# Finite-Time Stability Analysis of Switched Genetic Regulatory Networks with Time-Varying Delays via Wirtinger’s Integral Inequality

Authors: Shanmugam Saravanan; M. Syed Ali; Grienggrai Rajchakit; Bussakorn Hammachukiattikul; Bandana Priya; Ganesh Kumar Thakur
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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 Section2, 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,ℝ, ℝn, and ℝn×m denote, respectively, the set of all real numbers, real n-dimensional space, and real n×m-dimensional space. · denote the Euclidean norms in ℝn. For a vector or matrix P, PT denotes its transpose. For a square matrix P, λmaxP and λminP denote the maximum eigenvalue and minimum eigenvalue of matrix P, respectively, and sym P is used to represent P+PT. 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 by(1)e˙1t=−Ae1t+Bfe2t−τt+I,e˙2t=−Ce2t+De1t−σt,where e1t=e11t,e12t,…,e1ntT∈ℝn, e2t=e21t,e22t,…,e2ntT∈ℝn:, e2it∈ℝ are the concentrations of mRNA and protein, respectively; f⋅=f1⋅,…,fn⋅T is the regulatory functions of mRNAs, A=diaga1,a2,…,an and C=diagc1,c2,…,cn are the constant matrices, and they are the rates of degradation; D=diagd1,d2,…,dn represents the translation rate; B=bij is the regulative matrix, and τt and σt are the time-varying delays.For obtaining our conclusions, we make the following assumptions.Assumption 1.
fs:ℝ⟶ℝ, s = 1, 2, …, n are monotonically increasing functions with saturation and satisfy(2)0≤fsa−fsba−b≤us,∀a,b∈ℝ,s=1,2,…,n,where us, s = 1, 2, …, n are the nonnegative constants.Assumption 2.
τt and σt are the time-varying delays satisfying(3)0≤τ1≤τt≤τ2,τ˙t≤τd<∞,0≤σ1≤σt≤σ2,σ˙t≤σd<∞,τ12=τ2−τ1,σ12=σ2−σ1,where τ1,τ2,σ1,σ2 are the constants. The initial condition of system (1) is assumed to be(4)−ρ≤t≤0,ρ=maxτ2,σ2.
Use the following transformation:(5)xt=e1t−e1t∗,yt=e2t−e2t∗,where e1∗=e11∗,e12∗,…,e1n∗T and e2∗=e21∗,e22∗,…,e2n∗T 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 be(6)x˙t=−Axt+Bgyt−τt,y˙t=−Cyt+Dxt−σt,where g⋅=g1⋅,…,gn⋅T, and gsyt=fsyt+e2∗−fse2∗.
According to Assumption1 and the definition of gs⋅, we know that gs⋅ is bounded, that is, ∃F>0, such that gs⋅≤F, s = 1, 2, …, n and satisfies the following sector condition:(7)0≤gsaa≤us,∀a∈ℝ/0,s=1,2,…,n.
LetU=diagu1,u2,…,un.
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:(8)x˙t=−Aptxt+Bptgyt−τt,y˙t=−Cptyt+Dptxt−σt,where pt:0,∞⟶ℕ=1,2,…,N is the switching signal, which is a piece constant function depending on time t. For each i∈ℕ, the matrices are constant matrices of appropriate dimensions.
For the switching signalpt, we have the following switching sequence: i0,t0,…,ik,tk,…,|ik∈ℕ,k=0,1,…; in other words, when t∈tk,tk+1, ikth subsystem is activated. To assume xt=ϕt, yt=ψt.
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 c1,c2,T, if(9)Φt2+Ψt2≤c1⇒xt2+yt2≤c2,t∈0,T,where(10)Φt=sup−ρ≤t≤0ϕt,ϕ˙t,Ψt=sup−ρ≤t≤0ψt,ψ˙t.Definition 2 (See [51]).
For anyT2>T1≥0, let NpT1,T2 denote the switching number of pt on an interval T1,T2. If(11)NpT1,T2≤N0+T2−T1τaholds for given N0≥0, τa>0, then the constant τa is called the average dwell time and N0 is the chatter bound. Without loss of generality, we choose N0=0 throughout this study.Lemma 1 (See [52]).
For any constant matrixM∈ℝn×n, M=MT>0, scalars η2>η1>0, and vector function w:η1,η2⟶ℝn such that the integrations concerned are well defined, and the following inequality holds(12)∫t−η2t−η1wsdsTM∫t−η2t−η1wsds≤η2−η1∫t−η2t−η1wTsMwsds.∫−η2−η1∫t+θtwsdsTM∫−η2−η1wsds≤η22−η122∫−η2−η1∫t+θtwTsMwsds.Lemma 2 (See [53]).
