# Improved Delay-Dependent Stability Criterion for Genetic Regulatory Networks with Interval Time-Varying Delays via New Lyapunov Functionals

Authors: Aphirak Boonpikum; Thongchai Botmart; Piyapong Niamsup; Wajaree Weera
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this work, the stability analysis problem of the genetic regulatory networks (GRNs) with interval time-varying delays is presented. In the previous works, the constructions of Lyapunov functional have usually been in simple Lyapunov functional, augmented Lyapunov functional, and multiple integral Lyapunov functional. Therefore, we introduce new Lyapunov functionals expressed in terms of delay product functions. New delay-dependent sufficient conditions for the genetic regulatory networks (GRNs) are established in the terms of linear matrix inequalities (LMIS). In addition, a numerical example is provided to illustrate the effectiveness of the theoretical results.


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## Body

## 1. Introduction

In recent years, genetic regulatory networks (GRNs) have achieved popularity in the biological and biomedical applications since it can effectively reflect the living organisms of molecular and cellular levels [1, 2]. The researchers in various fields have wildly emphasized on GRNs. The study of living organism processes is one of the main challenges in our postgenomic era. Mathematical modelling of GRNs is a tool in the study of mechanism of control gene expression in an organism. The platforms of the studied model have various types such as Boolean models [3], differential equation model [4], stochastic equations [5, 6], and fractional-order dynamical systems [7]. In Boolean models, the expression of each gene is described to be either ON or OFF, no intermediate activity level is ever taken into consideration, and the state of gene is determined by the Boolean function of the states of other related genes [3]. In contrast, differential equation models described the continuous dynamical behaviors of GRNS between the concentration of gene product such as messenger ribonucleic acids (mRNAs) and proteins as the unknown functions. Consequently, differential equation models have drawn the attention of research studies to describe the gene regulatory process of organisms on the molecular level.In experimental study in biochemistry, the gene expression was studied which consists of transcription and translation processes. As we known that the transcription and translation processes are slow reaction, time delay is inevitable in genetic regulatory network. In a biochemistry experiment on mice, it has been proved that there exists the time lag of about 15 minutes in the peaks between the mRNA molecules and the proteins of the gene Hes1 [8]. Thus, time delay has been attended in the study of mathematical modelling of GRNs.Time delays in mathematical systems have received attention from researchers in the stability analysis which provide a poor performance or even instability of the relevant system [9–15]. In this reason, the stability analysis problems of genetic regulatory network with time delay have been reported in the literature [6, 8, 16–28]. They proposed new stability condition and reduced the possible conservativeness caused by the time delay. First, the local stability criteria for GRNs with constant delay were established in [8]. However, it is not sufficient to describe the behavior dynamic of nonlinear GRNs. The global asymptotic stability of GRN with SUM regulatory functions has been studied [16, 20, 21, 25, 26]. An increasing number of research studies about more complex modelled GRNs has been received attention in the following years, e.g., GRNs with distributed delayed [27], GRNs with nonlinear disturbance [17], leakage delay involved GRNS [22], and impulsive perturbation [24]. The stability problems of GRNs, such as stochastic stability [19], robust stability [18], exponential convergence analysis [28], and discrete-time stochastic stability [6, 23] have been also reported in the literature.The stability analysis of systems with time-varying delays is the major problem of stability in systems. The conservatism stability condition and the maximum allowable delay bound have been recognized as the most important index. As we know that there are many methods to reduce the conservatism in stability criterions such as the development of the integral inequalities, i.e., the Jensens inequality, the Wirtinger-based inequality [16, 20], delay partitioning approach, and free-weighting matrix were used in [29], generalized zero equalities [30], and Lyapunov function approach [31]. As we know, in order to get the less conservative results, the main efforts were concentrated on two directions. One of the most popular approach is Lyapunov function. The construction of Lyapunov functions generally has two types, i.e., augmented Lyapunov functional approach (ALFA) and multiple integral Lyapunov functional approach (MILFA), which have been mainly utilized to propose new Lyapunov functionals [30–38]. The ALFA is more state information into the vector of the positive quadratic term, for example, ξTtPξt with ξt including xt, yt, fxt, ∫t−τ1txsds, and ∫t−σ1txsds, while the MILFA is used in multiple integral terms of a positive quadratic term as a Lyapunov function. In [32, 33], they established a new Lyapunov functionals which are delay product-type functionals and lead to less conservative result in time-varying delay systems. Furthermore, the construction of MILFA also led to least conservatism and decision variables for uncertain systems with interval time-varying delay [30–38].Motivated by the above discussion, the most important concerns are the improved stability conditions for genetic regulatory networks with time-varying delays. In this paper, we will present new Lyapunov functions which are extended from [32, 33], by adding two quadratic terms. One is double integral terms of a positive quadratic term and the other one is a quadratic term which does not need to meet positive definite and would relax the stability conditions. Moreover, the development of double integral inequality is utilized to derive sufficient conditions in terms of linear matrix inequalities (LMIS). Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results.Notations: throughout this paper,ℝn and ℝm×n denote the n-dimensional Euclidean space and the set of all m×n real matrices, respectively. S+n denotes a set of positive definite matrices with n x n dimensions. X>0 means that the matrix X is a real symmetric positive definite matrix. I denotes the identity matrix with appropriate dimensions, diag… denotes block diagonal matrix. The superscript T denotes the matrix transposition. Col… denotes column matrix. The ∗ in the matrix represents the elements below the main diagonal of a symmetric matrix. SymX indicates indicates X+XT For X∈ℝm×n.


