exponential stability analysis for genetic regulatory networks with both time varying and continuous distributed delays

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 897280, 10 pages http://dx.doi.org/10.1155/2014/897280 Research Article Exponential Stability Analysis for Genetic Regulatory Networks with Both Time-Varying and Continuous Distributed Delays Lizi Yin1,2 and Yungang Liu1 School of Control Science and Engineering, Shandong University, Jinan 250061, China School of Mathematical Sciences, University of Jinan, Jinan 250022, China Correspondence should be addressed to Yungang Liu; lygfr@sdu.edu.cn Received 23 December 2013; Revised March 2014; Accepted March 2014; Published May 2014 Academic Editor: Sanyi Tang Copyright © 2014 L Yin and Y Liu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited 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 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 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 timevarying 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 timevarying delays, whose derivatives’ upper bound is less than In our models, the derivatives’ upper bound of interval time-varying delays has no limit of less than 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 Finally, conclusions are drawn in Section Notations Throughout this paper, R, R𝑛 , and R𝑛×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 R𝑛 𝐼 and denote, respectively, the identity matrix and the zero matrix with appropriate dimension For a vector or matrix 𝐴, 𝐴T denotes its transpose For a square matrix 𝐴, 𝜆 max (𝐴) and 𝜆 (𝐴) denote the maximum eigenvalue and minimum eigenvalue of matrix 𝐴, respectively, and sym(𝐴) is used to represent 𝐴 + 𝐴T For simplicity, in symmetric block matrices, we often use ∗ to represent the term that is induced by symmetry Abstract and Applied Analysis 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) = −𝑎𝑖 𝑚𝑖 (𝑡) + 𝜔𝑖 (𝑝1 (𝑡 − 𝜎 (𝑡)) 𝑝𝑛 (𝑡 − 𝜎 (𝑡))) , 𝑖 = 1, , 𝑛, 𝑝𝑖̇ (𝑡) = −𝑐𝑖 𝑝𝑖 (𝑡) + 𝑑𝑖 𝑚𝑖 (𝑡 − 𝜏 (𝑡)) , (1) 𝑖 = 1, , 𝑛, where 𝑚𝑖 (𝑡)’s and 𝑝𝑖 (𝑡)’s are the concentrations of mRNAs and proteins, respectively; 𝑎𝑖 ’s and 𝑐𝑖 ’s are the degradation rates of mRNAs and proteins, respectively; 𝑑𝑖 ’s are the translation rates of proteins; 𝜔𝑖 (⋅)’s are the regulatory functions of mRNAs, being generally monotonic to each argument; and 𝜎(𝑡) and 𝜏(𝑡) are time-varying delays Assumption 𝜎(𝑡) and 𝜏(𝑡) are the time-varying delay satisfying ≤ 𝜎1 ≤ 𝜎 (𝑡) ≤ 𝜎2 , 𝜎3 ≤ 𝜎̇ (𝑡) ≤ 𝜎4 < ∞, ≤ 𝜏1 ≤ 𝜏 (𝑡) ≤ 𝜏2 , 𝜏3 ≤ 𝜏̇ (𝑡) ≤ 𝜏4 < ∞, (2) where 𝜎1 , 𝜎2 , 𝜎3 , 𝜎4 , 𝜏1 , 𝜏2 , 𝜏3 , 𝜏4 are some constants Remark 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 𝜔𝑖 (𝑝1 (𝑡), , 𝑝𝑛 (𝑡)) = ∑𝑛𝑗=1 𝜔𝑖𝑗 (𝑝𝑗 (𝑡)), and 𝜔𝑖𝑗 (𝑝𝑗 (𝑡)) is a monotonic function with the following Hill form [21]: 𝐻𝑗 (𝑝𝑗 (𝑡) /𝛽𝑗 ) { { { 𝛼𝑖𝑗 { 𝐻 { { { + (𝑝𝑗 (𝑡) /𝛽𝑗 ) 𝑗 { { { { if transcription factor 𝑗 is an { { { { 𝜔𝑖𝑗 (𝑝𝑗 (𝑡)) = { activator of gene 𝑖, { { { 𝛼𝑖𝑗 { { { + (𝑝 (𝑡) /𝛽 )𝐻𝑗 { { 𝑗 𝑗 { { { { if transcription factor 𝑗 is a { { { repressor of gene 𝑖, (3) Abstract and Applied Analysis where 𝐻𝑗 is the Hill coefficient, 𝛽𝑗 is a positive constant, and 𝛼𝑖𝑗 is the constant transcriptional rate of 𝑗th transcriptional factor to 𝑖th gene Therefore, (1) can be rewritten as The vectors 𝑚∗ , 𝑝∗ are said to be an equilibrium point of system (6), if they satisfy = −𝐴𝑚∗ + 𝐵𝑓 (𝑝∗ (𝑡 − 𝜎 (𝑡))) + 𝐸, = −𝐶𝑃∗ + 𝐷𝑚∗ (𝑡 − 𝜏 (𝑡)) 𝑛 𝑚̇ 𝑖 (𝑡) = −𝑎𝑖 𝑚𝑖 (𝑡) + ∑𝑏𝑖𝑗 𝑓𝑗 (𝑝𝑗 (𝑡 − 𝜎 (𝑡))) + 𝑒𝑖 , 𝑗=1 𝑖 = 1, , 𝑛, 𝑝𝑖̇ (𝑡) = −𝑐𝑖 𝑝𝑖 (𝑡) + 𝑑𝑖 𝑚𝑖 (𝑡 − 𝜏 (𝑡)) , (4) Let 𝑥(𝑡) = 𝑚(𝑡) − 𝑚∗ and let 𝑦(𝑡) = 𝑝(𝑡) − 𝑝∗ ; we get 𝑥̇ (𝑡) = −𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) , 𝑖 = 1, , 𝑛, 𝑦̇ (𝑡) = −𝐶𝑦 (𝑡) + 𝐷𝑥 (𝑡 − 𝜏 (𝑡)) , where 𝑓𝑗 (𝑥) = (𝑥/𝛽𝑗 )𝐻𝑗 /(1 + (𝑥/𝛽𝑗 ))𝐻𝑗 is monotonically increasing function; 𝑒𝑖 ’s are basal rate defined by 𝑒𝑖 = ∑𝑗∈𝑈𝑘 𝛼𝑖𝑗 with 𝑈𝑘 = {𝑗 | the 𝑗th transcription factor being a repressor of the kth gene, 𝑗 = 1, , 𝑛}; and matrix 𝐵 = (𝑏𝑖𝑗 ) ∈ R𝑛×𝑛 is defined as 𝛼𝑖𝑗 , { { { { { { if transcription factor 𝑗 is an activator { { { of gene 𝑖, { { { { {0, 𝑏𝑖𝑗 = { { if there is no link from node 𝑗 to node 𝑖, { { { { −𝛼 { 𝑖𝑗 , { { { { if transcription factor 𝑗 is a repressor { { { of gene 𝑖 { 𝑝̇ (𝑡) = −𝐶𝑃 (𝑡) + 𝐷𝑚 (𝑡 − 𝜏 (𝑡)) , 𝑚𝑖− ≤ 𝑔𝑖 (𝑎) − 𝑚𝑖− 𝑎 ≥ 0, 𝑎 (5) 𝑥 (𝑡) = 𝜑1 (𝑡) , (6) (12) 𝑦 (𝑡) = 𝜓1 (𝑡) , −𝜌 ≤ 𝑡 ≤ 0, (13) Based on system (10), we also consider the genetic regulatory networks with continuous distributed delays: 𝑥̇ (𝑡) = −𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) + 𝑊 ∫ 𝑡 𝑡−𝜗(𝑡) 𝑔 (𝑦 (𝑠)) 𝑑𝑠, 𝑦̇ (𝑡) = −𝐶𝑦 (𝑡) + 𝐷𝑥 (𝑡 − 𝜏 (𝑡)) , 𝑇 𝑝 (𝑡) = [𝑝1 (𝑡) , , 𝑝𝑛 (𝑡)] , (14) 𝑇 𝑓 (𝑝 (𝑡)) = [𝑓1 (𝑝1 (𝑡)) , , 𝑓𝑛 (𝑝𝑛 (𝑡))] , (7) 𝑇 𝐼 = [𝑒1 , , 𝑒𝑛 ] , 𝐷 = diag (𝑑1 , , 𝑑𝑛 ) In order to get the stability results, the following assumption is necessarily imposed on (6) Assumption 𝑓𝑖 : R → R, 𝑖 = 1, , 𝑛, are monotonically increasing functions with saturation and moreover satisfy 𝑓 (𝑎) − 𝑓𝑖 (𝑏) ≤ 𝑖 ≤ 𝑚𝑖+ , 𝑎−𝑏 (11) 𝑚𝑖+ 𝑎 − 𝑔𝑖 (𝑎) ≥ 𝑎 𝜌 = max {𝜎2 , 𝜏2 } 𝑚 (𝑡) = [𝑚1 (𝑡) , , 𝑚𝑛 (𝑡)] , 𝑚𝑖− 𝑖 = 1, , 𝑛, Let 𝑀0 = diag(𝑚1− , , 𝑚𝑛− ), 𝑀1 = diag(𝑚1+ , , 𝑚𝑛+ ), and 𝑚 = max{|𝑚1− |, , |𝑚𝑛− |, |𝑚1+ |, , |𝑚𝑛+ |} The initial condition of system (10) is assumed to be 𝑇 𝐶 = diag (𝑐1 , , 𝑐𝑛 ) , 𝑔𝑖 (𝑎) ≤ 𝑚𝑖+ , 𝑎 which implies that where 𝐴 = diag (𝑎1 , , 𝑎𝑛 ) , (10) where 𝑥(𝑡) = [𝑥1 (𝑡), , 𝑥𝑛 (𝑡)]𝑇 , 𝑦(𝑡) = [𝑦1 (𝑡), , 𝑦𝑛 (𝑡)]𝑇 , 𝑔(𝑦(𝑡)) = [𝑔(𝑦1 (𝑡)), , 𝑔(𝑦𝑛 (𝑡))]𝑇 , and 𝑔𝑖 (𝑦𝑖 (𝑡)) = 𝑓𝑖 (𝑦𝑖 (𝑡) + 𝑝𝑖∗ ) − 𝑓𝑖 (𝑝𝑖∗ ) By the definition of 𝑔𝑖 (⋅), it satisfies sector condition: In the compact matrix form, (4) can be rewritten as 𝑚̇ (𝑡) = −𝐴𝑚 (𝑡) + 𝐵𝑓 (𝑝 (𝑡 − 𝜎 (𝑡))) + 𝐸, (9) ∀𝑎, 𝑏 ∈ R, 𝑖 = 1, , 𝑛, (8) where 𝑚𝑖− and 𝑚𝑖+ are constants Remark In the most existing literature, (8) was strengthened to ≤ ((𝑓𝑖 (𝑎) − 𝑓𝑖 (𝑏))/(𝑎 − 𝑏)) ≤ 𝑙𝑖 , for all 𝑎, 𝑏 ∈ R, where 𝑙𝑖 ’s are positive constant Therefore, Assumption