Letf1,f2,…,fN:ℝm⟶ℝ have positive values in an open subset D of ℝm. Then, the reciprocally convex combination of fi over D satisfies(13)minβi|βi>0∑iβi=1∑ifit+maxgi,jt∑i≠jgi,jt,subjected to(14)gij:ℝm⟶ℝ,gj,it=gijt,fitgi,jtgi,jtfjt≥0.Lemma 3 (See [54]).
For a positive definite matrixM>0, the following inequality holds for all continuously differentiable function xt in a,b∈ℝn×n:(15)−b−a∫abx˙TsMx˙sds≤−Φ1TMΦ1−3−Φ2TMΦ2,=−Φ1Φ2TM003MΦ1Φ2,where Φ1=xb−xa, and Φ2=xb+xa−2/b−a∫abxsds.


## 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 c1,c2,Tf and constants σ1,σ2,τ1, and τ2; if there exist symmetric positive definite matrices P1i,P2i,Qnin=1,2,3,Rnin=1,2,…,12,Snin=1,2,…,4 for all i∈ℕ, the diagonal matrix Lm=diagl1m,l2m,…,lnm≥0,m=1,2, and positive scalars μ≥1 and αi=α>0 such that the following LMIs hold(16)R3iM1i∗R4i>0,R6iR¯6i∗R6i≥0,R8iR¯8i∗R8i≥0,R10iR¯10i∗R10i>0,R12iM12i∗R12i>0,(17)Θpt=Θij22×22<0,(18)Pli≤μPlj,l=1,2,Qli≤μQlj,l=1,2,3,Rli≤μRlj,l=1,2,…,12,M1i≤μM1j,Sli≤μSlj,l=1,2,3,(19)e∑i=1NαiTdc1≤c2λ1+λ2,and the average dwell time of the switching signal p (t) satisfies(20)τai>τai∗=Tlnμilnc2λ1+λ2−lneαiTdc1,where(21)Θ11=−P1iA−ATP1iT+Q1i+Q3i+σ12R5i+σ122R6i−3e−ασ1R7i−e−ασ1R7i+αP1i,Θ13=e−ασ1R7i−3e−ασ1R7i,Θ110=P1iB,Θ113=6σ1e−ασ1R7i,Θ22=R1i+R3i−P2iC−CTP2iT+τ12R9i+τ122R10i−e−ατ1R11i−3e−ατ1R11i+αP2i,Θ24=P2iD,Θ26=e−ατ1R11i−3e−ατ1R11i,Θ29=M1i+UTL1T,Θ218=6τ1e−ατ1R11i,Θ33=−e−ασ1Q1i+e−ασ1Q2i−e−ασ1R7i−3e−ασ1R7i−e−ασ2R8i,Θ34=−e−ασ2R¯8iT+e−ασ2R8i,Θ35=e−ασ2R¯8iT,Θ313=6σ1e−ασ1R7i,Θ44=−1−σde−ασtQ3i+2e−ασ2R¯8i−2e−ασ2R8i,Θ45=e−ασ2R¯8iT+e−ασ2R8i,Θ55=e−ασ2Q2iT−e−ασ2R8i,Θ66=−e−ατ1R1i+e−ατ1R2i−e−ατ1R11i−3e−ατ1R11i−e−ατ2R12i,Θ67=−e−ατ2R¯12iT+e−ατ2R12i,Θ68=−e−ατ2R¯12iT,Θ618=6τ1e−ατ1R11i,Θ77=−1−τde−ατtR3i+2e−ατ2R¯12i−2e−ατ2R12i,Θ78=e−ατ2R¯12iT+e−ατ2R12i,Θ710=−1−τde−ατtM1i+UL2,Θ88=−e−ατ2R2iT−e−ατ2R12i,Θ99=R