡)^𝑇Γ₁𝑔𝑦(𝑡)−𝑀₀𝑦(𝑡𝑔𝑦((𝑡)−2𝑡−𝜎))−𝑀₁𝑦(𝑡−𝜎(𝑡))^𝑇×Γ₂𝑔𝑦(𝑡−𝜎(𝑡))−𝑀₀𝑦(𝑡−𝜎(𝑡))≥0. (30) For any constant matrices of appropriate dimensions 𝐻_𝑖, 𝑖=1,…,4, and from (10), we can obtain that 0=2𝑥^𝑇(𝑡)𝐻^𝑇₁+𝑥̇^𝑇(𝑡)𝐻^𝑇₂×−𝑥̇(𝑡)(𝑡)𝑦((𝑡,𝑦−𝐴𝑥+𝐵𝑔𝑡−𝜎))0=2^𝑇(𝑡)𝐻^𝑇₃+𝑦̇^𝑇(𝑡)𝐻^𝑇₄×−𝑦̇(𝑡)−𝐶𝑦(𝑡)+𝐷𝑥(𝑡−𝜏(𝑡)). (31) Combining (21)–(31), we have 𝑉̇𝑥(𝑡),𝑦(𝑡)+𝜆𝑉𝑥(𝑡),𝑦(𝑡)≤𝜉^T(𝑡)Ω+𝑂^𝑇₃𝑒^−𝜆𝜏2𝜏̇(𝑡)−𝜏₃𝑄₂+𝑒^−𝜆𝜏1𝜏₄−𝜏̇(𝑡)𝑄₃𝑂₃+𝑂^𝑇₈𝑒^−𝜆𝜎2𝜎̇(𝑡)−𝜎₃𝑄₅+𝑒^−𝜆𝜎1𝜎₄−𝜎̇(𝑡)𝑄₆𝑂₈𝜉(𝑡), (32) where 𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇𝑡−𝜏₁,𝑥^𝑇(𝑡−𝜏(𝑡)),𝑥^𝑇𝑡−𝜏₂,𝑥̇^𝑇(𝑡),𝑦^𝑇(𝑡),𝑦^𝑇𝑡−𝜎₁,𝑦^𝑇((𝑡,𝑦𝑡−𝜎))_𝑇𝑡−𝜎₂,𝑦̇^𝑇(𝑡),𝑔^𝑇𝑦(𝑡),𝑔^𝑇𝑦(𝑡−𝜎(𝑡)). (33) From (19) and (20), we can see that 𝑉̇𝑥(𝑡),𝑦(𝑡)+𝜆𝑉𝑥(𝑡),𝑦(𝑡)<0, (34) for all nonzero 𝜉(𝑡). Integrating the above inequality (34) from 𝑡₀ to 𝑡 gives 𝑉𝑥(𝑡),𝑦(𝑡)≤𝑒^{−𝜆(𝑡−𝑡0)}𝑉𝑥𝑡₀,𝑦𝑡₀. (35) From (21), we know that 𝑉𝑥(𝑡),𝑦(𝑡)≥𝑉₁𝑥(𝑡)(𝑡)𝜆,𝑦≥min_min𝑃₁,𝜆_min𝑃₂‖𝑧(𝑡)‖²,𝑉𝑥𝑡₀,𝑦𝑡₀≤𝜆_max𝑃₁+𝜆_max𝑃₂+𝜏₁𝜆_max𝑄₁+𝜏₂−𝜏₁𝜆_max𝑄₂+𝜆_max𝑄₃+𝜎₂−𝜎₁𝜆_max𝑄₅+𝜆_max𝑄₆+𝜎₁𝜆_max𝑄₄+𝜎₂𝑚²𝜆_max𝑄₇+12𝜏³₁𝜆_max𝑊₁+12𝜏₂−𝜏₁³𝜆_max𝑊₂+12𝜎³₁𝜆_max𝑊₃+12𝜎₂−𝜎₁³𝜆_max𝑊₄𝑧(𝑡₀)²_𝐶1. (36) Let 𝜆₁=min𝜆_min𝑃₁,𝜆_min𝑃₂,𝜆₂=𝜆_max𝑃₁+𝜆_max𝑃₂+𝜏₁𝜆_max𝑄₁+𝜏₂−𝜏₁𝜆_max𝑄₂+𝜆_max𝑄₃+𝜎₂−𝜎₁𝜆_max𝑄₅+𝜆_max𝑄₆+𝜎₁𝜆_max𝑄₄+𝜎₂𝑚²𝜆_max𝑄₇+12𝜏³₁𝜆_max𝑊₁+12𝜏₂−𝜏₁³𝜆_max𝑊₂+12𝜎³₁𝜆_max𝑊₃+12𝜎₂−𝜎₁³𝜆_max𝑊₄. (37) Then, by (35) and (36), we have 𝜆₁‖𝑧(𝑡)‖²≤𝑉𝑥(𝑡),𝑦(𝑡)≤𝜆₂𝑒^{−𝜆(𝑡−𝑡0)}𝑧(𝑡₀)²_𝐶1. (38) By (38), we get that ‖𝑧(𝑡)‖²≤𝜆₂𝜆₁𝑒^{−𝜆(𝑡−𝑡0)}𝑧(𝑡₀)²_𝐶1. (39) Let 𝛼=(𝜆₂/𝜆₁)^1/2, and, by Definition 5, the genetic regulatory networks in (10) are exponentially stable.
The proof is completed.