is somewhat general and, in fact, similar to that of [12]; see the discussion below where ≤ 𝜗(𝑡) ≤ 𝜗1 For completeness, we recall the following definition and lemmas Definition System (10) or (14) is said to be globally exponentially stable, if there exist constants 𝜆 > and 𝑀 ≥ 1, such that, for any initial value 𝑧𝑡0 , 󵄩 󵄩 ‖𝑧 (𝑡)‖ ≤ 𝑀󵄩󵄩󵄩󵄩𝑧𝑡0 󵄩󵄩󵄩󵄩𝐶1 𝑒−𝜆(𝑡−𝑡0 ) (15) hold, for all 𝑡 ≥ 0, where 𝑧(𝑡) = [𝑥(𝑡), 𝑦(𝑡)]𝑇 and ‖𝑧(𝑡)‖𝐶1 = ̇ + 𝜃)‖} sup−𝜌≤𝜃≤0 {‖𝑧(𝑡 + 𝜃)‖, ‖𝑧(𝑡 Lemma (see [22]) For any positive definite matrix 𝑀 ∈ R𝑛×𝑛 , there exists a scalar 𝑞 > and a vector-valued function 𝜔 : [0, 𝑞] → R𝑛 such that 𝑞 𝑇 𝑞 𝑞 0 (∫ 𝜔(𝑠)𝑑𝑠) 𝑀 (∫ 𝜔 (𝑠) 𝑑𝑠) ≤ 𝑞 ∫ 𝜔𝑇 (𝑠) 𝑀𝜔 (𝑠) 𝑑𝑠 (16) Abstract and Applied Analysis Lemma (see [23]) Let ℎ1 , , ℎ𝑁 : R𝑚 → R take 𝑚 positive values in an open subset D of R Then, the reciprocally convex combination of ℎ𝑖 over D satisfies ∑ ℎ𝑖 (𝜂) = ∑ℎ𝑖 (𝜂) + max ∑ 𝑘𝑖,𝑗 (𝜂) 𝑘𝑖,𝑗 (𝜂) 𝛼 {𝛼𝑖 |𝛼𝑖 >0,∑𝑖 𝛼𝑖 =1} 𝑖 𝑖 𝑖 𝑖 ≠ 𝑗 Proof Based on system (10), we construct the following Lyapunov-Krasovskii functional: 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) = 𝑉1 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉2 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉3 (𝑥 (𝑡) , 𝑦 (𝑡)) , (17) (21) where 𝑉1 (𝑥 (𝑡) , 𝑦 (𝑡)) = 𝑥𝑇 (𝑡) 𝑃1 𝑥 (𝑡) + 𝑦𝑇 (𝑡) 𝑃2 𝑦 (𝑡) , subject to 𝑉2 (𝑥 (𝑡) , 𝑦 (𝑡)) = ∫ 𝑡−𝜏1 {𝑘𝑖,𝑗 :R𝑚 󳨃󳨀→ R, 𝑘𝑗,𝑖 (𝜂) = 𝑘𝑖,𝑗 (𝜂) , +∫ (18) ℎ (𝜂) 𝑘𝑖,𝑗 (𝜂) [ 𝑖 ] ≥ 0} 𝑘𝑖,𝑗 (𝜂) ℎ𝑗 (𝜂) 𝑡 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠) 𝑄1 𝑥 (𝑠) 𝑑𝑠 𝑡−𝜏1 𝑡−𝜏(𝑡) +∫ 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠) 𝑄2 𝑥 (𝑠) 𝑑𝑠 𝑡−𝜏(𝑡) 𝑡−𝜏2 Main Results +∫ 𝑊 𝑋 ) ≥ 0, ( 𝑋 𝑊2 𝑊 𝑌 ( ) ≥ 0, 𝑌 𝑊4 Ω1 + Ω2 + 𝑂3𝑇 [𝑒−𝜆𝜏2 𝑄2 + 𝑒−𝜆𝜏1 𝑄3 ] 𝑂3 + 𝑂8𝑇 [𝑒−𝜆𝜎2 𝑄5 + 𝑒−𝜆𝜎1 𝑄6 ] 𝑂8 < 0, (19) +∫ +∫ 𝐻1𝑇 𝐴 𝑡−𝜎(𝑡) 𝑡−𝜎2 +∫ 𝑡 𝑡−𝜎(𝑡) 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 (𝑠) 𝑄5 𝑦 (𝑠) 𝑑𝑠 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 (𝑠) 𝑄6 𝑦 (𝑠) 𝑑𝑠 𝑒𝜆(𝑠−𝑡) 𝑔𝑇 (𝑦 (𝑠)) 𝑄7 𝑔 (𝑦 (𝑠)) 𝑑𝑠, 𝑉3 (𝑥 (𝑡) , 𝑦 (𝑡)) =∫ −𝜏1 +∫ ∫ 𝑡 −𝜏1 −𝜏2 −𝜎1 +∫ 𝜏1 𝑒𝜆(𝑠−𝑡) 𝑥̇𝑇 (𝑠) 𝑊1 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝑡+𝜃 −𝜎1 −𝜎2 −𝜆𝜏1 𝑡−𝜎1 𝑡−𝜎(𝑡) +∫ (20) 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 (𝑠) 𝑄4 𝑦 (𝑠) 𝑑𝑠 𝑡−𝜎1 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 For system (10) with Assumptions 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 , 𝑄𝑖 , 𝑖 = 1, , 7, 𝑊𝑖 , 𝑖 = 1, , 4, Γ1 = diag(𝛾11 , , 𝛾1𝑛 ), and Γ2 = diag(𝛾21 , 𝛾22 , , 𝛾2𝑛 ), such that, for any appropriate dimensions constant matrices 𝐻1 , 𝐻3 , 𝑋, 𝑌, the following LMIs hold: 𝑡 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠) 𝑄3 𝑥 (𝑠) 𝑑𝑠 𝑡 ∫ 𝑡+𝜃 ∫ 𝑡 𝑡+𝜃 𝑡 ∫ 𝑡+𝜃 (𝜏2 − 𝜏1 ) 𝑒𝜆(𝑠−𝑡) 𝑥̇𝑇 (𝑠) 𝑊2 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝜎1 𝑒𝜆(𝑠−𝑡) 𝑦𝑇̇ (𝑠) 𝑊3 𝑦̇ (𝑠) 𝑑𝑠 𝑑𝜃 (𝜎2 − 𝜎1 ) 𝑒𝜆(𝑠−𝑡) 𝑦𝑇̇ (𝑠) 𝑊4 𝑦̇ (𝑠) 𝑑𝑠 𝑑𝜃 (22) 𝑇 where Ω1 = diag(𝜆𝑃1 + 𝑄1 − 𝑒 𝑊1 − − 𝐴 𝐻1 , 𝑒−𝜆𝜏1 (𝑄1 −𝑄2 )−𝑒−𝜆𝜏1 𝑊1 −𝑒−𝜆𝜏2 𝑊2 , 𝑒−𝜆𝜏1 (1−𝜏4 )𝑄3 −𝑒−𝜆𝜏2 [(1− 𝜏3 )𝑄2 +2𝑊2 −𝑋𝑇 −𝑋], −𝑒−𝜆𝜏2 𝑊2 , 𝜏12 𝑊1 +(𝜏2 −𝜏1 )2 𝑊2 −𝐻2𝑇 −𝐻2 , 𝜆𝑃2 +𝑄4 −𝑒−𝜆𝜎1 𝑊3 −2𝑀1𝑇 Γ1 𝑀0 −𝐻3𝑇 𝐶−𝐶𝑇 𝐻3 , −𝑒−𝜆𝜎1 (𝑄4 −𝑄5 + 𝑊3 ) − 𝑒−𝜆𝜎2 𝑊4 , 𝑒−𝜆𝜎1 (1 − 𝜎4 )𝑄6 − 𝑒−𝜆𝜎2 [(1 − 𝜎3 )𝑄5 + 2𝑊4 − 𝑌𝑇 − 𝑌] − 𝑀1𝑇 Γ2 𝑀0 , −𝑒−𝜆𝜎2 (𝑄4 + 𝑊4 ), 𝜎12 𝑊3 + (𝜎2 − 𝜎1 )2 𝑊4 − 𝐻4𝑇 − 𝐻4 , 𝑄7 −2Γ1 , −𝑒−𝜆𝜎2 𝑄7 (1−𝜎𝑑 )−2Γ2 ), Ω2 = sym(𝑂1𝑇 𝑒−𝜆𝜏1 𝑊1 𝑂2 + 𝑂1𝑇 [𝑃1 − 𝐻1𝑇 − 𝐻2𝑇 𝐴]𝑂5 + 𝑂1𝑇 𝐻1𝑇 𝐵𝑂12 + 𝑂2𝑇 𝑒−𝜆𝜏2 (𝑊2 − 𝑋)𝑂3 + 𝑂2𝑇 𝑒−𝜆𝜏2 𝑋𝑂4 + 𝑂3𝑇 𝑒−𝜆𝜏2 (𝑊2 − 𝑋)𝑂4 + 𝑂3𝑇 𝐻3𝑇 𝐷𝑂6 + 𝑂3𝑇 𝐻4𝑇 𝐷𝑂10 + 𝑂5𝑇 𝐻2𝑇 𝐵𝑂12 + 𝑂6𝑇 𝑒−𝜆𝜎1 𝑊3 𝑂7 + 𝑂6𝑇 [𝑃2 − 𝐻3𝑇 − 𝐻4𝑇 𝐶]𝑂10 + 𝑂6𝑇 [Γ1 𝑀0 + 𝑀1𝑇 Γ1 ]𝑂11 + 𝑂7𝑇 𝑒−𝜆𝜎2 (𝑊4 − 𝑌)𝑂8 + 𝑂7𝑇 𝑒−𝜆𝜎2 𝑌𝑂9 + 𝑂8𝑇 𝑒−𝜆𝜎2 (𝑊4 − 𝑌)𝑂9 + 𝑂8𝑇 [Γ2 𝑀0 + 𝑀1𝑇 Γ2 ]𝑂12 ), and 𝑂𝑖 = [0𝑛×(𝑖−1)𝑛 , 𝐼𝑛×𝑛 , 0𝑛×(13−𝑖)𝑛 ], 𝑖 = 1, , 12 Taking the derivatives of 𝑉𝑖 , 𝑖 = 1, 2, 3, we have 𝑉1̇ (𝑥 (𝑡) , 𝑦 (𝑡)) = 2𝑥𝑇 (𝑡) 𝑃1 𝑥̇ (𝑡) + 2𝑦𝑇 (𝑡) 𝑃2 𝑦̇ (𝑡) , 𝑉2̇ (𝑥 (𝑡) , 𝑦 (𝑡)) 𝑡 = −𝜆 ∫ 𝑡−𝜏1 − 𝜆∫ 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠) 𝑄1 𝑥 (𝑠) 𝑑𝑠 𝑡−𝜏1 𝑡−𝜏(𝑡) − 𝜆∫ 𝑡−𝜏(𝑡) 𝑡−𝜏2 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠) 𝑄2 𝑥 (𝑠) 𝑑𝑠 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠) 𝑄3 𝑥 (𝑠) 𝑑𝑠 Abstract and Applied Analysis − 𝜆∫ 𝑡 𝑡−𝜎1 − 𝜆∫ −∫ 𝑡−𝜎1 𝑡 𝑡−𝜎(𝑡) 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 (𝑠) 𝑄6 𝑦 (𝑠) 𝑑𝑠 𝑒𝜆(𝑠−𝑡) 𝑔𝑇 (𝑦 (𝑠)) 𝑄7 𝑔 (𝑦 (𝑠)) 𝑑𝑠 + 𝑥𝑇 (𝑡) 𝑄1 𝑥 (𝑡) − 𝑒−𝜆𝜏1 𝑥𝑇 (𝑡 − 𝜏1 ) 𝑄1 𝑥 (𝑡 − 𝜏1 ) −𝜆𝜏1 𝑇 𝑥 (𝑡 − 𝜏1 ) 𝑄2 𝑥 (𝑡 − 𝜏1 ) +𝑒 From Lemma 6, we have −∫ 𝑡 𝑡−𝜏1 𝜏1 𝑒𝜆(𝜃−𝑡) 𝑥̇𝑇 (𝜃) 𝑊1 𝑥̇ (𝜃) 𝑑𝜃 𝑇 ≤ −𝑒−𝜆𝜏1 [𝑥 (𝑡) − 𝑥 (𝑡 − 𝜏1 )] 𝑊1 [𝑥 (𝑡) − 𝑥 (𝑡 − 𝜏1 )] , −∫ 𝑡 + 𝑦𝑇 (𝑡) 𝑄4 𝑦 (𝑡) − 𝑒−𝜆𝜎1 𝑦𝑇 (𝑡 − 𝜎1 ) 𝑄4 𝑦 (𝑡 − 𝜎1 ) 𝜎1 𝑒𝜆(𝜃−𝑡) 𝑦𝑇̇ (𝜃) 𝑊3 𝑦̇ (𝜃) 𝑑𝜃 𝑇 ≤ −𝑒−𝜆𝜎1 [𝑦 (𝑡) − 𝑦 (𝑡 − 𝜎1 )] 𝑊3 [𝑦 (𝑡) − 𝑦 (𝑡 − 𝜎1 )] (24) − 𝑒−𝜆𝜏2 𝑥𝑇 (𝑡 − 𝜏 (𝑡)) 𝑄2 𝑥 (𝑡 − 𝜏 (𝑡)) (1 − 𝜏̇ (𝑡)) + 𝑒−𝜆𝜏1 𝑥𝑇 (𝑡 − 𝜏 (𝑡)) 𝑄3 𝑥 (𝑡 − 𝜏 (𝑡)) (1 − 𝜏̇ (𝑡)) Meanwhile, −∫ 𝑡−𝜏1 𝑡−𝜏2 (𝜏2 − 𝜏1 ) 𝑒𝜆(𝜃−𝑡) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 ≤ −𝑒−𝜆𝜏2 [∫ + 𝑒−𝜆𝜎1 𝑦𝑇 (𝑡 − 𝜎1 ) 𝑄5 𝑦 (𝑡 − 𝜎1 ) (𝜎2 − 𝜎1 ) 𝑒𝜆(𝜃−𝑡) 𝑦𝑇̇ (𝜃) 𝑊4 𝑦̇ (𝜃) 𝑑𝜃 (23) 𝑡−𝜎1 − 𝑒−𝜆𝜏2 𝑥𝑇 (𝑡 − 𝜏2 ) 𝑄3 𝑥 (𝑡 − 𝜏2 ) 𝑡−𝜎1 𝑡−𝜎2 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 (𝑠) 𝑄5 𝑦 (𝑠) 𝑑𝑠 𝑡−𝜎(𝑡) 𝑡−𝜎2 − 𝜆∫ + (𝜎2 − 𝜎1 ) 𝑦𝑇̇ (𝑡) 𝑊4 𝑦̇ (𝑡) 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 (𝑠) 𝑄4 𝑦 (𝑠) 𝑑𝑠 𝑡−𝜎(𝑡) − 𝜆∫ 𝑡−𝜏(𝑡) 𝑡−𝜏2 (𝜏2 − 𝜏1 ) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 −𝜆𝜎2 𝑇 𝑦 (𝑡 − 𝜎 (𝑡)) 𝑄5 𝑦 (𝑡 − 𝜎 (𝑡)) (1 − 𝜎̇ (𝑡)) −𝑒 +∫ −𝜆𝜎2 𝑇 −𝑒 𝑡−𝜏(𝑡) 𝑦 (𝑡 − 𝜎2 ) 𝑄6 𝑦 (𝑡 − 𝜎2 ) + 𝑒−𝜆𝜎1 𝑦𝑇 (𝑡 − 𝜎 (𝑡)) 𝑄6 𝑦 (𝑡 − 𝜎 (𝑡)) (1 − 𝜎̇ (𝑡)) + 𝑔𝑇 (𝑦 (𝑡)) 𝑄7 𝑔 (𝑦 (𝑡)) − 𝑒−𝜆𝜎2 𝑔𝑇 = −𝑒−𝜆𝜏2 [∫ 𝑡−𝜏(𝑡) 𝑡−𝜏2 = −𝜆 ∫ −𝜏1 − 𝜆∫ −𝜎1 − 𝜆∫ −𝜎1 −𝜎2 ∫ 𝜏1 𝑒 𝑡 𝑡+𝜃 ∫ 𝑡 𝑡+𝜃 𝑡 ∫ 𝑡+𝜃 𝑥̇ (𝑠) 𝑊1 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 +∫ (𝜏2 − 𝜏1 ) 𝑒𝜆(𝑠−𝑡) 𝑥̇𝑇 (𝑠) 𝑊2 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝜎1 𝑒𝜆(𝑠−𝑡) 𝑦𝑇̇ (𝑠) 𝑊3 𝑦̇ (𝑠) 𝑑𝑠𝑑𝜃 (𝜎2 − 𝜎1 ) 𝑒𝜆(𝑠−𝑡) 𝑦𝑇̇ (𝑠) 𝑊4 𝑦̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝑡 𝑡−𝜏1 𝜏1 𝑒𝜆(𝜃−𝑡) 𝑥̇𝑇 (𝜃) 𝑊1 𝑥̇ (𝜃) 𝑑𝜃 = −𝑒−𝜆𝜏2 [∫ 𝑡−𝜏(𝑡) 𝑡−𝜏2 𝑡−𝜏2 ×∫ + −∫ 𝑡 𝑡−𝜎1 𝜆(𝜃−𝑡) 𝑇 𝜎1 𝑒 𝑦̇ (𝜃) 𝑊3 𝑦̇ (𝜃) 