## 2. Problem Formulations

Genetic regulatory networks (GRNs) are composed of a number of genes that interact and regulate the expression of other genes by proteins (the gene product). The dynamic behavior of genetic regulatory networks with variable delays can be described by the following state equations [4]:(1)m.it=−aimit+fip1t−σt,p2t−σt,…,pnt−σt,p.it=−cipit+dimit−τt,where mit and pit denote the concentration of mRNA and protein of the ith node at time t. ai and ci are positive real numbers that present the degradation rates of mRNA and protein, respectively. di is the translation rate. σt and τt are transcriptional and translational delays, respectively, and function fi denotes the regulatory function or transcription function, which is generally nonlinear function but has a form of monotonicity of each variable. It is usual to assume that the regulatory function satisfies the following SUM logic [26]:(2)fip1t−σt,p2t−σt,…,pnt−σt=∑j=1nbijpjt−σt.And the functionbijpjt is generally expressed by a monotonic function of the Hill form:(3)bijpjt=αijpjt/βjhj1+pjt/βjhj,if transcription factorjis an activator of genei,αij1/1+pjt/βjhj,if transcription factorjis a repressor of genei,where hj is the Hill coefficient, βj is a positive scalar, and αij is a bounded constant denoting the dimensionless transcriptional rate of transcription factor j to gene i. We note that(4)αij11+pjt/βjhj=αij1−pjt/βjhj1+pjt/βjhj.Thus, GRNs (1) can be rewritten as(5)m˙it=−aimit+∑j=1nwijgjpjt−σt+Ji,p.it=−cipit+dimit−τt,where gjx=x/βihj/1+x/βihj is a monotonically increasing function, Ji=∑j∈Viαij, and Vi is the set of all the transcription factor j which is a repressor of gene i. The matrix W=wij∈ℝn×n is the coupling matrix of GRNs, which is defined as follows:(6)wijpjt=αij,if transcription factorjactivates genei,0,if transcription factorjdoes not regulate genei,−αij,if transcription factorjrepresses genei.In compact matrix form, (5) can be written as(7)m.t=−Amt+Wgpt−σt+J,p.t=−Cpt+Dmt−τt,where(8)mt=m1t,m2t,…,mntT,pt=p1t,p2t,…,pntT,gpt=g1p1t,g2p2t,…,gnpntT,J=j1,j2,…,jnT,A=diagkm1,km2,…,kmn,C=diagkp1,kp2,…,kpn,D=diagr1,r2,…,rn.Letm∗,p∗ be the equilibrium point of GRNs (7), where m∗=m1∗,m2∗,…,mn∗T and p∗=p1∗,p2∗,…,pn∗T; then, we obtain(9)−Am∗+Wgp∗=0,−Cp∗+Dm∗=0.Shifting equilibrium pointm∗,p∗ to the origin, using the following transformation xt=mt−m∗ and yt=pt−p∗, model (7) can be transformed into the following form:(10)x.t=−Axt+Wfyt−σt,y.t=−Cyt+Dxt−τt,where fyt=gpt+p∗−gp∗. Since the function g is monotonically increasing function with saturation, it satisfies that, for all a,b∈ℝ with a≠b,(11)0≤ga−gba−b≤k.Wheng⋅ is differentiable, the above inequality is equivalent to 0≤gda/da≤k. From the relationship of f⋅ and g⋅, we know that f⋅ satisfies the sector condition [39], or equivalently(12)fafa−ka≤0.Assumption 1.
The assumptions of time-delay conditions are(13)0<τ1≤τt≤τ2,0<σ1≤σt≤σ2,0<τ.t≤τdand0<σ.t≤σd,where 0<τ1≤τ2, 0<σ1≤σ2, τd>0, and σd>0.Assumption 2.
Letgi:ℝ⟶ℝ, i=1,…,n, be monotonically increasing functions with saturation and moreover satisfy(14)0≤gixx≤li,li=maxu≥0fj′u=hi−1hi−1/hihi+1hi−1/hi4bihi.Remark 1.
The modelling of GRNs is largely dependent on powerful tools of mathematics theory. The differential equation model has drawn a lot of research attention since variables in gene dynamics are usually the concentrations of gene product (messenger ribonucleic acids (mRNAs) and proteins). It has been shown that the time delays may play an important role in the predictions of the dynamics of the mRNA and protein concentrations; GRNs models without consideration of time delays may provide wrong predictions. Therefore, it is significant to study the stability of delayed GRNs and sufficient stability conditions. However, the conservatism stability condition and the maximum allowable delay bound have been recognized as the most important index. Therefore, in this research, we have improved delay dependent stability criterion for genetic regulatory network (GRNs) with interval-time varying delays via new Lyapunov function by constructing a new Lyapunov functionals in Section4. In addition, we have established one integral and double integral inequalities to estimate. Therefore, the results had been established that the stability criterion lead to less conservativeness.
To proceed, the following lemmas are introduced which will be useful for further derivations.Lemma 1 (Jensen integral inequalities, see [40]).
Letw:a,b⟶ℝn be a differentiable function. For any given matrix R>0 and a≤s≤b, the following inequalities hold:(15)b−a∫abw.TsRw.sds≥∫abw.sdsTR∫abw.sds,b−a22∫ab∫sbw.TuRw.ududs≥∫ab∫sbw.ududsTR∫ab∫sbw.ududs.Lemma 2 (Wirtinger-based integral inequality, see [41]).
Letw:a,b⟶ℝn be a differentiable function. For any given matrix R>0 and a≤s≤b, the following inequality holds:(16)b−a∫abwTsRw.sds≥χ1TRχ1+3χ2TRχ2+5χ3TRχ3,where(17)χ1=wb−wa,χ2=wb+wa−2b−a∫abwsds,χ3=wb−wa+6b−a∫abwsds−12b−a2∫ab∫sbwududs.Lemma 3 (relaxed double integral inequality, see [35]).
Letw:a,b⟶ℝn be a differentiable function. For any given matrix R>0 and a≤s≤b, the following inequality holds:(18)∫ab∫sbw.TuRw.ududs≥2χ1TRχ1+4χ2TRχ2,where(19)χ1=wb−1b−a∫abwsds,χ2=wb+2b−a∫abwsds−6b−a2∫ab∫sbwududs.Lemma 4 (Schur complement, see [42]).
For a symmetric matrixS=S11S12∗S22, the following conditions are equivalent:(a)
S<0(b)
S11<0 and S22−S12TS11−1S12<0(c)
S22<0 and S11−S12S22−1S12T<0