4i−2L1,Θ1010=−1−τde−ατ2R4i−2L2,Θ1111=σ12R7i+σ122R8i+eαteασ2−ασ2−1α2S1i+eαteασ2−eασ1−ασ2−σ1α2S2i,Θ1212=τ12R11i+τ122R12i+eαteατ2−ατ2−1α2S3i+eαteατ2−eατ1−ατ2−τ1α2S4i,Θ1313=−eασ1R5i−12σ12e−ασ1R7i,Θ1414=−eασ2R6i,Θ1415=−eασ2R¯6i,Θ1515=−eασ2R6i,Θ1616=−eαtS1i,Θ1717=−eαtS2i,Θ1818=−eατ1R9i−12τ12e−ατ1R11i,Θ1919=−eατ2R10i,Θ1920=−eατ2R¯10i,Θ2020=−eατ2R10i,Θ2121=−eαtS3i,Θ2222=−eαtS4i,d=λ3+λ4+λ5σ1e−ασ1+λ6σ2−σ1e−ασ2+λ7σ2e−ασ2+λ8τ1e−ατ1+λ9τ2−τ1e−ατ1+λ10τ2e−ατ2+2λ11τ2e−ατ2+λ12τ2e−ατ2+λ13σ132e−ασ1+λ14σ2−σ132e−ασ2+λ15σ132e−ασ1+λ16τ2−τ132e−ατ2+λ17τ132e−ατ1+λ18τ2−τ132e−ατ2+λ19τ132e−ατ1+λ20τ2−τ132e−ατ2+λ21σ154e−ασ1+λ22σ22−σ12σ2−σ134e−ασ2+λ23τ154e−ατ1+λ24τ22−τ12τ2−τ134e−ατ2,λ1=mini∈ℕλminP1i,λ2=mini∈ℕλminP2i,λ3=maxi∈ℕλmaxP1i,λ4=maxi∈ℕλmaxP2i,λ5=maxi∈ℕλmaxQ1i,λ6=maxi∈ℕλmaxQ2i,λ7=maxi∈ℕλmaxQ3i,λ8=maxi∈ℕλmaxR1i,λ10=maxi∈ℕλmaxR2i,λ11=maxi∈ℕλmaxR3i,λ12=maxi∈ℕλmaxR4i,λ13=maxi∈ℕλmaxR5i,λ14=maxi∈ℕλmaxR6i,λ15=maxi∈ℕλmaxR7i,λ16=maxi∈ℕλmaxR8i,λ17=maxi∈ℕλmaxR9i,λ18=maxi∈ℕλmaxR10i,λ19=maxi∈ℕλmaxR11i,λ20=maxi∈ℕλmaxR12i,λ21=maxi∈ℕλmaxS1i,λ22=maxi∈ℕλmaxS2i,λ23=maxi∈ℕλmaxS3i,λ24=maxi∈ℕλmaxS4i.Proof.
Choose the Lyapunov functional candidate as(22)Vptt=V1pt+V2pt+V3pt+V4pt+V5pt+V6pt,where(23)V1pt=xTtP1ixt+yTtP2iyt,V2pt=∫t−σ1teαs−txTsQ1ixsds+∫t−σ2t−σ1eαs−txTsQ2ixsds+∫t−σtteαs−txTsQ3ixsds,V3pt=∫t−τ1teαs−tyTsR1iysds+∫t−τ2t−τ1eαs−tyTsR2iysds+∫t−τtteαs−tysgysTR3iM1i∗R4iysgysds,V4pt=∫−σ10∫t+θtσ1eαs−txTsR5ixsdsdθ+∫−σ2−σ1∫t+θtσ12eαs−txTsR6ixsdsdθ+∫−σ10∫t+θtσ1eαs−tx˙TsR7ix˙sdsdθ+∫−σ2−σ1∫t+θtσ12eαs−tx˙TsR8ix˙sdsdθ,V5pt=∫−τ10∫t+θtτ1eαs−tyTsR9iysdsdθ+∫−τ2−τ1∫t+θtτ12eαs−tyTsR10iysdsdθ+∫−τ10∫t+θtτ1eαs−ty˙TsR11iy˙sdsdθ+∫−τ2−τ1∫t+θtτ12eαs−ty˙TsR12iy˙sdsdθ,V6pt=∫−σ20∫θ0∫t+νtσ122eαs−θx˙TsS1ix˙sdsdνdθ+∫−σ2−σ1∫θ0∫t+νt12σ22−σ12eαs−θx˙TsS2ix˙sdsdνdθ+∫−τ20∫θ0∫t+νtτ122eαs−θy˙TsS3iy˙sdsdνdθ+∫−τ2−τ1∫θ0∫t+νt12τ22−τ12eαs−θy˙TsS4iy˙sdsdνdθ.