𝑑𝜃 + (𝜏 (𝑡) − 𝜏1 ) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 𝜏 (𝑡) − 𝜏1 𝜏2 − 𝜏 (𝑡) 𝑡−𝜏(𝑡) 𝑡−𝜏2 (𝜏2 − 𝜏1 ) 𝑒𝜆(𝜃−𝑡) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 𝜎12 𝑦𝑇̇ (𝑡) 𝑊3 𝑦̇ (𝑡) 𝑡−𝜏1 𝑡−𝜏(𝑡) + (𝜏2 − 𝜏 (𝑡)) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃] (𝜏2 − 𝜏 (𝑡)) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 +∫ + (𝜏2 − 𝜏1 ) 𝑥̇𝑇 (𝑡) 𝑊2 𝑥̇ (𝑡) −∫ 𝑡−𝜏1 𝑡−𝜏(𝑡) + 𝜏12 𝑥̇𝑇 (𝑡) 𝑊1 𝑥̇ (𝑡) − ∫ 𝑡−𝜏1 (𝜏 (𝑡) − 𝜏1 ) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 𝑡−𝜏2 𝑡+𝜃 (𝜏 (𝑡) − 𝜏1 ) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 𝑡−𝜏(𝑡) 𝜆(𝑠−𝑡) 𝑇 ∫ −𝜏2 − 𝜆∫ +∫ 𝑡 −𝜏1 𝑡−𝜏1 𝑡−𝜏(𝑡) 𝑉3̇ (𝑥 (𝑡) , 𝑦 (𝑡)) (𝜏2 − 𝜏1 ) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃] (𝜏2 − 𝜏 (𝑡)) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 +∫ × (𝑦 (𝑡 − 𝜎 (𝑡))) 𝑄7 𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) (1 − 𝜎̇ (𝑡)) , 𝑡−𝜏1 (𝜏2 − 𝜏 (𝑡)) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 𝜏2 − 𝜏 (𝑡) 𝜏 (𝑡) − 𝜏1 ×∫ 𝑡−𝜏1 𝑡−𝜏(𝑡) (𝜏 (𝑡) − 𝜏1 ) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃] (25) Abstract and Applied Analysis By Lemma 6, we obtain that −∫ 𝑡−𝜏1 𝑡−𝜏2 It can be rewritten as (𝜏2 − 𝜏1 ) 𝑒𝜆(𝜃−𝑡) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 ≤ −𝑒−𝜆𝜏2 { 𝑇 − 2[𝑔 (𝑦 (𝑡)) − 𝑀1 𝑦 (𝑡)] Γ1 [𝑔 (𝑦 (𝑡)) − 𝑀0 𝑦 (𝑡)] − 2[𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) − 𝑀1 𝑦 (𝑡 − 𝜎 (𝑡))] 𝜏2 − 𝜏1 𝑇 [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2 )] 𝜏2 − 𝜏 (𝑡) 𝜏2 − 𝜏1 𝑇 [𝑥 (𝑡 − 𝜏1 ) − 𝑥 (𝑡 − 𝜏 (𝑡))] 𝜏 (𝑡) − 𝜏1 For any constant matrices of appropriate dimensions 𝐻𝑖 , 𝑖 = 1, , 4, and from (10), we can obtain that = [𝑥𝑇 (𝑡) 𝐻1𝑇 + 𝑥̇𝑇 (𝑡) 𝐻2𝑇 ] × 𝑊2 [𝑥 (𝑡 − 𝜏1 ) − 𝑥 (𝑡 − 𝜏 (𝑡))] } × [−𝑥̇ (𝑡) − 𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡)))] , (26) = [𝑦𝑇 (𝑡) 𝐻3𝑇 + 𝑦𝑇̇ (𝑡) 𝐻4𝑇 ] And by Lemma 7, we get that −∫ 𝑡−𝜏1 𝑡−𝜏2 Combining (21)–(31), we have 𝑇 𝑉̇ (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝜆𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) × 𝑊2 [𝑥 (𝑡 − 𝜏1 ) − 𝑥 (𝑡 − 𝜏 (𝑡))] (27) 𝑇 + [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2 )] ≤ 𝜉T (𝑡) {Ω + 𝑂3𝑇 [𝑒−𝜆𝜏2 (𝜏̇ (𝑡) − 𝜏3 ) 𝑄2 +𝑒−𝜆𝜏1 (𝜏4 − 𝜏̇ (𝑡)) 𝑄3 ] 𝑂3 × 𝑊2 [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2 )] + 2[𝑥 (𝑡 − 𝜏1 ) − 𝑥 (𝑡 − 𝜏 (𝑡))] 𝑇 + 𝑂8𝑇 [𝑒−𝜆𝜎2 (𝜎̇ (𝑡) − 𝜎3 ) 𝑄5 ×𝑋 [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2 )] } +𝑒−𝜆𝜎1 (𝜎4 − 𝜎̇ (𝑡)) 𝑄6 ] 𝑂8 } 𝜉 (𝑡) , (32) Similar to (27), −∫ 𝑡−𝜎1 𝑡−𝜎2 (31) × [−𝑦̇ (𝑡) − 𝐶𝑦 (𝑡) + 𝐷𝑥 (𝑡 − 𝜏 (𝑡))] (𝜏2 − 𝜏1 ) 𝑒𝜆(𝜃−𝑡) 𝑥̇𝑇 (𝜃) 𝑊2 𝑥̇ (𝜃) 𝑑𝜃 ≤ −𝑒−𝜆𝜏2 {[𝑥 (𝑡 − 𝜏1 ) − 𝑥 (𝑡 − 𝜏 (𝑡))] (30) × Γ2 [𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) − 𝑀0 𝑦 (𝑡 − 𝜎 (𝑡))] ≥ × 𝑊2 [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2 )] + 𝑇 𝑒𝜆(𝜃−𝑡) 𝑦𝑇̇ (𝜃) 𝑊4 𝑦̇ (𝜃) 𝑑𝜃 ≤ −𝑒−𝜆𝜎2 {[𝑦 (𝑡 − 𝜎1 ) − 𝑦 (𝑡 − 𝜎 (𝑡))] where 𝜉𝑇 (𝑡) = [𝑥𝑇 (𝑡) , 𝑥𝑇 (𝑡 − 𝜏1 ) , 𝑥𝑇 (𝑡 − 𝜏 (𝑡)) , 𝑥𝑇 (𝑡 − 𝜏2 ) , 𝑇 𝑥̇𝑇 (𝑡) , 𝑦𝑇 (𝑡) , 𝑦𝑇 (𝑡 − 𝜎1 ) , 𝑦𝑇 (𝑡 − 𝜎 (𝑡)) , × 𝑊4 [𝑦 (𝑡 − 𝜎1 ) − 𝑦 (𝑡 − 𝜎 (𝑡))] (28) 𝑇 + [𝑦 (𝑡 − 𝜎 (𝑡)) − 𝑦 (𝑡 − 𝜎2 )] × 𝑊4 [𝑦 (𝑡 − 𝜎 (𝑡)) − 𝑦 (𝑡 − 𝜎2 )] + 2[𝑦 (𝑡 − 𝜎1 ) − 𝑦 (𝑡 − 𝜎 (𝑡))] 𝑇 × 𝑌 [𝑦 (𝑡 − 𝜎 (𝑡)) − 𝑦 (𝑡 − 𝜎2 )] } By Assumption 3, for any Γ𝑖 = diag(𝛾𝑖1 , , 𝛾𝑖𝑛 ) ≥ 0, 𝑖 = 1, 2, the following inequality is true: 𝑦𝑇 (𝑡 − 𝜎2 ) , 𝑦𝑇̇ (𝑡) , 𝑔𝑇 (𝑦 (𝑡)) , 𝑔𝑇 (𝑦 (𝑡 − 𝜎 (𝑡)))] (33) From (19) and (20), we can see that 𝑉̇ (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝜆𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) < 0, (34) for all nonzero 𝜉(𝑡) Integrating the above inequality (34) from 𝑡0 to 𝑡 gives 𝑛 − 2∑𝛾𝑖1 [𝑔𝑖 (𝑦𝑖 (𝑡)) − 𝑚𝑖+ 𝑦𝑖 (𝑡)] [𝑔𝑖 (𝑦𝑖 (𝑡)) − 𝑚𝑖− 𝑦𝑖 (𝑡)] 𝑖=1 𝑛 − 2∑𝛾𝑖2 [𝑔𝑖 (𝑦𝑖 (𝑡 − 𝜏 (𝑡))) − 𝑚𝑖+ 𝑦𝑖 (𝑡 − 𝜏 (𝑡))] 𝑖=1 × [𝑔𝑖 (𝑦𝑖 (𝑡 − 𝜏 (𝑡))) − 𝑚𝑖− 𝑦𝑖 (𝑡 − 𝜏 (𝑡))] ≥ (29) 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ 𝑒−𝜆(𝑡−𝑡0 ) 𝑉 (𝑥 (𝑡0 ) , 𝑦 (𝑡0 )) From (21), we know that 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≥ 𝑉1 (𝑥 (𝑡) , 𝑦 (𝑡)) ≥ {𝜆 (𝑃1 ) , 𝜆 (𝑃2 )} ‖𝑧 (𝑡)‖2 , (35) Abstract and Applied Analysis 𝑉 (𝑥 (𝑡0 ) , 𝑦 (𝑡0 )) for any appropriate dimensions constant matrices 𝐻1 , 𝐻3 , 𝑋, 𝑌, the following LMIs hold: ≤ [𝜆 max (𝑃1 ) + 𝜆 max (𝑃2 ) + 𝜏1 𝜆 max (𝑄1 ) 𝑊 𝑋 ( ) ≥ 0, 𝑋 𝑊2 + (𝜏2 − 𝜏1 ) [𝜆 max (𝑄2 ) + 𝜆 max (𝑄3 )] + (𝜎2 − 𝜎1 ) [𝜆 max (𝑄5 ) + 𝜆 max (𝑄6 )] Ω󸀠1 + Ω󸀠2 + 𝐿𝑇3 [𝑒−𝜆𝜏2 𝑄2 + 𝑒−𝜆𝜏1 𝑄3 ] 𝐿 + 𝜎1 𝜆 max (𝑄4 ) + 𝜎2 𝑚 𝜆 max (𝑄7 ) + 𝐿𝑇8 [𝑒−𝜆𝜎2 𝑄5 + 𝑒−𝜆𝜎1 𝑄6 ] 𝐿 < 0, 1 + 𝜏13 𝜆 max (𝑊1 ) + (𝜏2 − 𝜏1 ) 𝜆 max (𝑊2 ) 2 1 󵄩 󵄩2 + 𝜎13 𝜆 max (𝑊3 ) + (𝜎2 − 𝜎1 ) 𝜆 max (𝑊4 )] 󵄩󵄩󵄩𝑧(𝑡0 )󵄩󵄩󵄩𝐶1 2 (36) Let 𝜆 = {𝜆 (𝑃1 ) , 𝜆 (𝑃2 )} , 𝜆 = 𝜆 max (𝑃1 ) + 𝜆 max (𝑃2 ) + 𝜏1 𝜆 max (𝑄1 ) + (𝜏2 − 𝜏1 ) [𝜆 max (𝑄2 ) + 𝜆 max (𝑄3 )] + (𝜎2 − 𝜎1 ) [𝜆 max (𝑄5 ) + 𝜆 max (𝑄6 )] + 𝜎1 𝜆 max (𝑄4 ) + 𝜎2 𝑚2 𝜆 max (𝑄7 ) (37) 1 + 𝜏13 𝜆 max (𝑊1 ) + (𝜏2 − 𝜏1 ) 𝜆 max (𝑊2 ) 2 + 𝑉3 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉4 (𝑥 (𝑡) , 𝑦 (𝑡)) , (38) (42) where 𝑉1 (𝑥(𝑡), 𝑦(𝑡)), 𝑉2 (𝑥(𝑡), 𝑦(𝑡)), and 𝑉3 (𝑥(𝑡), 𝑦(𝑡)) are defined as in Theorem and By (38), we get that ‖𝑧 (𝑡)‖2 ≤ (41) where Ω󸀠1 = diag(𝜆𝑃1 +𝑄1 −𝑒−𝜆𝜏1 𝑊1 −𝐻1𝑇 𝐴−𝐴𝑇 𝐻1 , 𝑒−𝜆𝜏1 (𝑄1 − 𝑄2 ) − 𝑒−𝜆𝜏1 𝑊1 − 𝑒−𝜆𝜏2 𝑊2 , 𝑒−𝜆𝜏1 (1 − 𝜏4 )𝑄3 − 𝑒−𝜆𝜏2 [(1 − 𝜏3 )𝑄2 + 2𝑊2 − 𝑋𝑇 − 𝑋], −𝑒−𝜆𝜏2 𝑊2 , 𝜏12 𝑊1 + (𝜏2 − 𝜏1 )2 𝑊2 − 𝐻2𝑇 − 𝐻2 , 𝜆𝑃2 + 𝑄4 − 𝑒−𝜆𝜎1 𝑊3 − 2𝑀1𝑇 Γ1 𝑀0 − 𝐻3𝑇 𝐶 − 𝐶𝑇 𝐻3 , −𝑒−𝜆𝜎1 (𝑄4 − 𝑄5 + 𝑊3 ) − 𝑒−𝜆𝜎2 𝑊4 , 𝑒−𝜆𝜎1 (1 − 𝜎4 )𝑄6 − 𝑒−𝜆𝜎2 [(1 − 𝜎3 )𝑄5 + 2𝑊4 − 𝑌𝑇 − 𝑌] − 𝑀1𝑇 Γ2 𝑀0 , −𝑒−𝜆𝜎2 (𝑄4 + 𝑊4 ), 𝜎12 𝑊3 + (𝜎2 − 𝜎1 )2 𝑊4 − 𝐻4𝑇 − 𝐻4 , 𝑄7 − 2Γ1 + 𝜗12 𝑆, −𝑒−𝜆𝜎2 𝑄7 (1 − 𝜎4 ) − 2Γ2 , −𝑒−𝜆𝜗1 𝑆, −𝑒−𝜆𝜗1 𝑆), Ω󸀠2 = sym(𝐿𝑇1 𝑒−𝜆𝜏1 𝑊1 𝐿 + 𝐿𝑇1 [𝑃1 − 𝐻1𝑇 − 𝐻2𝑇 𝐴]𝐿 + 𝐼1𝑇 𝐻1𝑇 𝐵𝐿 12 + 𝐿𝑇2 𝑒−𝜆𝜏2 (𝑊2 − 𝑋)𝐿 + 𝐿𝑇2 𝑒−𝜆𝜏2 𝑋𝐿 + 𝐿𝑇3 𝑒−𝜆𝜏2 (𝑊2 − 𝑋)𝐿 + 𝐿𝑇3 𝐻3𝑇 𝐷𝐿 + 𝐿𝑇3 𝐻4𝑇 𝐷𝐿 10 + 𝐼5𝑇 𝐻2𝑇 𝐵𝐿 12 + 𝐿𝑇6 𝑒−𝜆𝜎1 𝑊3 𝐿 + 𝐿𝑇6 [𝑃2 −𝐻3𝑇 −𝐻4𝑇 𝐶]𝐿 10 +𝐿𝑇6 [Γ1 𝑀0 +𝑀1𝑇 Γ1 ]𝐿 11 + 𝐿𝑇7 𝑒−𝜆𝜎2 (𝑊4 − 𝑌)𝐿 + 𝐿𝑇7 𝑒−𝜆𝜎2 𝑌𝐿 + 𝐿𝑇8 𝑒−𝜆𝜎2 (𝑊4 − 𝑌)𝐿 + 𝐿𝑇8 [Γ2 𝑀0 + 𝑀1𝑇 Γ2 ]𝐿 12 ), and 𝐿 𝑖 = [0𝑛×(𝑖−1)𝑛 , 𝐼𝑛×𝑛 , 0𝑛×(15−𝑖)𝑛 ], 𝑖 = 1, , 12 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) = 𝑉1 