## 3. Improved Integral Inequalities

In this section, we propose the following results which will be used to the development of the inequality.Lemma 5 (extended relaxed one integral inequality, see [42]).
Letw:α1,α2⟶ℝn be a differentiable function, for a time-varying scalar αt∈α1,α2, symmetric matrices R˜i=diagRi,3Ri, with Ri>0, and any matrices Si∈ℝ2n×2n, i=1,2, the following integral inequality holds:(20)∫t−αtt−α1w.TsR1w.sds+∫t−αtt−α1tw.TsR2w.sds≥1α21E1E2T1+βR˜1βS1+αS21+αR˜2−βS2R˜2−1S2T00αS1TR˜1−1S1E1E2,where(21)α21=α2−α1,α=αt−α1α21=α1tα21,β=α2−αtα21=α2tα21,E1=wt−α1−wt−αtwt−α1+wt−αt−2α1t∫t−αtt−α1wsds,E2=wt−αt−wt−α2wt−αt+wt−α2−2α2t∫t−α2t−αtwsds.Proof.
By setting a functionfs,a,b=2s−b−a/b−a, where a and b are constants, then the following equations are derived:(22)∫abw.sds=wb−wa,∫abfs,a,bw.sds=wb+wa−2b−a∫bawsds,∫abfs,a,bds=0,∫abf2s,a,bds=b−a3.
Furthermore, based on Schur complement, for symmetric matricesRi>0, i=1,2, and any matrices Xj, j=1,2,3,4, with appropriate dimensions, the following inequalities hold:(23)X1R1−1X1−1X1R1−1X2−1X1X2R1−1X2−1X2∗R1≥0,X3R2−1X3−1X3R2−1X4−1X3X4R2−1X4−1X4∗R2≥0,which lead to(24)Π1=∫t−αtt−α1gλ1gw.sTX1R1−1X1−1X1R1−1X2−1X1X2R1−1X2−1X2∗R1gλ1gw.sds≥0,Π2=∫t−α2t−αtgλ2gw.sTX3R2−1X3−1X3R2−1X4−1X3X4R2−1X4−1X4∗R2gλ2gw.sds≥0,where g=E1T,E2TT, λ1=fs,t−α1,t−αt, and λ2=fs,t−αt,t−α2.
Define matricesXi, i=1,2,3,4, as follows:(25)X1=−1α21R10L1,X2=−1α2103R1L2,X3=−1α21L3R20,X4=−1α21L403R2,where Li, i=1,2,3,4, are appropriate dimensional matrices, S1=L1,L2T and S2=L3,L4. Then, the following equivalent relations are taken into account similar to [42]:(26)∫t−αtt−α1gλ1gTX1R1−1X1−1X2R1−1X2−1X2R1−1X2−1gλ1gds=α1tα212E1E2TR10L1T03R1L2TL1L2L1R1−1L1T+L23R1−1L2TE1E2=1α21E1E2Tα1tα21R˜1S1∗S1TR˜1−1S1E1E2,2∫t−αtt−α1gλ1gTX1X2x.sds=−1α21E1E2T2R10L1T06R1L2TL1L20E1E2=−1α21E1E2T2R˜1S1∗0E1E2,∫t−αtt−α1gλ1gTX3R2−1X3−1X3R2−1X4−1X4R2−1X4−1gλ2gds=α2tα21E1E2TL3R2−1L3T+L43R˜2−1L4TL3L4L3TR20L4T03R2E1E2=1α21E1E2Tα2tα21S2R˜2−1S2TS2∗R˜2E1E2,2∫t−α2t−αtgλ2gTX3X4x.sds=−1α21E1E2T0L3L4L3T2R20L4T06R2E1E2=−1α21E1E2T2R˜2S2∗0E1E2.
To sum up with multiplication, we have(27)0≤Π1+Π2=∫t−αtt−α1w.sR1w.sds+∫t−α2t−αtw.sR2w.sdsE1E2−1α21E1E2T1+1−αR˜11−αS1+αS2∗1+αR˜2−1−αS2R˜2−1S2T00αS1TR˜1−1S1E1E2.
Thus,(28)Π1+Π2≥1α21E1E2T1+1−αR˜11−αS1+αS2∗1+αR˜2−1−αS2R˜2−1S2T00αS1TR˜1−1S1E1E2.
This completes the proof inequality (20).Remark 2.
Consider the inequality in Lemma4 [42] can be written in the following form:(29)∫t−httw.TsRw.sds+∫t−ht−htw.TsRw.sds≥1hE1E2TR˜+βR˜−SR˜−1STS∗R˜+αR˜−STR˜−1SE1E2.
The advantage of Lemma5 can be concluded as follows:(1)
Equation (20) can be reduced to Lemma 4 [42] when S1=S2 and the lower bound of time delay is zeros, i.e., αt∈0,h. However, Lemma 4 [42] cannot be used to estimate in the time-delay interval αt∈α1,α2(2)
The matrix S in [42] can be written as βS1+αS2 which leads to exactitude in the estimation of equation (20).(3)
The matricesR1=R2 can deal with Lemma 4 [42]. Lemma 5 can be applied in case of the different matrices, in which Lemma 4 [42] cannot be estimated in this case.Remark 3.
From the estimation of Corollary 5 [41] and reciprocally convex combination lemma [43], the estimation of integral (20) is as follows:(30)∫t−αtt−α1w.TsR1w.sds+∫t−α2t−αtw.TsR2w.sds≥1α21E1E2TR˜1S∗R˜2E1E2.
WhenR˜1S∗R˜2>0, compared with inequality (20), two aspects can be shown:(1)
The requirementR˜1S∗R˜2>0 and the matrix S which is relaxed to βS1+αS2 is substituted in (20).(2)
The estimation gap between (20) and (30), which is calculated from the right-hand side of (20) and (30), is, respectively, defined by Γ1 and Γ2; then,(31)Γ1−Γ2=βR˜1−S2R˜2−1S2T00αR˜2−S1R˜1−1S1T.(i)
By using Schur complement, thenΓ1−Γ2>0. This can be shown that an upper bound of inequality (20) is more than (30). So, we can see that (20) is less conservative than (30).Lemma 6 (extended relaxed double integral inequality).
Letw:α1,α2⟶ℝn be a differentiable function, for a time-varying scalar αt∈α1,α2, symmetric matrices R˜i=diag2Ri,4Ri, with Ri>0, i=1,2 and any matrices S, and the following integral inequality holds:(32)∫t−αtt−α1∫θt−α1w.TsR1w.sdsdθ+∫t−α2t−αt∫θt−αtw.TsR2w.sdsdθ≥F1F2TR˜100R˜2−α22tS2R˜2−1S2T00α12tS1TR˜1−1S1F1F2,where(33)α21=α2−α1,α1t=αt−α1,α2t=α2−αt,F1=wt−α1−1α1t∫t−αtt−α1wsdswt−α1+2α1t∫t−αtt−α1wsds−6α12t∫t−αtt−α1∫θt−α1wsds,F2=wt−αt−1α2t∫t−α2t−αtwsdswt−αt+2α2t∫t−α2t−αtwsds−6α22t∫t−α2t−αt∫θt−αtwsdsdθ.Proof.
By setting a functionfs,a,b=3s−a−2b/2b−a, where a and b are constants, then the following equations are derived:(34)∫ab∫θbw˙sdsdθ=b−awb−1b−a∫abwsds,∫ab∫θbfs,a,bw˙sdsdθ=b−a2wb+2b−a∫abwsds−6b−a2∫ab∫θbwsdsdθ,∫ab∫θbfs,a,bdsdθ=0,∫ab∫θbf2s,a,bdsdθ=b−a216.
Furthermore, based on Schur complement, for symmetric matricesRi>0, i=1,2, and any matrices Xj, j=1,2,3,4, with appropriate dimensions, the following inequalities hold:(35)X1R1−1X1−1X1R1−1X2−1X1X2R1−1X2−1X2∗R1≥0,X3R2−1X3−1X3R2−1X4−1X3X4R2−1X4−1X4∗R2≥0.which lead to(36)Π1=∫t−αtt−α1∗∫θt−α1gλ1gw.sTX1R1−1X1−1X1R1−1X2−1X1∗X2R1−1X2−1X2∗∗R1gλ1gw.sdsdθ≥0,Π2=∫t−α2t−αt∫θt−αtgλ2gw.sTX3R2−1X3−1X3R2−1X4−1X3∗X4R2−1X4−1X4∗∗R2gλ2gw.sdsdθ≥0,where g=F1T,F2TT, λ1=fs,t−αt,t−α1, and λ2=fs,t−α2,t−αt. Define matrices Xi, i=1,2,3,4, as follows:(37)X1=−col2α1tR1,0,L1,X2=−col0,8α1tR1,2L2,X3=−colL3,2α2tR2,0,X4=−col2L4,0,8α2tR2,where Li, i=1,2,3,4, are appropriate dimensional matrices, S1=L1,L2T and S2=L3,L4. Carrying out simple algebraic calculation, then(38)∫t−αtt−α1∫θt−α1gλ1gTX1R1−1X1−1X2R1−1X2−1X2R1−1X2−1gλ1gdsdθ=F1F2T2R10α1tL1T04R1α1tL2Tα1tL1α1tL2α12tL12R1−1L1T+α12tL24R1−1L2TF1F2,2∫t−αtt−α1∫θt−α1gλ1gTX1X2w.sdsdθ=−α1tF1F2T4α1tR10L1T08α1tR1L2TL1L20F1F2=−F1F2T2R˜1α1tS1∗0F1F2,2∫t−α2t−αt∫θt−αtgλ1gTX3R2−1X3−1X3R2−1X4−1X4R2−1X4−1gλ2gdsdθ=F1F2Tα22tL32R2−1L3T+α22tL44R2−1L4Tα2tL3α2tL4α2tL3T2R20α2tL4T04R2=F1F2Tα22tS2R˜2−1S2Tα2tS2R˜2F1F2,2∫t−α2t−αt∫θt−αtgλ2gX3X4w.sdsdθ=−α2tF1F2T0L3L4L3T4α2tR20L4T08α2tR2F1F2=−F1F2T0α2tS22R˜2F1F2.
Thus,(39)Π1+Π2≥F1F2TR˜100R˜2−α22tS2R˜2−1S2T00α12tS1TR˜1−1S1F1F2.
This completes the proof of inequality.Remark 4.
From Lemma3, the following double integral inequality can be written as(40)∫t−αtt−α1∫θt−α1w.sR1w.sdsdθ+∫t−α2t−αt∫θt−αtw.TsR2w˙sdsdθ≥F1F2TR˜100R˜2F1F2.
Compared with inequality (32), two aspects can be shown:(1)
IfS1=0 and S2=0, then (32) is reduced to (40) which it is a more general form. Based on Schur complement, (32) can be written as(41)R˜10α2tF1TS2α1tF2TS1T∗R˜200∗∗R˜20∗∗∗R˜1>0.(i)
It provides the freedom matricesS1 and S2, which lead to reduced conservativeness.(2)
The right-hand side of (32) has information of nonlinear delayed terms α12t and α22t. Time-delay terms appear in this term. It is the advantage to deal with a larger time delay system which can help to reduce conservatism.