Taking the derivatives ofVpt along the trajectory of system (8), we have that(24)V˙1pt=2xTtP1ix˙t+2yTtP2iy˙t,(25)V˙2pt=−αV2pt+xTtQ1ixt−e−ασ1xTt−σ1Q1ixt−σ1+e−ασ1xTt−σ1Q2ixt−σ1−e−ασ2xTt−σ2Q2ixt−σ2+xTtQ3ixt−1−σ˙te−ασtxTt−σtQ3ixt−σt,(26)V˙3pt=−αV3pt+yTtR1iyt−e−ατ1yTt−τ1R1iyt−τ1+e−ατ1yTt−τ1R2iyt−τ1−e−ατ2yTt−τ2R2iyt−τ2+ytgytTR3iM1i∗R4iytgyt−1−τ˙te−ατtyt−τtgyt−τtTR3iM1i∗R4iyt−τtgyt−τt≤−αV3pt+yTtR1iyt−e−ατ1yTt−τ1R1iyt−τ1+e−ατ1yTt−τ1R2iyt−τ1−e−ατ2yTt−τ2R2iyt−τ2+ytgytTR3iM1i∗R4iytgyt−1−τ˙te−ατ2yt−τtgyt−τtTR3iM1i∗R4iyt−τtgyt−τt,(27)V˙4pt=−αV4pt+σ12xTtR5ixt−∫t−σ1tσ1eαs−txTsR5ixsds+σ122xTtR6ixt−∫t−σ2t−σ1σ12eαs−txTsR6ixsds+σ12x˙TtR7ix˙t−∫t−σ1tσ1eαs−tx˙TsR7ix˙sds+σ122x˙TtR8ix˙t−∫t−σ2t−σ1σ12eαs−tx˙TsR8ix˙sds,(28)V˙5pt=−αV5pt+τ12yTtR9iyt−∫t−τ1tτ1eαs−tyTsR9iysds+τ122yTtR10iyt−∫t−τ2t−τ1τ12eαs−tyTsR10iysds+τ12y˙TtR11iy˙t−∫t−τ1tτ1eαs−ty˙TsR11iy˙sds+τ122y˙TtR12iy˙t−∫t−τ2t−τ1τ12eαs−ty˙TsR12iy˙sds,(29)V˙6pt=−αV6pt+eαtx˙Tteασ2−ασ2−1α2S1ix˙t−eαt∫−σ20∫t+θtx˙TsS1ix˙sdsdθ+eαtx˙Tteασ2−eασ1−ασ2−σ1α2S2ix˙t−eαt∫−σ2−σ1∫t+θtx˙TsS2ix˙sdsdθ+eαtx˙Tteατ2−ατ2−1α2S3ix˙t−eαt∫−τ20∫t+θtx˙TsS3ix˙sdsdθ+eαtx˙Tteατ2−eατ1−ατ2−τ1α2S4ix˙t−eαt∫−τ2−τ1∫t+θtx˙TsS4ix˙sdsdθ.
From Lemmas1 and 3, we have(30)−σ1∫t−σ1teαs−txTsR5ixsds≤−e−ασ1∫t−σ1txsdsTR5i∫t−σ1txsds,−σ1∫t−σ1teαs−tx˙TsR7ix˙sds≤−e−ασ1Φ1tΦ2tTR7i0∗3R7iΦ1tΦ2t,−τ1∫t−τ1teαs−tyTsR9iysds≤−e−ατ1∫t−τ1tysdsTR9i∫t−τ1tysds,−τ1∫t−τ1teαs−ty˙TsR11iy˙sds≤e−ατ1Φ3tΦ4tTR11i0∗3R11iΦ3tΦ4t,and from Lemma 2, we can obtain(31)−σ12∫t−σ2t−σ1eαs−txTsR6ixsds=−σ12e−ασ2∫t−σ2t−σtxTsR6ixsds−σ12e−ασ2∫t−σtt−σ1xTsR6ixsds≤e−ασ2ψ1ψ2T−R6iR¯6i∗−R6iψ1ψ2.
Similarly, we have(32)−σ12∫t−σ2t−σ1eαs−txTsR8ixsds≤e−ασ2ψ3ψ4T−R8iR¯8i∗−R8iψ3ψ4,(33)−τ12∫t−τ2t−τ1eαs−txTsR10ixsds=−τ12e−ατ2∫t−τ2t−τtxTsR10ixsds−τ12e−ατ2∫t−τtt−τ1xTsR10ixsds≤e−ατ2ψ5ψ6T−R10iR¯10i∗−R10iψ5ψ6,(34)−τ12∫t−τ2t−τ1eτs−txTsR12ixsds≤e−ατ2ψ7ψ8T−R12iR¯12i∗−R12iψ7ψ8,where(35)ψ1=∫t−σ2t−σtxsds,ψ2=∫t−σtt−σ1xsds,ψ3=xt−σt−xt−σ2,ψ4=xt−σ1−xt−σt,ψ5=∫t−τ2t−τtysds,ψ6=∫t−τtt−τ1ysds,ψ7=yt−τt−yt−τ2,ψ8=yt−τ1−yt−τt,Φ1t=xt−xt−σ1,Φ2t=xt+xt−σ1−2σ1∫t−σ1txsds,Φ3t=yt−yt−τ1,Φ4t=yt+yt−τ1−2τ1∫t−τ1tysds.
Meanwhile, for anyLm=diagl1m,l2m,…,lnm≥0,m=1,2, the following inequality is true from Assumption 1.(36)−2∑i=1nl1igiyitgiyit−uiyit−2∑i=1nl2igiyit−τtgiyit−τt−uiyit−τt≥0.
It can written as(37)−2gTytL1gyt+2yTtUL1gyt−2gTyt−τtL2gyt−τt+2yTt−τtUL2gyt−τt≥0.
What is more, the following equations are true for any matricesN1,N2 with appropriate dimensions from system (8).(38)2x˙TtN1−Axt+Bgyt−τt−x˙t=0,(39)2y˙TtN2−Cyt+Dxt−τt−y˙t=0.