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉2 (𝑥 (𝑡) , 𝑦 (𝑡)) Then, by (35) and (36), we have 󵄩 󵄩2 𝜆 𝑒−𝜆(𝑡−𝑡0 ) 󵄩󵄩󵄩𝑧(𝑡0 )󵄩󵄩󵄩𝐶1 (40) Proof Based on system (14), we construct the following Lyapunov-Krasovskii functional: 1 + 𝜎13 𝜆 max (𝑊3 ) + (𝜎2 − 𝜎1 ) 𝜆 max (𝑊4 ) 2 𝜆 ‖𝑧 (𝑡)‖2 ≤ 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ 𝑊 𝑌 ( ) ≥ 0, 𝑌 𝑊4 𝜆 −𝜆(𝑡−𝑡0 ) 󵄩󵄩 󵄩2 𝑒 󵄩󵄩𝑧(𝑡0 )󵄩󵄩󵄩𝐶1 𝜆1 (39) Let 𝛼 = (𝜆 /𝜆 )1/2 , and, by Definition 5, the genetic regulatory networks in (10) are exponentially stable The proof is completed 𝑉4 (𝑥 (𝑡) , 𝑦 (𝑡)) =∫ −𝜗1 ∫ 𝑡 𝑡+𝜃 𝜗1 𝑒𝜆(𝑠−𝑡) 𝑔𝑇 (𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 (43) Taking the derivative of 𝑉4 , In the following, we consider the globally exponential stability of the genetic regulatory networks with time-varying delays and continuous distributed delays Theorem For system (14) with Assumptions 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, 𝑄𝑖 , 𝑖 = 1, , 7, 𝑊𝑖 , 𝑖 = 1, , 4, Γ1 = diag(𝛾11 , , 𝛾1𝑛 ), and Γ2 = diag(𝛾21 , , 𝛾2𝑛 ), such that, 𝑉4̇ (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ −𝜆 ∫ −𝜗1 𝑡 ∫ 𝑡+𝜃 𝜗1 𝑒𝜆(𝑠−𝑡) 𝑔𝑇 (𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 + 𝜗12 𝑔T (𝑦 (𝑡)) 𝑆𝑔 (𝑦 (𝑡)) 𝑡 − 𝑒−𝜆𝜗1 ∫ 𝑡−𝜗1 𝜗1 𝑔𝑇 (𝑦 (𝜃)) 𝑆𝑔 (𝑦 (𝜃)) 𝑑𝜃 (44) Abstract and Applied Analysis By Lemma 6, we get that 𝑇 𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) , (∫ 𝑡−𝜗1 𝑉4̇ (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ −𝜆 ∫ −𝜗1 ∫ 𝑡 𝑡+𝜃 (∫ 𝑡−𝜗(𝑡) 𝑡−𝜗1 − 𝑒−𝜆𝜗1 × (∫ −𝜗1 𝑔(𝑦(𝜃))𝑑𝜃) 𝑆 (∫ 𝑡+𝜃 𝑡−𝜗(𝑡) 𝑡−𝜗1 𝑡 𝑔 (𝑦 (𝜃)) 𝑑𝜃) 𝑇 𝑔(𝑦(𝜃))𝑑𝜃) 𝑆 (∫ 𝑡−𝜗(𝑡) 𝑡−𝜗1 𝑡−𝜗(𝑡) 𝜗 (𝑡) ) 𝜗1 𝑔 (𝑦 (𝜃)) 𝑑𝜃) 𝑔(𝑦(𝜃))𝑑𝜃) 𝑆 (∫ 𝑡 𝑡−𝜗(𝑡) 𝑔 (𝑦 (𝜃)) 𝑑𝜃) (45) Combining (23)–(31), (42), and (45), we get 𝑉̇ (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝜆𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ 𝜉1𝑇 (𝑡) {Ω + 𝐿𝑇3 [𝑒−𝜆𝜏2 (𝜏̇ (𝑡) − 𝜏3 ) 𝑄2 −𝜆𝜏1 +𝑒 + (48) where 𝜆󸀠2 = 𝜆 + (1/2)𝜗13 𝑚2 𝜆 max (𝑆) and 𝜆 , 𝜆 are defined Remark 11 When of time-varying 𝜗1 − 𝜗 (𝑡) ) 𝜗1 𝑇 𝑡 𝜆󸀠2 −𝜆(𝑡−𝑡0 ) 󵄩󵄩 󵄩2 𝑒 󵄩󵄩𝑧(𝑡0 )󵄩󵄩󵄩𝐶1 𝜆1 Remark 10 In the proof of Theorems 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 𝜗1 𝑒𝜆(𝑠−𝑡) 𝑔𝑇 (𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 − 𝑒−𝜆𝜗1 (1 + × (∫ ‖𝑧 (𝑡)‖2 ≤ 𝑔 (𝑦 (𝜃)) 𝑑𝜃) + 𝜗12 𝑔𝑇 (𝑦 (𝑡)) 𝑆𝑔 (𝑦 (𝑡)) − 𝑒−𝜆𝜗1 (1 + × (∫ 𝑔(𝑦(𝜃))𝑑𝜃) ] as Theorem Let 𝛼󸀠 = (𝜆󸀠2 /𝜆 ) , and, by Definition 5, the genetic regulatory network (14) is exponentially stable The proof is completed 𝑡−𝜗(𝑡) 𝑡 (47) 1/2 𝑇 ∫ 𝑔(𝑦(𝜃))𝑑𝜃) , ̇ By (40) and (41), we get that 𝑉(𝑥(𝑡), 𝑦(𝑡))+𝜆𝑉(𝑥(𝑡), 𝑦(𝑡)) < Similar to the proof of Theorem we have 𝜗1 𝜗 (𝑡) 𝑡 𝑔(𝑦(𝜃))𝑑𝜃) 𝑆 (∫ 𝑡−𝜗(𝑡) 𝑡−𝜗1 𝑡−𝜗(𝑡) ≤ −𝜆 ∫ 𝑇 𝑇 𝑇 𝑡 𝑡−𝜗(𝑡) 𝜗1 𝑒𝜆(𝑠−𝑡) 𝑔𝑇 (𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 𝜗1 + 𝜗12 𝑔𝑇 (𝑦 (𝑡)) 𝑆𝑔 (𝑦 (𝑡)) − 𝑒−𝜆𝜗1 𝜗1 − 𝜗 (𝑡) × (∫ 𝑡−𝜗(𝑡) 𝐿𝑇8 [𝑒−𝜆𝜎2 (𝜏4 − 𝜏̇ (𝑡)) 𝑄3 ] 𝐿 (𝜎̇ (𝑡) − 𝜎3 ) 𝑄5 +𝑒−𝜆𝜎1 (𝜎4 − 𝜎̇ (𝑡)) 𝑄6 ] 𝐿 } 𝜉1 (𝑡) , (46) 𝑡−𝜏(𝑡) the lower bound of derivatives delays is immeasurable, let 𝑡−𝜎(𝑡) = and ∫𝑡−𝜎 