## 4. Novel Lyapunov Functionals

In this section, we propose two novel Lyapunov functionals, which are the main contributions of this paper.Proposition 1.
For system (10) with given scalars σ1, σ2, τ1, and τ2 and positive definite matrices Mi and Ni, i=1,2,3,4, the following function can be Lyapunov functional candidate:(42)VJt=VJ1t+VJ2t,where(43)VJ1t=τ1t∫t−τtt−τ1x.TsM1x.sds+τ2t∫t−τ2t−τtx.TsM2x.sds+σ1t∫t−σtt−σ1y.TsN1y.sds+σ2t∫t−σ2t−σty.TsN2y.sds−τ1−xt−τ1−xt−τtTM1xt−xt−τt−xt−τt−xt−τ2TM2xt−τt−xt−τ2−yt−σ1−yt−σtTN1yt−σ1−yt−σt−yt−σt−yt−σ2TN2yt−σt−yt−σ2,VJ2t=τ1t∫t−τtt−τ1∫θt−τ1x.TsM3x.sdsdθ−η1TtM3η1t+τ2t∫t−τ2t−τt∫θt−τtx.TsM4x.sdsdθ−η2tM4η2t+σ1t∫t−σtt−σ1∫θt−σ1y.TsN3y.sdsdθ−η3TtN3η3t+σ2t∫t−σ2t−σt∫θt−σty.TsN4y.sdsdθ−η4TtN4η4t,η1t=xt−τ1−∫t−τtt−τ1xsτ1tds,η2t=xt−τt−∫t−τtt−τ2xsτ2tds,η3t=yt−σ1−∫t−σtt−σ1ysσ1tds,η4t=yt−σt−∫t−σ2t−σtysσ2tds.Proof.
By Lemma1, VJt is a positive definite function which completes the proof.Remark 5.
The construction of Lyapunov functionals is usually in the form of ALFA and MILFA. The special formed function (42) in Proposition 1 is based on Jensen integral inequality and relax double integral inequality, respectively. It is obtained in the following points:(i)
The proposed functions are the product of the time-varying delays; the time-varying delays and rate of the change of the time-varying delays are associated withτt, σt, τ.t, and σ.t. Thus, the proposed functional may be of some advantage to find the stability conditions for genetic regulatory networks with interval time-varying delays.(ii)
There are nonpositive definite terms, i.e.,–xt−τ1−xt−τtTM1xt−τ1−xt−τt, −xt−τt−xt−τ2TM2xt−τt−xt−τ2, –yt−σ1−yt−σtTN1yt−σ1−yt−σt, and −yt−σt−yt−σ2TN2yt−σt−yt−σ2 in VJ1t and −τ1tη1TtM3η1t,−τ2tη2TtM4η2t, −σ1tη3TtN3η3t, and −σ2tη4TtN4η4t in VJ2t. These terms would play an important role in relaxing the stability condition.Remark 6.
The new Lyapunov functional has different forms compared with [32, 33], i.e., there are two double integral terms, τ1t∫t−τtt−τ1∫θt−τ1x.TsM3x.sdsdθ, τ2t∫t−τ2t−τt∫θt−τtx.TsM4x.sdsdθ, σ1t∫t−σtt−σ1∫θt−σ1y.TsN3y.sdsdθ, and σ2t∫t−σ2t−σt∫θt−σty.TsN4y.sdsdθ. These terms are increasing the information in stability conditions which can help in delay conditions for GRNs with interval time-varying delays.