In the following, we consider the globally exponential stability of the genetic regulatory networks with time-varying delays and continuous distributed delays.

Theorem 9. For system (14) with Assumptions 1 and 3, the equilibrium point is globally exponentially stable (that is, there are two positive constants 𝛼^′ and 𝜆 such that ‖𝑧(𝑡)‖≤𝛼𝑒^−𝜆𝑡‖𝑧(𝑡₀)‖_𝐶1, for all 𝑡≥𝑡₀) if there exist positive definite matrices 𝐻₂,𝐻₄,𝑃₁,𝑃₂,𝑆,𝑄_𝑖,𝑖=1,…,7,𝑊_𝑖, 𝑖=1,…,4, Γ₁=diag(𝛾₁₁,…,𝛾_1𝑛), and Γ₂=diag(𝛾₂₁,…,𝛾_2𝑛), such that, for any appropriate dimensions constant matrices 𝐻₁,𝐻₃,𝑋,𝑌, the following LMIs hold: 𝑊₂𝑋𝑋𝑊₂𝑊≥0,⁴𝑌𝑌𝑊₄≥0,Ω^′₁+Ω^′₂+𝐿^𝑇₃𝑒^−𝜆𝜏2𝑄₂+𝑒^−𝜆𝜏1𝑄₃𝐿₃+𝐿^𝑇₈𝑒^−𝜆𝜎2𝑄₅+𝑒^−𝜆𝜎1𝑄₆𝐿₈<0, (40)(41) where Ω^′₁=diag(𝜆𝑃₁+𝑄₁−𝑒^−𝜆𝜏1𝑊₁−𝐻^𝑇₁𝐴−𝐴^𝑇𝐻₁, 𝑒^−𝜆𝜏1(𝑄₁−𝑄₂)−𝑒^−𝜆𝜏1𝑊₁−𝑒^−𝜆𝜏2𝑊₂, 𝑒^−𝜆𝜏1(1−𝜏₄)𝑄₃− 𝑒^−𝜆𝜏2[(1−𝜏₃)𝑄₂+2𝑊₂−𝑋^𝑇−𝑋], −𝑒^−𝜆𝜏2𝑊₂,𝜏²₁𝑊₁+(𝜏₂−𝜏₁)²𝑊₂−𝐻^𝑇₂−𝐻₂, 𝜆𝑃₂+𝑄₄−𝑒^−𝜆𝜎1𝑊₃− 2𝑀^𝑇₁Γ₁𝑀₀−𝐻^𝑇₃𝐶−𝐶^𝑇𝐻₃, −𝑒^−𝜆𝜎1(𝑄₄−𝑄₅+𝑊₃)−𝑒^−𝜆𝜎2𝑊₄, 𝑒^−𝜆𝜎1(1−𝜎₄)𝑄₆−𝑒^−𝜆𝜎2[(1−𝜎₃)𝑄₅+2𝑊₄− 𝑌^𝑇−𝑌]−𝑀^𝑇₁Γ₂𝑀₀, −𝑒^−𝜆𝜎2(𝑄₄+𝑊₄), 𝜎²₁𝑊₃+(𝜎₂−𝜎₁)²𝑊₄−𝐻^𝑇₄−𝐻₄, 𝑄₇−2Γ₁+𝜗²₁𝑆, −𝑒^−𝜆𝜎2𝑄₇(1−𝜎₄)−2Γ₂, −𝑒^−𝜆𝜗1𝑆,−𝑒^−𝜆𝜗1𝑆), Ω^′₂=sym(𝐿^𝑇₁𝑒^−𝜆𝜏1𝑊₁𝐿₂+𝐿^𝑇₁[𝑃₁−𝐻^𝑇₁−𝐻^𝑇₂𝐴]𝐿₅+𝐼^𝑇₁𝐻^𝑇₁𝐵𝐿₁₂+ 𝐿^𝑇₂𝑒^−𝜆𝜏2(𝑊₂−𝑋)𝐿₃+𝐿^𝑇₂𝑒^−𝜆𝜏2𝑋𝐿₄+𝐿^𝑇₃𝑒^−𝜆𝜏2(𝑊₂−𝑋)𝐿₄+𝐿^𝑇₃𝐻^𝑇₃𝐷𝐿₆+𝐿^𝑇₃𝐻^𝑇₄𝐷𝐿₁₀+𝐼^𝑇₅𝐻^𝑇₂𝐵𝐿₁₂+𝐿^𝑇₆𝑒^−𝜆𝜎1𝑊₃𝐿₇+ 𝐿^𝑇₆[𝑃₂−𝐻^𝑇₃−𝐻^𝑇₄𝐶]𝐿₁₀+𝐿^𝑇₆[Γ₁𝑀₀+𝑀^𝑇₁Γ₁]𝐿₁₁+𝐿^𝑇₇𝑒^−𝜆𝜎2(𝑊₄−𝑌)𝐿₈+𝐿^𝑇₇𝑒^−𝜆𝜎2𝑌𝐿₉+ 𝐿^𝑇₈𝑒^−𝜆𝜎2(𝑊₄−𝑌)𝐿₉+ 𝐿^𝑇₈[Γ₂𝑀₀+𝑀^𝑇₁Γ₂]𝐿₁₂), and 𝐿_𝑖=[0_{𝑛×(𝑖−1)𝑛},𝐼_𝑛×𝑛,0_{𝑛×(15−𝑖)𝑛}], 𝑖=1,…,12.