From (24) to (39), we have that(40)V˙pt−αVpt≤ξTtΘiξt,where(41)ξTt=xTtyTtxTt−σ1xTt−σtxTt−σ2yTt−τ1yTt−τtyTt−τ2gTytgTyt−τtx˙Tty˙Tt∫t−σ1txTsds∫t−σ2t−σtxTsds∫t−σtt−σ1xTsds∫t−σ10∫t+θtx˙Tsdsdθ∫−σ2−σ1∫t+θtx˙Tsdsdθ∫t−τ1tyTsds∫t−τ2t−τtyTsds∫t−τtt−τ1yTsds∫t−τ10∫t+θty˙Tsdsdθ∫−τ2−τ1∫t+θty˙Tsdsdθ.
By condition (17), we have(42)V˙pt+αVpt<0.
Note that(43)ddte−αtVpt<0.
For anyT>0, let t0=0, and we denote t0,t1,t2,t3,…,ti,…,tNP0,T and t1,t2,…,ti,…,tNP0,T as the switching times on the interval 0,1, where(44)Np0,T=∑iMNpi0,T.
By integrating (43) for any t∈ti,ti+1, we find that(45)Vpt≤eαptit−tiVpt.
From (18), we can obtain(46)Vpt≤μVpti−,pti−,∀pti=i,pti−=j∈ℕ×ℕ,i≠j.
By substituting (46) into (45), we can obtain(47)Vpt≤eαptNpt0,tt−tNpt0,tVtNp0,t,ptNp0,t≤μptNp0,teαptNp0,tt−tNp0,tVxtNp0,t−,ptNp0,t−1≤μptNp0,TeαptNpt,Tt−tNp0,T+αptNp0,T−1tNp0,T−tNp0,T−1VxtNp0,T−1,ptNp0,T−1+μptNp0,T∫tNp0,T−1tNp0,TeαptNp0,TT−tNp0,T+⋯≤∏l=0Np0,T−1μptl+1e∑l=0Np0,T−1αptl+1+αptltl+1+αptNp0,TT+αpt0t0Vxt0,pt0≤e∑i=1NTi0,T/τailnμi+∑i=1NαiTi0,TVxt0,pt0≤e∑i=1Nlnμi/τai+αiTVxt0,pt0.
On the other hand, it follows from (22) that(48)Vp00=xT0P1ix0+yT0P2iy0+∫−σ10eαsxTsQ1ixsds+∫−σ2−σ1eαsxTsQ2ixsds+∫−σ00eαsxTsQ3ixsds+∫−τ10eαsyTsR1iysds+∫−τ2−τ1eαsyTsR2iysds+∫−τ00eαsysgysTR3iM1i∗R4iysgysds+∫−σ10∫θ0σ1eαsxTsR5ixsdsdθ+∫−σ2−σ2∫θ0σ12eαsxTsR6ixsdsdθ+∫−σ10∫θ0σ1eαsx˙TsR7ix˙sdsdθ+∫−σ2−σ2∫θ0σ12eαsx˙TsR8ix˙sdsdθ+∫−τ10∫θ0τ1eαsyTsR9iysdsdθ+∫−τ2−τ2∫θ0τ12eαsyTsR10iysdsdθ+∫−τ10∫θ0τ1eαsy˙TsR11iy˙sdsdθ+∫−τ2−τ2∫θ0τ12eαsy˙TsR12iy˙sdsdθ+∫−σ20∫θ0∫ν0σ122eαs−θx˙TsS1ix˙sdsdνdθ+∫−σ2−σ1∫θ0∫ν0σ13eαs−θx˙TsS2ix˙sdsdνdθ+∫−τ20∫θ0∫ν0τ122eαs−θy˙TsS3iy˙sdsdνdθ+∫−τ2−τ1∫θ0∫ν0τ13eαs−θy˙TsS4iy˙sdsdνdθ,Vp00≤maxi∈ℕλmaxP1i+maxi∈ℕλmaxP2i+σ1e−ασ1maxi∈ℕλmaxQ1i+σ2−σ1e−ασ2maxi∈ℕλmaxQ2i+σ2e−ασ2maxi∈ℕλmaxQ3i+τ1e−ατ1maxi∈ℕλmaxR1i+τ2−τ1e−ατ1maxi∈ℕλmaxR2i+τ2e−ατ2maxi∈ℕλmaxR3i+2τ2e−ατ2maxi∈ℕλmaxM1i+τ2e−ατ2maxi∈ℕλmaxR4imaxLi−,Li+2+σ132e−ασ1maxi∈ℕλmaxR5i+σ2−σ132e−ασ2maxi∈ℕλmaxR6i+σ132e−ασ1maxi∈ℕλmaxR7i+σ2−σ132e−ασ2maxi∈ℕλmaxR8i+τ132e−ατ1maxi∈ℕλmaxR9i+τ2−τ132e−ατ2maxi∈ℕλmaxR10i+τ132e−ατ1maxi∈ℕλmaxR11i+τ2−τ132e−ατ2maxi∈ℕλmaxR12i+σ154e−ασ1maxi∈ℕλmaxS1i+σ22−σ12σ2−σ134e−ασ2maxi∈ℕλmaxS2i+τ154e−ατ1maxi∈ℕλmaxS3i+τ22−τ12τ2−τ134e−ατ2maxi∈ℕλmaxS4i×sup−ρ≤t≤0Φt2,Ψt2,≤λ3+λ4+λ5σ1e−ασ1+λ6σ2−σ1e−ασ2+λ7σ2e−ασ2+λ8τ1e−ατ1+λ9τ2−τ1e−ατ1+λ10τ2e−ατ2+2λ11τ2e−ατ2+λ12τ2e−ατ2+λ13σ132e−ασ1+λ14σ2−σ132e−ασ2+λ15σ132e−ασ1+λ16τ2−τ132e−ατ2+λ17τ132e−ατ1+λ18τ2−τ132e−ατ2+λ19τ132e−ατ1+λ20τ2−τ132e−ατ2+λ21σ154e−ασ1+λ22σ22−σ12σ2−σ134e−ασ2+λ23τ154e−ατ1+λ24τ22−τ12τ2−τ134e−ατ2c1,=dc1.
Then, we can easily obtain(49)Vptt=e∑i=1Nlnμi/τai−αiTdc1,(50)Vptt=xTtP1ixt+yTtP2iyt≥mini∈ℕλminP1ixt2+λminP2iyt2=λ1+λ2xt2+yt2.
From (49) and (50), we obtain(51)xt2+yt2≤e∑i=1Nlnμi/τai−αiTdc1λ1+λ2.
Therefore, by Definition1, we conclude that xt2+yt2<c2. This completes the proof.Remark 1.