## 5. Main Results

In this section, we analyse the asymptotic stability of genetic regulatory networks (10) with time-varying delays. The main theorem given below shows that the stability criteria can be expressed in the terms of the feasibility of the linear matrix inequalities (LMIs).For simplification, the following vectors and matrices are defined for later use:(44)ξt=colxt,yt,xt−τ1,xt−τ2,xt−τt,yt−σ1,yt−σ2,yt−σtfyt,fyt−σt,∫t−τ1txsτ1ds,∫t−τtt−τ1xsτ1tds,∫t−τ2t−τtxsτ2tds,∫t−σ1tysσ1ds,∫t−σtt−σ1ysσ1tds,∫t−σ2t−σtysσ2tds,∫t−τ1t∫θtxsτ12dsdθ,∫t−τtt−τ1∫θtxsτ12tdsdθ,∫t−τ2t−τt∫θtxsτ22tdsdθ,∫t−σ1t∫θtysσ12dsdθ,∫t−σtt−σ1∫θtysσ12tdsdθ,∫t−σ2t−σt∫θtyuσ22tdsdθ,x.t−τ1,x.t−τ2,x.t−τt,y.t−σ1,y.t−σ2,y.t−σt,τ21=τ2−τ1,τ1t=τt−τ1,τ2t=τ2−τt,τ.d=1−τ.t,σ21=σ2−σ1,σ1t=σt−σ1,σ2t=σ2−σt,σ.d=1−σ.t,η1t=e2−e12,η2t=e5−e13,η3t=e6−e15,η4t=e8−e16,ψ1=e1−e3e1+e3−2e11e1−e3+6e11−12e17,ψ2=e2−e6e2+e6−2e24e2−e6+6e24−12e20,ψ3=e3−e5e3+e5−2e12e3−e5+6e12−12e18,ψ4=e6−e8e6+e8−2e15e6−e8+6e15−12e21,ψ5=e1−e11e1+2e11−6e17,ψ6=e2−e14e2+2e14−6e20,ψ7=e5−e4e5+e4−2e13e5−e4+6e13−12e18,ψ8=e8−e7e8+e7−2e16e8−e7+6e16−12e22,π1=E1E2=e3−e5e3+e5−2e12e5−e4e5+e4−2e13,π2=E3E4=e6−e8e6+e8−2e15e8−e7e8+e7−2e16,π3=F1F2=e3−e12e3+2e12−6e18e5−e13e5+2e13−6e19,π4=F3F4=e6−e15e6+2e15−6e21e8−e16e8+2e16−6e22.Andei∈ℝn×28n is defined as ei=0n×i−1nIn0n×28−infori=1,2,...,28.Theorem 1.
For given scalarsσ1, σ2, τ1, τ2, σd, and τd, the genetic regulatory networks (10) with time-varying delays subject to assumptions 1 and 2 which are asymptotically stable, if there exist matrices P∈S+n×n, Ri∈S+n×ni=1,2,3,4,6, R5∈S+2n×2n, Wi,Xi∈S+n×ni=1,2,3,4, and Mi,Ni∈S+n×ni=1,2,3,4, diagonal matrices Λi,∈S+n×n and Hk∈S+n×ni=1,2, and any matrices Si,Ti∈ℝ2n×2ni=1,2,3,4 such that the following linear matrix inequalities (LMIs) hold:(45)W2−τ.tM1>0,W2+τ.tM2>0,(46)W4−σ.tN1>0,W4+σ.tN2>0,(47)X2−τ.tM3>0,X2+τ.tM4>0,(48)X4−σ.tN3>0,X4+σ.tN4>0,(49)Σ1E1TS2τ21F1TT2∗−τ21W¯20∗∗−X¯2<0,Σ2E2TS1Tτ21F2TT1∗−τ21W¯10∗∗−X¯1<0,(50)Σ3E3TS4σ21F3TT4∗−σ21W¯40∗∗−X¯4<0,Σ4E4TS3Tσ21F4TT3∗−σ21W¯30∗∗−X¯3<0,(51)2Φ2F1TT2F2TT11∗−X¯20∗∗−X¯1>0,2Ψ2F3TT4F4TT31∗−X¯40∗∗−X¯3>0,where(52)Σ1=τ12Φ2+τ1Φ1+Φ0+Ωxτ.t−1τ21π1T2W¯1S1∗W¯2π1−π3TX¯100X¯2π3,(53)Σ2=τ22Φ2+τ2Φ1+Φ0+Ωxτ.t−1τ21π1TW¯1S2∗2W¯2π1−π3TX¯100X¯2π3,(54)Σ3=σ12Ψ2+σ1Ψ1+Ψ0+Ωyσ.t−1σ21π2T2W¯3S3∗W¯4π2−π4TX¯300X4π4,(55)Σ4=σ22Ψ2+σ2Ψ1+Ψ0+Ωyσ.t−1σ21π2TW¯3S4∗2W¯4π2−π4TX¯300X4π4,(56)Ωτ.t,σ.t=Ωxτ.t+Ωyσ.t,(57)Ωxτ.t=e1TPe1+e1TR1+R2e1+e3TR3−R1e3−τde5TR2e5−e4TR3e4+−Ae1+We10Tτ1W1+τ21W2−Ae1+We10+−Ae1+We10T×τ122X1+τ2122X2−Ae1+We10+ψ1TW˜1ψ1+ψ3TX˜2+M˜3ψ3+ψ5TX˜1ψ5+ψ7TM˜4ψ7,(58)Ωyτ.t=SymLe2−e9TΛ1−Ce2+De5+e9TΛ2−Ce2+De5+e1TPe1+e2TR4e2+e2e9TR5e2e9T+e6TR6e6−σde8e10TR5e8e10−e7TR6e7+−Ce2+De5T×σ1W3+σ21W4−Ce2+De5+−Ce2+De5Tσ122X3+σ2122X4×−Ce2+De5+ψ2TW˜3ψ2+ψ4TX˜4+N˜3ψ4+ψ6TX˜3ψ6+ψ8TN˜4ψ8−Syme9−Le2TH1+e10−Le8TH2e10,(59)Ω1τt,τ.t=Φ2τ2t+Φ1τt+Φ0,Φ2=e23TM3e23+τde25TM4e25,Φ1=e23TM1−2τ1M3e23−τde25TM1+M2+2τ2M4e25+e24TM2e24−Symη1TtM3e23−τdη2TtM4e25,Φ0=e23Tτ12M3+τ1M1e23+τde25Tτ22M4+τ2M2+τ1M1e25−τ2e24TM2e24+Symτ1η1TtM3e23−τ2τdη2TtM4e25−Syme3−e5TM1e23−τde25+e5−e4TM2τde25−e24−τ.tτ212η1TtM3η1t−η2TtM4η2t+Symη1TtM3e3−τde5+τ.te12+η2TtM4τde5−e4+τ.te13,(60)Ω2σt,σ.t=Ψ2σ2t+Ψ1σt+Ψ0,Ψ2=e26TN3e26+σde28TN4e26,Ψ1=e26TN1−2σ1N3e26−σde28TN1+N2+2σ2N4e28+e27TN2e27−Symη3TtN3e26−σdη4TtN4e28,Φ0=e26Tσ12N3+σ1N1e26+τde25Tσ22N4+σ2N2+σ1N1e25−σ2e27TN2e27+Symσ1η3TtN3e26−σ2σdη4TtN4e26−Syme6−e8TN1e26−σde28+e8−e7TN2σde28−e27−σ.tσ212η3TtN3η3tη4TtN4η4t+Symη3TtN3e6−σde8+τ.te15+η4TtN4σde8−e7+τ.te16,(61)W¯1=diagW2−τ.tM1,3W2−τ.tM1,W¯2=diagW2+τ.tM2,3W2+τ.tM2,W¯3=diagW4−σ.tN1,3W4−σ.tN1,W¯4=diagW4+σ.tN2,3W4+σ.tN2,X¯1=diag2X2−τ.tM3,4W2−τ.tM3,X¯2=diag2X2+τ.tM4,4X2+τ.tM4,X¯3=diag2X4−σ.tN3,4X4−σ.tN3,X¯4=diag2X4+σ.tN4,4X4+σ.tN4,Π5=diagW˜1,W˜3,X˜2+M˜3,X˜4+N˜3,X˜1,X˜3,M˜4,N˜4,W˜i=diagWi,3Wi,5Wi,X˜i+1=diagXi+1,3Xi+1,5Xi+1,X˜i=diag2Xi,4Xi,M˜4=diagM4,3M4,5M4,N˜4=diagN4,3N4,5N4,i=1,3,ψ=colψ1,ψ2,ψ3,ψ4,ψ5,ψ6,ψ7,ψ8.Proof.
We first consider the following Lyapunov functional candidate:(62)V1t=V0t+VJt,where V0t=∑i=14Vit and VJt is defined in Proposition 1:(63)V1t=xtytTP00Pxtyt−2∑i=1n∫0yitλ1ilis−fis+λ2ifisds,V2t=∫t−τ1txTsR1xsds+∫t−τttxTsR2xsds+∫t−τ2t−τ1xTsR3xsds+∫t−σ1txTsR4ysds+∫t−σttysfysTR5ysfysds+∫t−σ2t−σ1yTR6ysds,V3t=∫t−τ1t∫θtx.TsW1x.sdsdθ+∫t−τ2t−τ1∫θtx.TsW2x.sdsdθ+∫t−σ1t∫θty.sW3y.sdsdθ+∫t−σ2t−σ1∫θty.sW4y.sdsdθ,V4t=∫t−τ1t∫ut∫θtx.sX1x.sdsdθdu+∫t−τ2t−τ1∫ut−τ1∫θtx.sX2x.sdsdθdu+∫t−σ1t∫ut∫θty.TsX3y.sdsdθdu+∫t−σ2t−σ1∫ut−σ1∫θty.TsX4y.sdsdθdu.