𝑒𝜆(𝑠−𝑡) 𝑦𝑇 ∫𝑡−𝜏 𝑒𝜆(𝑠−𝑡) 𝑥𝑇 (𝑠)𝑄3 𝑥(𝑠)𝑑𝑠 2 (𝑠)𝑄6 𝑦(𝑠)𝑑𝑠 = in Theorem (20) or Theorem (41); our results are true still Numerical Examples In this section, two examples are given to illustrate the effectiveness of our theoretical results Example 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): 𝑑𝑥𝑖 𝛼 + 𝛼0 , = −𝑥𝑖 + 𝑑𝑡 + 𝑦𝑗𝑛 𝑑𝑦𝑖 = 𝛽 (𝑥𝑖 − 𝑦𝑖 ) 𝑑𝑡 Taking time-varying delays into account and shifting the equilibrium point to the origin, one gets the following model: 𝑥̇ (𝑡) = −𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) , where 𝑦̇ (𝑡) = −𝐶𝑦 (𝑡) + 𝐷𝑥 (𝑡 − 𝜏 (𝑡)) , 𝜉1𝑇 (𝑡) = [𝑥𝑇 (𝑡) , 𝑥𝑇 (𝑡 − 𝜏1 ) , 𝑥𝑇 (𝑡 − 𝜏 (𝑡)) , 𝑥𝑇 (𝑡 − 𝜏2 ) , 𝑥̇𝑇 (𝑡) , 𝑦𝑇 (𝑡) , 𝑦𝑇 (𝑡 − 𝜎1 ) , 𝑦𝑇 (𝑡 − 𝜎 (𝑡)) , 𝑦𝑇 (𝑡 − 𝜎2 ) , 𝑦𝑇̇ (𝑡) , 𝑔𝑇 (𝑦 (𝑡)) , (49) (50) where 𝐴 = diag(2, 2, 2), 𝐶 = diag(3, 3, 3), 𝐷 = diag(1, 1, 1), and the coupling matrix 0 −1 𝐵 = 1.5 × (−1 0 ) −1 (51) 0.5 0.6 0.4 0.5 0.3 0.4 Protein concentrations mRNA concentrations Abstract and Applied Analysis 0.2 0.1 −0.1 −0.2 0.3 0.2 0.1 0 10 15 t 20 25 30 −0.1 10 15 t (a) 20 25 30 (b) Figure 1: (a) mRNA concentrations 𝑥(𝑡) (b) Protein concentrations 𝑦(𝑡) 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) = + 2sin2 𝑡, 𝜏 (𝑡) = + cos2 𝑡 (52) 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 + 0.3sin2 𝑡, 𝜗 (𝑡) = 2sin2 (𝑡) We can get the parameters as follows: 𝜎1 = 1, 𝜎2 = 3, 𝜎3 = −2, 𝜎4 = 2, 𝜏1 = 1, 𝜏2 = 2, 𝜏3 = −1, 𝜏4 = 1, (53) 17.1041 7.9997 7.9997 𝑃1 = ( 7.9997 17.1041 7.9997 ) , 7.9997 7.9997 17.1041 13.8943 0.5573 0.5573 𝑃2 = ( 0.5573 13.8943 0.5573 ) 0.5573 0.5573 13.8943 (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 Example 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: 𝐷 = diag (0.3, 0.2, 0.4) , 𝐶 = diag (5, 4, 5) , 0.8 𝐵 = ( 0 0.8) , 0.8 0 (55) (56) We can get the parameters as follows: 𝜎1 = 0.5, 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: 𝐴 = diag (1, 2, 3) , 𝜏 (𝑡) = 0.4 + 0.1cos2 𝑡, 𝜎2 = 0.8, 𝜏1 = 0.4, 𝜎3 = −0.3, 𝜏2 = 0.5, 𝜏4 = 0.1, 𝜎4 = 0.3, 𝜏3 = −0.1, 𝜗1 = (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: 7.1735 −0.4713 −0.2152 𝑃1 = (−0.4713 7.2143 −0.3071) , −0.2152 −0.3071 6.9607 7.9363 0 8.0199 ) 𝑃2 = ( 0 7.9001 (59) 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 LyapunovKrasovskii 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 10 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) References [1] H Li and X Yang, “Asymptotic stability analysis of genetic regulatory networks with time-varying delay,” in Proceedings of 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delays Firstly, a globally exponential stability result is developed for the genetic regulatory network with. .. 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. .. consider the globally exponential stability of the genetic regulatory networks with time- varying delays and continuous distributed delays Theorem For system (14) with Assumptions and 3, the equilibrium