Proof. Based on system (14), we construct the following Lyapunov-Krasovskii functional: 𝑉𝑥(𝑡),𝑦(𝑡)=𝑉₁𝑥(𝑡),𝑦(𝑡)+𝑉₂𝑥(𝑡),𝑦(𝑡)+𝑉₃𝑥(𝑡),𝑦(𝑡)+𝑉₄𝑥(𝑡),𝑦(𝑡), (42) where 𝑉₁(𝑥(𝑡),𝑦(𝑡)), 𝑉₂(𝑥(𝑡),𝑦(𝑡)), and 𝑉₃(𝑥(𝑡),𝑦(𝑡)) are defined as in Theorem 8 and 𝑉₄𝑥(𝑡)(𝑡)=,𝑦_{0−𝜗1𝑡𝑡+𝜃}𝜗₁𝑒^{𝜆(𝑠−𝑡)}𝑔^𝑇𝑦(𝑠)𝑆𝑔𝑦(𝑠)𝑑𝑠𝑑𝜃. (43) Taking the derivative of 𝑉₄, 𝑉̇₄𝑥(𝑡),𝑦(𝑡)≤−𝜆⁰_−𝜗1^𝑡_𝑡+𝜃𝜗₁𝑒^{𝜆(𝑠−𝑡)}𝑔^𝑇𝑦(𝑠)𝑆𝑔𝑦(𝑠)𝑑𝑠𝑑𝜃+𝜗²₁𝑔^T𝑦(𝑡)𝑆𝑔𝑦(𝑡)−𝑒^{−𝜆𝜗1𝑡}_𝑡−𝜗1𝜗₁𝑔^𝑇𝑦(𝜃)𝑆𝑔𝑦(𝜃)𝑑𝜃. (44) By Lemma 6, we get that 𝑉̇₄𝑥(𝑡),𝑦(𝑡)≤−𝜆⁰_−𝜗1^𝑡_𝑡+𝜃𝜗₁𝑒^{𝜆(𝑠−𝑡)}𝑔^𝑇𝑦(𝑠)𝑆𝑔𝑦(𝑠)𝑑𝑠𝑑𝜃+𝜗²₁𝑔^𝑇𝑦(𝑡)𝑆𝑔𝑦(𝑡)−𝑒^−𝜆𝜗1𝜗₁𝜗₁(𝑡)×−𝜗_{𝑡−𝜗(𝑡)𝑡−𝜗1}𝑔(𝑦(𝜃))𝑑𝜃^𝑇𝑆^{𝑡−𝜗(𝑡)}_𝑡−𝜗1𝑔𝑦(𝜃)𝑑𝜃−𝑒^−𝜆𝜗1𝜗₁𝜗(𝑡)×^𝑡_{𝑡−𝜗(𝑡)}𝑔(𝑦(𝜃))𝑑𝜃^𝑇𝑆^𝑡_{𝑡−𝜗(𝑡)}𝑔𝑦(𝜃)𝑑𝜃≤−𝜆⁰_−𝜗1^𝑡_𝑡+𝜃𝜗₁𝑒^{𝜆(𝑠−𝑡)}𝑔^𝑇𝑦(𝑠)𝑆𝑔𝑦(𝑠)𝑑𝑠𝑑𝜃+𝜗²₁𝑔^𝑇𝑦(𝑡)𝑆𝑔𝑦(𝑡)−𝑒^−𝜆𝜗1𝜗(𝑡)1+𝜗₁×^{𝑡−𝜗(𝑡)}_𝑡−𝜗1𝑔(𝑦(𝜃))𝑑𝜃^𝑇𝑆^{𝑡−𝜗(𝑡)}_𝑡−𝜗1𝑔𝑦(𝜃)𝑑𝜃−𝑒^−𝜆𝜗1𝜗1+¹−𝜗(𝑡)𝜗₁×^𝑡_{𝑡−𝜗(𝑡)}𝑔(𝑦(𝜃))𝑑𝜃^𝑇𝑆^𝑡_{𝑡−𝜗(𝑡)}𝑔𝑦(𝜃)𝑑𝜃. (45) Combining (23)–(31), (42), and (45), we get 𝑉̇𝑥(𝑡),𝑦(𝑡)+𝜆𝑉𝑥(𝑡),𝑦(𝑡)≤𝜉^𝑇₁(𝑡)Ω+𝐿^𝑇₃𝑒^−𝜆𝜏2𝜏̇(𝑡)−𝜏₃𝑄₂+𝑒^−𝜆𝜏1𝜏₄−𝜏̇(𝑡)𝑄₃𝐿₃+𝐿^𝑇₈𝑒^−𝜆𝜎2𝜎̇(𝑡)−𝜎₃𝑄₅+𝑒^−𝜆𝜎1𝜎₄−𝜎̇(𝑡)𝑄₆𝐿₈𝜉₁(𝑡), (46) where 𝜉^𝑇₁(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇𝑡−𝜏₁,𝑥^𝑇((𝑡,𝑥𝑡−𝜏))_𝑇𝑡−𝜏₂,𝑥̇^𝑇(𝑡),𝑦^𝑇(𝑡),𝑦^𝑇𝑡−𝜎₁,𝑦^𝑇(𝑡−𝜎(𝑡)),𝑦^𝑇𝑡−𝜎₂,𝑦̇^𝑇(𝑡),𝑔^𝑇𝑦(𝑡),𝑔^𝑇𝑦(𝑡−𝜎(𝑡)),^{𝑡−𝜗(𝑡)}_𝑡−𝜗1𝑔(𝑦(𝜃))𝑑𝜃^𝑇,_{𝑡𝑡−𝜗(𝑡)}𝑔(𝑦(𝜃))𝑑𝜃^𝑇. (47) By (40) and (41), we get that 𝑉̇(𝑥(𝑡),𝑦(𝑡))+𝜆𝑉(𝑥(𝑡),𝑦(𝑡))<0. Similar to the proof of Theorem 8 we have ‖𝑧(𝑡)‖²≤𝜆^′₂𝜆₁𝑒^{−𝜆(𝑡−𝑡0)}𝑧(𝑡₀)²_𝐶1. (48) where 𝜆^′₂=𝜆₂+(1/2)𝜗³₁𝑚²𝜆_max(𝑆) and 𝜆₁, 𝜆₂ are defined as Theorem 8. Let 𝛼^′=(𝜆^′₂/𝜆₁)^1/2, and, by Definition 5, the genetic regulatory network (14) is exponentially stable.
The proof is completed.