Moreover, we consider the following simple calculation results:(64)ddtτ1tη1TtM3η1t=τ.tη1TtM3η1t+Symη1TtM3η5t,ddtτ2tη2TtM4η2t=−τ˙tη2TtM4η2t+Symη2TtM4η6t,ddtσ1tη3TtN3η3t=σ.tη3TtN3η3t+Symη3TtN3η7t,ddtσ2tη4TtN4η4t=−σ.tη4TtN4η4t+Symη4TtN4η8t,where(65)η5t=τ1tx.t−τ1−xt−τ1+τdxt−τt−τ.t∫t−τtt−τ1xsτ1tds,η6t=τ2tτdtx.t−τt−τdxt−τt+xt−τ2−τ.t∫t−τ2t−τtxsτ2tds,η7t=σ1ty.t−σ1−yt−σ1+σdyt−σt−σ.t∫t−σtt−σ1ysσ1tds,η8t=σ2tσdty.t−σt−σdyt−σt+yt−σ2−σ.t∫t−σ2t−σtysσ2tds.
Then, taking time derivatives ofVt along the trajectory of system (10) yields:(66)V.0t=∑i=14V.it,where(67)V.1t=xtytTP00Px.ty.t+2Lyt−fytTΛ1y.t+2fTytΛ2y.t=ξTtSymΠ1ξt,(68)V.2t=xTtR1+R2xt+xTt−τ1R3−R1xt−τ1−1−τ.txTt−τtR2xt−τt−xTt−τ2R3xt−τ2+yTtR4yt+ytfytTR5ytfyt+yTt−σ1R6−R4yt−σ1−1−σ.tyt−σtfyt−σtTR5yt−σtfyt−σt−yTt−σ2R6yt−σ2≤xTtR1+R2xt+xTt−τ1R3−R1xt−τ1−τdxTt−τtR2xt−τt−xTt−τ2R3xt−τ2+yTtR4yt+ytfytTR5ytfyt+yTt−σ1R6−R4yt−σ1−σdyt−σtfyt−σtTR5yt−σtfyt−σt−yTt−σ2R6yt−σ2=ξTtΠ2ξt,(69)V.3t=x.Ttτ1W1+τ21W2x.t+y.Ttσ1W3+σ21W4y.t−τ1∫t−τ1tx.sW1x.sds−∫t−τ2t−τ1xTsW2x.sds−σ1∫t−σ1tyTsW2y.sds−∫t−σ2t−σ1yTsW4y.sds=ξTtΠ3ξt−τ1∫t−τ1txTsW1x.sds−∫t−τ2t−τ1xTsW2x.sds−σ1∫t−σ1tyTsW3y.sds−∫t−σ2t−σ1yTsW4y.sds,(70)V.4t=x.Ttτ122X1+τ2122X2x.t+y.Ttσ122X3+σ2122X4y.t−∫t−τ1t∫θtx.TsX1x.sdsdθ−∫t−τ2t−τ1∫θt−τ1x.TsX2x.sdsdθ−∫t−σ1t∫θty.TsX3y.sdsdθ−∫t−σ2t−σ1∫θt−σ1y.TsX4y.sdsdθ=ξTtΠ4ξt−∫t−τ1t∫θtx.TsX1x.sdsdθ−∫t−σ1t∫θty.TsX3y.sdsdθ−∫t−τtt−τ1∫θt−τ1x.TsX2x.sdsdθ−∫t−τ2t−τt∫θt−τtx.TsX2x.sdsdθ−∫t−σtt−σ1∫θt−σ1y.TsX4y.sdsdθ−∫t−σ2t−σt∫θt−σty.TsX4y.sdsdθ−τ2t∫t−τtt−τ1x.TsX2x.sds−σ2t∫t−σtt−σ1y.TsX4y.sds.
And(71)V.J=V.J1t+V.J2t,where(72)V.J1t=τ.t∫t−τtt−τ1x.TsM1x.sds∫t−τ2t−τtx.TsM2x.sds+τ1tx.Tt−τ1M1x.t−τ1−τdx.Tt−τtM1x.t−τt+τ2tτdx.Tt−τtM2x.t−τt−x.Tt−τ2M2x.t−τ2−2xt−τ1−xt−τtTM1x.t−τ1−τdx.t−τt−2xt−τt−xt−τ2TM2τdx.t−τt−x.t−τ2+τ˙t∫t−τtt−τ1∫st−τ1x.TuM3x.ududs−∫t−τ2t−τt∫st−τtx.TuM4x.ududs+τ1tτ1tx.Tt−τ1M3x.t−τ1−∫t−τtt−τ1x.TsM3x.sds+τ2tτdτ2tx.Tt−τtM4x.t−τt−∫t−τ2t−τtx.TsM4x.sds−1τ212τ˙tη1TtM3η1t−τ.tη2TtM4η2t+Symη1TtM3η5t+Symη2TtM4η6t,(73)V.J2t=σ.t∫t−σtt−σ1y.TsN1y.sds−∫t−σ2t−σty.TsN2y.sds+σ1ty.Tt−σ1N1y.t−σ1−σdy.Tt−σtN1y.t−σt+σ2tσdy.Tt−σtN2y.t−σt−y.Tt−σ2N2y.t−σ2−2yt−σ1−yt−σtTN1y.t−σ1−σdy.t−σt−2yt−σt−yt−σ2TN2σdy.t−σt−y.t−σ2+σ.t∫t−σtt−σ1∫st−σ1y.TuN3y.ududs−∫t−σ2t−σt∫st−σty.TuN4y.ududs+σ1tσ1ty.Tt−σ1N3y.t−σ1−∫t−σtt−σ1y.TsN3y.sds+σ2tσdσ2ty.Tt−σtN4y.t−σt−∫t−σ2t−σty.TsN4y.sds−1σ212σ.tη3TtN3η3t−σ.tη4TtN4η4t+Symη3TtN3η7t+Symη4TtN4η8t.
So, the time derivative ofVJt can be rewritten as follows:(74)V.Jt=ξTtΩ1τt,τ.t+Ω2σt,σ.tξt+τ.t∫t−τtt−τ1xTsM1x.sds−∫t−τ2t−τtxTsM2x.sds+τ.t∫t−τtt−τ1∫st−τ1x.TuM3x.ududs−∫t−τ2t−τt∫st−τtx.TuM4x.ududs−τ1t∫t−τtt−τ1xTsM3x.sds−τ2t∫t−τ2t−τtxTsM4x.sds+σ.t∫t−σtt−σ1yTsN1y.sds−∫t−σ2t−σty.TsN2y.sds+σ.t∫t−σtt−σ1∫st−σ1yTuN3y.ududs−∫t−σ2t−σt∫st−σty.TuN4y.ududs−σ1t∫t−σtt−σ1yTsN3y.sds−σ2t∫t−σ2t−σty.TsN4y.sds,where(75)Ω1τt,τ.t=τ1tx.Tt−τ1M1x.t−τ1−τdx.Tt−τtM1x.t−τt+τ2tτdx.Tt−τtM2x.t−τt−x.Tt−τ2M2x.t−τ2−2xt−τ1−xt−τtTM1x.t−τ1−τdx.t−τt−2xt−τt−xt−τ2TM2τdx.t−τt−x.t−τ2+τ12tx.Tt−τ1M3x.t−τ1+τdτ22tx.Tt−τtM4x.t−τt−τ.tη1TtM3η1t+τ.tη2TtM4η2t−Symη1TtM3η5t−Symη2TtM4η6t=Φ2τ2t+Φ1τt+Φ0,(76)Ω2σt,σ.t=σ1ty.Tt−σ1N1y.t−σ1−σdy.Tt−σtN1y.t−σt+σ2tσdy.Tt−σtN2y.t−σt−y.Tt−σ2N2y.t−σ2−2yt−σ1−yt−σtTN1y.t−σ1−σdy.t−σt−2yt−σt−yt−σ2TN2σdy.t−σt−y.t−σ2+σ12ty.Tt−σ1N3y.t−σ1+σdσ22ty.Tt−σtN4y.t−σt−σ.tη3TtN3η3t+σ.tη4TtN4η4t−Symη3TtN3η7t−Symη4TtN4η8t=Ψ2σ2t+Ψ1σt+Ψ0.
Utilizing Lemmas2 and 3, the single integral terms and the double integral terms of V.3t, V.4t, and V.Jt, respectively, can be as follows:(77)−τ∫t−τ1txTsW1x.sds−σ1∫t−σ1tyTsW3y.sds−τ2t∫t−τtt−τ1xTsX2x.sds−σ2t∫t−σtt−σ1yTsX4y.sds−∫t−τ1t∫θtx.TsX1x.sdsdθ−∫t−σ1t∫θty.TsX3y.sdsdθ−τ1t∫t−τtt−τ1xTsM3x.sds−τ2t∫t−τ2t−τtxTsM4x.sds−σ1t∫t−σtt−σ1y.TsN3y.sds−σ2t∫t−σ2t−σtyTsN4y.sds≤ξTtψTΠ5ψξt,where(78)Π5=diagW˜1,W˜3,X˜2+M˜3,X˜4+N˜3,X˜1,X˜3,M˜4,N˜4,W˜i=diagWi,3Wi,5Wi,X˜i+1=diagXi+1,3Xi+1,5Xi+1,X˜i=diag2Xi,4Xi,i=1,3,M˜4=diagM4,3M4,5M4,N˜4=diagN4,3N4,5N4,ψ=colψ1,ψ2,ψ3,ψ4,ψ5,ψ6,ψ7,ψ8.
According to (11), it follows that there exist diagonal matrices H1 and H2; then,(79)0≤−2fyt−LytTH1fyt−2fyt−σt−Lyt−σtTH2fyt−σt=ξTtSymΠ6ξt.
Therefore, combining with (66), (74), (75), and (79), we have(80)V.t=ξTtΩτt,σt,τ.t,σ.tξt−∫t−τtt−τ1xTsW2−τ.tM1x.sds−∫t−τ2t−τtxTsW2−τ.tM2x.sds−∫t−σtt−σ1yTsW4−σ.tN1y.sds−∫t−σ2t−σtyTsW4−σ.tN2y.sds−∫t−τtt−τ1∫θt−τ1x.TsX2−τ.tM3x.sdsdθ−∫t−τ2t−τt∫θt−τtx.TsX2−τ.tM4x.sdsdθ−∫t−σtt−σ1∫θt−σ1y.TuX4−σ.tN3y.ududθ−∫t−σ2t−σt∫θt−σty.TuX4−σ.tN4y.ududθ,where(81)Ωτt,σt,τ.t,σ.t=SymΠ1+Π2+Π3+Π4+Π5+SymΠ6+Ω1τ1t,τ2t+Ω2σ1t,σ2t=Ωxτ.t+Ωyσ.t.