We introduce∫−σ20∫θ0∫t+νtσ12/2eαs−θx˙TsS1ix˙sdsdνdθ+∫−σ2−σ1∫θ0∫t+νtσ13eαs−θx˙TsS2ix˙sdsdνdθ+∫−τ20∫θ0∫t+νtτ12/2eαs−θy˙TsS3iy˙sdsdνdθ+∫−τ2−τ1∫θ0∫t+νtτ13eαs−θy˙TsS4iy˙sdsdνdθ in our Lyapunov–Krasovskii functional. The novel Lyapunov–Krasovskii functional can make the stability criteria applicable to both fast and slow time-varying delays directly. Besides, by using the convex combination technique together with the Jensen inequality lemma, less conservative criteria are obtained.
Case: we consider the following genetic regulatory networks without switched term:(52)x˙t=−Axt+Bgyt−τt,y˙t=−Cyt+Dxt−σt.
Based on Theorem1, the next rate-independent corollary is derived.Corollary 1.
The genetic regulatory network (52) is asymptotically stable with respect to positive real numbers σ1,σ2,τ1,τ2; if there exist symmetric positive definite matrices P1,P2,Qnn=1,2,3,Rnn=1,2,…,12,Snn=1,2,…,4, the diagonal matrix Lm=diagl1m,l2m,…,lnm≥0,m=1,2 such that the following LMIs hold(53)R3M1∗R4≥0,R6R¯6∗R6≥0,R8R¯8∗R8≥0,R10R¯10∗R10≥0,R12R¯12∗R12≥0,Θ¯=Θ¯ij22×22<0,where,(54)Θ¯11=−P1A−ATP1T+Q1+Q3+σ12R5+σ122R6−3e−ασ1R7−e−ασ1R7+αP1,Θ¯13=e−ασ1R7−3e−ασ1R7,Θ¯110=P1B,Θ¯113=6σ1e−ασ1R7,Θ¯22=R1+R3−P2C−CTP2T+τ12R9+τ122R10−e−ατ1R11−3e−ατ1R11+αP2,Θ¯24=P2D,Θ¯26=e−ατ1R11−3e−ατ1R11,Θ¯29=M1+UTL1T,Θ¯218=6τ1e−ατ1R11,Θ¯33=−e−ασ1Q1+e−ασ1Q2−e−ασ1R7−3e−ασ1R7−e−ασ2R8,Θ¯34=−e−ασ2R¯8T+e−ασ2R8,Θ¯35=e−ασ2R¯8T,Θ¯313=6σ1e−ασ1R7,Θ¯44=−1−σde−ασtQ3+2e−ασ2R¯8−2e−ασ2R8,Θ¯45=e−ασ2R¯8T+e−ασ2R8,Θ¯55=e−ασ2Q2T−e−ασ2R8,Θ¯66=−e−ατ1R1+e−ατ1R2−e−ατ1R11−3e−ατ1R11−e−ατ2R12,Θ¯67=−e−ατ2R¯12T+e−ατ2R12,Θ¯68=−e−ατ2R¯12T,Θ¯618=6τ1e−ατ1R11,Θ¯77=−1−τde−ατtR3+2e−ατ2R¯12−2e−ατ2R12,Θ¯78=e−ατ2R¯12T+e−ατ2R12,Θ¯710=−1−τde−ατtM1+UL2,Θ¯88=−e−ατ2R2T−e−ατ2R12,Θ¯99=R4−2L1,Θ¯1010=−1−τde−ατ2R4−2L2,Θ¯1111=σ12R7+σ122R8+eαteασ2−ασ2−1α2S1+eαteασ2−eασ1−ασ2−σ1α2S2,Θ¯1212=τ12R11+τ122R12+eαteατ2−ατ2−1α2S3+eαteατ2−eατ1−ατ2−τ1α2S4,Θ¯1313=−eασ1R5−12σ12e−ασ1R7,Θ¯1414=−eασ2R6,Θ¯1415=−eασ2R¯6,Θ¯1515=−eασ2R6,Θ¯1616=−eαtS1,Θ¯1717=−eαtS2,Θ¯1818=−eατ1R9−12τ12e−ατ1R11,Θ¯1919=−eατ2R10,Θ¯1920=−eατ2R¯10,Θ¯2020=−eατ2R10,Θ¯2121=−eαtS3,Θ¯2222=−eαtS4.Proof.
The corollary follows by a similar argument as that in proof of Theorem1.


## 4. Numerical Example

In this section, numerical examples are provided to illustrate the validity and the advantage of the proposed finite-time stability of switched GRNs with time-varying delays.Example 1.
Consider switched GRNs with time-varying delay (8) as(55)x˙t=−Aptxt+Bptgyt−τt,y˙t=−Cptyt+Dptxt−σt,with(56)A1=200020002,B1=00−2−2000−20,C1=300030003,D1=0.90000.90000.9,A2=300030003,B2=0−1.50−1.5000−1.50,C2=400040004,D2=1.20001.20001.2.
The activation function is chosen asU=diag0.65,0.65,0.65, and the values of c1,c2,T are given as follows:(57)σ1=0.7,σ2=3.5,σd=0.4,τ1=0.6,τ2=3.2,τd=0.2,c1=1.5,c2=4.5,T=6,μ=1.5.