Remark 10. In the proof of Theorems 8 and 9, we use convex combination and interactive convex combination definition to estimate the upper bound of derivative function of the Lyapunov-Krasovskii functional and obtain some new conservative weaker sufficient conditions.

Remark 11. When the lower bound of derivatives of time-varying delays is immeasurable, let ∫^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑒^{𝜆(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑𝑠=0 and ∫^{𝑡−𝜎(𝑡)}_𝑡−𝜎2𝑒^{𝜆(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑄₆𝑦(𝑠)𝑑𝑠=0 in Theorem 8 (20) or Theorem 9 (41); our results are true still.

4. Numerical Examples

In this section, two examples are given to illustrate the effectiveness of our theoretical results.

Example 1. Consider a genetic regulatory network model reported by Elowitz and Leiber [24], which studied the dynamics of repressilator which is cyclic negative-feedback loop comprising three repressor genes (lacl, tetR, and cl) and their promoters (cl, lacl, and tetR): 𝑑𝑥_𝑖𝑑𝑡=−𝑥_𝑖+𝛼1+𝑦^𝑛_𝑗+𝛼₀,𝑑𝑦_𝑖𝑥𝑑𝑡=𝛽_𝑖−𝑦_𝑖. (49) Taking time-varying delays into account and shifting the equilibrium point to the origin, one gets the following model: 𝑥̇(𝑡)(𝑡)𝑦((𝑡,𝑦̇(𝑡)(𝑡)((𝑡,=−𝐴𝑥+𝐵𝑔𝑡−𝜎))=−𝐶𝑦+𝐷𝑥𝑡−𝜏)) (50) where 𝐴=diag(2,2,2), 𝐶=diag(3,3,3), 𝐷=diag(1,1,1), and the coupling matrix 𝐵=1.5×.00−1−1000−10 (51)

The gene regulation function is taken as 𝑔(𝑥)=𝑥²/(1+𝑥²), 𝑀₀=diag(0,0,0), and 𝑀₁=diag(0.65,0.65,0.65). The time delays 𝜎(𝑡) and 𝜏(𝑡) are assumed to be 𝜎(𝑡)=1+2sin²𝑡,𝜏(𝑡)=1+cos²𝑡. (52) We can get the parameters as follows: 𝜎₁=1,𝜎₂=3,𝜎₃=−2,𝜎₄𝜏=2,₁=1,𝜏₂=2,𝜏₃=−1,𝜏₄=1, (53) and choose 𝜆=0.5. In accordance with the condition in Theorem 8, system (50) is exponentially stable. By using the MATLAB LMI toolbox, we can get the feasible solutions. Due to the space limitation, we only list matrices 𝑃₁ and 𝑃₂ here as follows: 𝑃₁=,𝑃17.10417.99977.99977.999717.10417.99977.99977.999717.1041₂=.13.89430.55730.55730.557313.89430.55730.55730.557313.8943 (54) The initial condition is 𝑥(0)=(0.3,0.5,0.4)^𝑇 and 𝑦(0)=(0.2,0.4,0.6)^𝑇. The simulation results of the trajectories are shown in Figure 1.

(a)

(b)

Figure 1 

(a) mRNA concentrations 𝑥(𝑡). (b) Protein concentrations 𝑦(𝑡).

Example 2. In this example, we consider the genetic regulatory network (14) with time-varying delays and continuous distributed delays, in which the parameters are listed as follows: ()(),(),𝐴=diag1,2,3,𝐶=diag5,4,5𝐷=diag0.3,0.2,0.4,𝐵=00.80000.80.800 (55) and 𝑔(𝑥)=𝑥²/(1+𝑥²), 𝑀₀=diag(0,0,0), and 𝑀₁=diag(0.65,0.65,0.65). The time delays 𝜎(𝑡), 𝜏(𝑡), and 𝜗(𝑡) are assumed to be 𝜎(𝑡)=0.5+0.3sin²𝑡,𝜏(𝑡)=0.4+0.1cos²𝜗(𝑡)𝑡,=2sin²(𝑡). (56) We can get the parameters as follows: 𝜎₁=0.5,𝜎₂=0.8,𝜎₃=−0.3,𝜎₄𝜏=0.3,₁=0.4,𝜏₂=0.5,𝜏₃=−0.1,𝜏₄=0.1,𝜗₁=2 (57)(58) and choose 𝜆=0.2. By using the MATLAB LMI toolbox, we can get the feasible solutions. Due to the space limitation, we only list matrices 𝑃₁ and 𝑃₂ here as follows: 𝑃₁=,𝑃7.1735−0.4713−0.2152−0.47137.2143−0.3071−0.2152−0.30716.9607₂=.7.93630008.01990007.9001 (59)

5. Concluding Remarks

This paper has investigated the exponential stability of genetic regulatory networks with time-varying delays and continuous distributed delays. By using the novel Lyapunov-Krasovskii functions and employing the Jensen inequality and the interactive convex combination method, some sufficient criteria are given to ensure the exponential stability with less conservative. All the obtained conditions are dependent on the delays and on linear matrix inequalities. Two examples are provided to illustrate the effectiveness of our results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (61273084, 61233014, and 61174217), the Natural Science Foundation for Distinguished Young Scholar of Shandong Province of China (JQ200919), the Independent Innovation Foundation of Shandong University (2012JC014), the Natural Science Foundation of Shandong Province of China (ZR2010AL016, ZR2011AL007), and the Doctoral Foundation of University of Jinan (XBS1244).