Utilizing Lemmas5 and 6, then the single and double integrals in (80) are estimated as follows:(82)−∫t−τtt−τ1xTsW2−τ.tM1x.sds+∫t−τ2t−τtxTsW2−τ.tM2x.sds≤−1τ12ξTtE1E2TK1τt,τ.t−ℋ1τt,τ.tE1E2ξt,(83)−∫t−σtt−σ1yTsW4−σ.tN1y.sds+∫t−σ2t−σtyTsW4−σ.tN2y.sds≤−1σ12ξTtE3E4TK2σt,σ.t−ℋ2σt,σ.tE3E4ξt,(84)−∫t−τtt−τ1∫θt−τ1x.TsX2−τ.tM3x.sdsdθ+∫t−τ2t−τt∫θt−τtx.TsX2−τ.tM4x.sdsdθ≤−ξTtF1F2TK3τt,τ.t−ℋ3τtτ.tF1F2ξt,(85)−∫t−σtt−σ1∫θt−σ1y.TsX4−σ.tN3y.sdsdθ+∫t−σ2t−σt∫θt−σty.TsX4−σ.tN4y.sdsdθ≤−ξTtF3F4TK4σt,σ.t−ℋ4σt,σ.tF3F4ξt,where(86)K1τt,τ.t=1+τ2tτ21W¯1τ2tτ21S1+τ1tτ21S2∗1+τ1tτ21W¯2,ℋ1τt,τ.t=τ2tτ21S2W¯2−1S2T00τ1tτ21S1TW¯1−1S1,K2σt,σ.t=1+σ2tτ21W¯3σ2tτ21S3+σ1tτ21S4∗1+σ1tτ21W¯4,ℋ2σt,σ.t=σ2tτ21S4W¯4−1S4T00σ1tτ21S3TW¯3−1S3,K3τt,τ.t=X¯100X¯2,ℋ3τt,τ.t=τ22tT2X¯2−1T2T00τ12tT1TX¯1−1T1,K4σt,σ.t=X¯300X¯4,ℋ4σt,σ.t=α22tF4X¯4−1F4T00α12tF3TX¯3−1F3,W¯1=diagW2−τ.tM1,3W2−τ.tM1,W¯2=diagW2+τ.tM2,3W2+τ.tM2,W¯3=diagW4−σ.tN1,3W4−σ.tN1,W¯4=diagW4+σ.tN2,3W4+σ.tN2,X¯1=diag2X2−τ.tM3,4W2−τ.tM3,X¯2=diag2X2+τ.tM4,4X2+τ.tM4,X¯3=diag2X4−σ.tN3,4X4−σ.tN3,X¯4=diag2X4+σ.tN4,4X4+σ.tN4.
Combining (82)–(84), we have(87)V.t=ξTtΥxτt,τ.t+Υyσt,σ.tξt,where(88)Υxτt,τ.t=a2τ2t+a1τt+a0,a2=Φ2+F1F2TT2X¯2−1T2T00T1TX¯1−1T1F1F2,a2=Φ2+F1F2TT2X¯2−1T2T00T1TX¯1−1T1F1F2−F1F2T2τ2T2X¯2−1T2T002τ1T1TX¯1−1T1F1F2,a0=Ωxτ.t+Φ0−E1E2T1+τ2τ21W¯1−τ2τ21S2W¯2−1S2Tτ2τ21S1+1−τ2τ21S2∗τ2τ21W¯2−1−τ2τ21S1TW¯1−1S1E1E2−F1F2TX¯100X¯2F1F2+F1F2Tτ22T2X¯2−1T2T00τ12T1TX¯1−1T1F1F2,Υyσt,σ.t=b2σ2t+b1σt+b0,b2=Ψ2+F3F4TT4X¯4−1T4T00T3TX¯3−1T3F3F4,b1=Ψ1−1σ21E3E4TS4W¯4−1S4T−W¯3S4−S3∗W¯4−S3TW¯3−1S3E3E4−F3F4T2σ2T4X¯4−1T4T002σ1T3TX¯3−1T3F3F4,b0=Ωyσ.t+Ψ0−E3E4T1+σ2σ21W¯3−σ2σ21S4W¯4−1S4Tσ2σ21S3+1−σ2σ21S4∗σ2σ21W¯4−1−σ2σ21S3TW¯3−1S3E3E4−F3F4TX¯300X¯4F3F4+F3F4Tσ22T4X¯4−1T4T00σ12T3TX¯3−1T3F3F4.
SinceΥxτt,τ.t and Υyσt,σ.t are convex functions when 2a2>0 and 2b2>0, it can be guaranteed that if the following conditions hold Υxτt,τ.t<0, for all τt∈τ1,τ2 when(89)Υxτ1,τ.t<0,Υxτ1,τ.t<0,2a2>0.and Υyσt,σ.t<0, for all σt∈σ1,σ2, when(90)Υyσ1,σ.t<0,Υyσ2,σ.t<0,2b2>0.
Thus, it can be concluded that(91)Υxτt,τ.t+Υyσt,σ.t<0,for all τt∈τ1,τ2 and σt∈σ1,σ2.
Finally, by applying Schur complement to (87), then LMIs are equivalent to (91), which implies V.t<0 for a sufficient small scalar ϵ>0. Therefore, if LMIs (45)–(51) hold, then GRNs (10) are asymptotically stable. This completes our proof.Remark 7.
In the proof of Theorem1, the function V0t is usually shown in [20, 31, 35] for the proof of sufficient condition which leads to a less conservative stability condition. Although, Li et al. [35] added function V0t that related information interval of time-varying delays τt and σt, adding information of time-varying delays may reduce conservativeness but the number of decision variables in LMI are more than the original function V0t. Thus, our Lyapunov functional V0t is not necessary to add information related in time-varying delays τt and σt.Remark 8.
Utilizing Lemma5 in double integral terms (84) and (85) affects LMI condition (91) which is in the form of nonlinear time-delayed terms τ2t and σ2t. The stability criterion is obtained unfeasible by MATLAB LMI tools. However, it can be easy to find negative conditions for a quadratic function with time-varying delays by using the convex function [44].Remark 9.
The estimation of the derivative of the new Lyapunov functionsV3t and V4t are considered together which appear in (82)–(85). By employing extended relaxed one integral inequality (Lemma 5) and the extended relaxed double integral inequality (Lemma 6), it ensures that the conservativeness in stability condition is reduced compared with [32, 33]. Moreover, the estimation of the double integral terms in the derivative forms (84) and (85) guarantees our approach to be less conservative than one in [35]. The effectiveness of this method will be demonstrated in numerical simulation.Remark 10.
The stability criterion of Theorem1 in the form LMIs (45)–(51) can be easy to examine by using LMI toolbox in MATLAB [45].Remark 11.
The improved stability conditions by constructing new Lyapunov functionals are based on LMIs and the dimension of the LMIs depends on the number of the genes in GRNs. Thus, the computational burden problem goes up. This problem is the issue in studying needs of LMI optimization in applied mathematics and the optimization research. Hence, further new techniques are developed to reduce the conservativeness caused by the time delays such as the delay-fractioning approach.