We show the simulation result of the trajectories of the variablesxt and yt in Figures 1 and 2. It should be point out that the condition is feasible when employing the LMI toolbox in MATLAB, solve LMIs (16)–(20), and then, we can reach feasible solution. Hence, the switched GRNs (8) is finite-time stable.(58)P11=0.1837−0.0007−0.0007−0.00070.1837−0.0007−0.0007−0.00070.1837,P21=0.53360.00080.00080.00080.53360.00080.00080.00080.5336,Q11=0.0419−0.0000−0.0000−0.00000.0419−0.0000−0.0000−0.00000.0419,Q21=0.02250.00000.00000.00000.02250.00000.00000.00000.0225,Q31=0.15560.00030.00030.00030.15560.00030.00030.00030.1556,R11=0.06690.00010.00010.00010.06690.00010.00010.00010.0669,R21=0.02870.00000.00000.00000.02870.00000.00000.00000.0287,R31=0.1518−0.0001−0.0001−0.00010.1518−0.0001−0.0001−0.00010.1518,R41=0.03890.00000.00000.00000.03890.00000.00000.00000.0389,R51=0.0570−0.0000−0.0000−0.00000.0570−0.0000−0.0000−0.00000.0570,R61=0.00530.00000.00000.00000.00530.00000.00000.00000.0053,R71=0.05620.00000.00000.00000.05620.00000.00000.00000.0562,R81=0.0057−0.0000−0.0000−0.00000.0057−0.0000−0.0000−0.00000.0057,R91=0.10060.00000.00000.00000.10060.00000.00000.00000.1006,R101=0.00940.00000.00000.00000.00940.00000.00000.00000.0094,R111=0.0627−0.0001−0.0001−0.00010.0627−0.0001−0.0001−0.00010.0627,R121=0.0057−0.0000−0.0000−0.00000.0057−0.0000−0.0000−0.00000.0057,S11=0.0016−0.0000−0.0000−0.00000.0016−0.0000−0.0000−0.00000.0016,S21=10−3×0.29150.00050.00050.00050.29150.00050.00050.00050.2915,S31=0.00230.00000.00000.00000.00230.00000.00000.00000.0023,S41=10−3×0.49600.00070.00070.00070.49600.00070.00070.00070.4960,P12=0.17470.00010.00010.00010.17470.00010.00010.00010.1747,P22=0.12670.00030.00030.00030.12670.00030.00030.00030.1267,Q12=0.12650.00040.00040.00040.12650.00040.00040.00040.1265,Q22=0.1121−0.0004−0.0004−0.00040.1121−0.0004−0.0004−0.00040.1121,Q32=0.17450.00140.00140.00140.17450.00140.00140.00140.1745,R12=0.12310.00080.00080.00080.12310.00080.00080.00080.1231,R22=0.1145−0.0005−0.0005−0.00050.1145−0.0005−0.0005−0.00050.1145,R32=0.14340.00200.00200.00200.14340.00200.00200.00200.1434,R42=0.08630.00010.00010.00010.08630.00010.00010.00010.0863,R52=0.1176−0.0002−0.0002−0.00020.1176−0.0002−0.0002−0.00020.1176,R62=0.0641−0.0005−0.0005−0.00050.0641−0.0005−0.0005−0.00050.0641,R72=0.0886−0.0005−0.0005−0.00050.0886−0.0005−0.0005−0.00050.0886,R82=0.1177−0.0004−0.0004−0.00040.1177−0.0004−0.0004−0.00040.1177,R92=0.1230−0.0001−0.0001−0.00010.1230−0.0001−0.0001−0.00010.1230,R102=0.0675−0.0005−0.0005−0.00050.0675−0.0005−0.0005−0.00050.0675,R112=0.0878−0.0006−0.0006−0.00060.0878−0.0006−0.0006−0.00060.0878,R122=0.1055−0.0008−0.0008−0.00080.1055−0.0008−0.0008−0.00080.1055,S12=0.1889−0.0000−0.0000−0.00000.1889−0.0000−0.0000−0.00000.1889,S22=0.07360.00000.00000.00000.07360.00000.00000.00000.0736,S32=0.1937−0.0000−0.0000−0.00000.1937−0.0000−0.0000−0.00000.1937,S42=0.06850.00000.00000.00000.06850.00000.00000.00000.0685.Figure 1