## 6. Numerical Example

In this section, we provide numerical example with a simulation to demonstrate the effectiveness of our results.Example 1.
Consider the genetic regulatory networks (10) with the following parameters [46]:(92)A=diag3,3,3,C=diag2.5,2.5,2.5,W=00−2.5−2.5000−2.50.
D=diag0.8,0.8,0.8, fiyi=yi2/1+yi2, i=1,2,3, and K=diag0.65,0.65,0.65.
Assumeσ1=0.1, σ2=0.3, τd=1.5, and σd=0.7, and the maximum allowable delay bound (MADB) of τ2 with respect to various τ1 which is obtained in Table 1. Moreover, the time-varying delays τt and σt are assumed to τt=5.4sin25/18t+0.1 and σt=0.2sin23.5t+0.1. So, the trajectories of mRNA and proteins are given in Figure 1 with the initial conditions m=0.6,1,1.5T, t∈−5.5,0 and p=1,2,0.8T, t∈−0.3,0. And the unique equilibrium point m∗,p∗ of (5) is m∗=0.780.780.78T and p∗=0.250.250.25T.Table 1
The maximum allowable delay bound (MADB) ofτt for different τ1.
Methods0.10.51Corollary 3.1, [34]5.35.455.94Theorem1, [20]5.55.916.41Theorem1, [35]9.269.6610.16Theorem19.159.7510.35Figure 1
The trajectories of mRNAs and proteins.


## 7. Conclusions

In this paper, the stability analysis problem for genetic regulatory networks (GRNs) with time-varying delays is studied. The new Lyapunov functionals have been established for deriving the stability criterion for genetic regulatory networks with time-varying delays to reduce the conservativeness of the stability condition. This paper focuses on the construction of new Lyapunov functionals based on Jensen's inequality and relaxed double integral inequality. By employing Lemmas5 and 6, new delay dependent sufficient conditions are expressed in the terms of linear matrix inequalities (LMIs) to ensure that it is asymptotically stable for GRNs with time-varying delays. Finally, a numerical example was given to illustrate the effectiveness of the theoretical result and to show less conservativeness than some existing results in the literature.



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