The mRNA concentrationsxt in Example 1.Figure 2
Protein concentrationsyt in Example 2.Example 2.
(59)A=300030003,B=00−2.5−2.5000−2.50,C=2.50002.50002.5,D=0.80000.80000.8,gy=y2/1+y2 and U=diag0.65,0.65,0.65.
Furthermore, for the parameters listed above, letτd=0.5 and σd=0.5.
In order to compare the results in [15–18], using Corollary 1, the comparison results are listed in Table 1 for τ2. Clearly, the results proposed in this study provide a larger admissible upper bound delay to guarantee the asymptotically stable system (52). In addition, the trajectories of the genetic regulatory network (52) are shown in Figures 3 and 4.Table 1
The maximum allowable time delay upper bound forτ2 with different values of σ12.
σ120.1250.250.551.01.1[15]0.5——−8—[16]——1.0——[17]2.82732.16611.15440.49040.3845[18]3.29573.19322.94552.56612.4799Corollary 3.35.49245.02574.64124.46703.9249Figure 3
The trajectories ofxt in Example 2.Figure 4
The trajectories ofyt in Example 2.Remark 2.
The discussion in Example2 illustrates that the conditions in this study (Corollary 1) is less conservative than those in [15–18], which shows the superiority of our method compared with that in [15–18].Example 3.
Consider the following switched GRNs with time-varying delay:(60)x˙t=−Aptxt+Bptgyt−τt,y˙t=−Cptyt+Dptxt−σt,with(61)A1=3003,B1=0.81−0.200.100.64,C1=3003,D1=1001,A2=4004,B2=0.1−1−10.1,C2=4004,D2=0.8000.8.The activation function is chosen as U=diag0.2,0.2, and the values of c1,c2,T are given as follows:(62)σ1=0.2,σ2=3.2,σd=1,τ1=0.1,τ2=0.3,τd=0.1,c1=1,c2=3.2,T=4,μ=0.9.
By employing the LMI toolbox in MATLAB, solve LMIs (16)–(20), and the feasible solutions are then reached.


## 5. Conclusion

In this study, a finite-time stability analysis for switched GRNs with time-varying delays has been investigated. We utilized the reciprocally convex combination method, Wirtinger’s integral inequality, and new triple integral with exponential function in Lyapunov–Krasovskii functionals; a less conservative LMI-based finite-time stability criterion is obtained with the switched ADT approach to reduce the conservatism of our results, compared with existing ones. A numerical example has been given to demonstrate the effectiveness and the advantage of our proposed methods. State estimation as well as other research topics such as switched lure systems and complex networks [55, 56] and stabilization of probabilistic Boolean networks [57, 58] of the time delay systems will be further investigated based on the methods proposed in this study.



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