# Finite-Time Robust Stability of Uncertain Genetic Regulatory Networks with Time-Varying Delays and Reaction-Diffusion Terms

Authors: Wenqin Wang; Yali Dong; Shouming Zhong; Feng Liu
Journal: Complexity
 (2019)

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## Abstract

This study seeks to address the finite-time robust stability of delayed genetic regulatory networks (GRNs) with uncertain parameters and reaction-diffusion terms. We employ an appropriate Lyapunov-Krasovskii functional to derive some less conservative stability criteria for GRNs under Dirichlet boundary conditions, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. The time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. In addition, we define some new variables to deal with uncertain parameters. Moreover, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed to handle integral terms. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the obtained stability criteria.


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## Body

## 1. Introduction

Genetic regulatory networks (GRNs) are collections of DNA segments, which interact with each other and with other substances in the cell to control the gene expression levels of mRNA and protein. To better understand the complex structure of biological mechanism is very important in biological and biomedical sciences [1–3]. To date, it has attracted considerable attention for its extensive applications.In the practical GRNs, due to the slow speed of transcription and translation, time delay is inevitable in modeling gene regulation process. The existence of time delay is a primary factor impacting the dynamic behavior of whole GRNs, which may result in poor performance or even system instability, bifurcation, and oscillation. To describe the mathematic model more accurately, time delay has been considered in the genetic regulatory processes in the past few years [4–10]. For example, [4] presents the asymptotic stability analysis of delayed GRNs with SUM regulatory logic. The sampled-data stabilization problem for Takagi-Sugeno fuzzy GRNs with leakage delays is concerned in [5]. Reference [8] reports the stability analysis of Markovian switching GRNs with leakage and mode-dependent time-varying delays along Brownian motions.In addition to the time delay, it should be noticed that parameter uncertainties are also unavoidable in the modelling process. This is owing to the intrinsic fluctuations, extrinsic disturbances, data errors, using an approximate system model for simplicity, and so on. Hence, an increasing number of studies have investigated the robust problem of uncertain GRNs with time delays [11–24]. For instance, the robust stability analysis for various kinds of delayed GRNs with parametric uncertainties are studied in [11–20], including T-S fuzzy GRNs, Markovian jumping GRNs, stochastic GRNs, etc. Besides, by using H∞ control and sample-data method, [21–24] investigate the robust H∞ control and state estimation problem for uncertain delayed GRNs.Moreover, it is necessary to consider the diffusion of regulatory proteins or metabolites from one compartment to another [25, 26] since the concentrations of mRNAs and proteins are not homogenous in space at all times. However, the reaction-diffusion term has not been taken into consideration in the above-mentioned works [4–24]. As far as we know, the stability problems for GRNs with reaction-diffusion terms are first proposed in [27, 28]. Due to the reaction-diffusion terms considered in GRNs, it is much more difficult to obtain stability criteria in terms of linear matrix inequalities (LMIs) than it would be without them. A new inequality introduced in [29] solves this problem and makes the diffusion-dependent stability criteria in terms of LMIs. However, there are still a few papers [30–34] that have considered the dynamic properties of GRNs with reaction-diffusion terms. References [30, 31] investigate the asymptotic stability problem for delayed GRNs with reaction-diffusion terms under Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions. From [28, 30, 31], we can find that the reaction-diffusion terms have no effect on the stability of GRNs under Neumann and Robin boundary conditions. However, it can be still retained in the stability criterion under Dirichlet boundary conditions, which reduces the conservatism. The finite-time stability problem of the delayed GRNs with reaction-diffusion terms under Dirichlet boundary conditions is concerned in [32] by employing the Wirtinger-type integral inequality, Gronwall inequality, convex technique, and reciprocally convex technique to obtain less conservative results. In [27, 28, 30–34], the lower bound of time-varying delay is all regarded as zero, which should be improved. Meanwhile, the reaction-diffusion GRNs with parameters uncertainty are only considered in [27, 28, 31].Furthermore, the stability analysis of GRNs is a significant field for understanding the living organisms at both molecular and cellular levels. In real biological system, the GRNs regulate the concentrations of mRNA and protein in a steady state, which is fatal for normal life activities. Otherwise, the instability of those molecular concentrations can lead to very serious consequences. Take cancer, for example; an important factor that causes cancer is the high amount of antiapoptosis proteins in the cancer patient’s body, which exceeds the ability of regulation by GRNs; see [35]. However, in the healthy human body, there is a fine-tuned balance in the process of apoptosis, which can force the cells containing genetic mutations to die and protect the whole living organisms. Therefore, the stability analysis of GRNs is an important area of research in the biological and biomedical sciences.Motivated by the work discussed above, the main purpose of this study is to investigate the finite-time robust stability of delayed GRNs with uncertain parameters and reaction-diffusion terms. In order to obtain less conservative stability criteria, the time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. Besides, the approach for conducting uncertain parameters is to define parts of them as some new variables. Moreover, the Lyapunov stability theory, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed in the proof of Theorem11. Finally, sufficient stability conditions are formed for GRNs in terms of LMIs, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. The numerical example is provided to illustrate the effectiveness of the proposed theoretical results.The rest of the paper is organized as follows. The problem formulation and some preliminaries are given in Section2 firstly. Second, Section 3 presents the finite-time robust stability analysis for GRNs with reaction-diffusion terms under Dirichlet boundary conditions. Then, a numerical example is given to illustrate the effectiveness of the main stability criteria in Section 4. Finally, we conclude this paper in Section 5.Notations. The notations used throughout the paper are fairly standard. The superscript “T” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space; Rn×m is the set of all n×m real matrices; the notation P>0 means that P is a positive definite matrix; In and 0n×n represent identity matrix and zero matrix with dimension n, respectively; diag⁡(·) denotes the diagonal matrix. λmin(P) and λmax(P) denote the minimum and maximum eigenvalue of matrix P, respectively. In symmetric block matrices, we use an asterisk (∗) to represent a term which is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. Ω is a compact set in space Rn with smooth boundary ∂Ω and mes⁡Ω>0 denotes the measure of Ω; dS is the element of ∂Ω. Cn(Ω) represents the space of derivable n-order real functions with compact support on Ω. Let L2(R×Ω,Rn) denotes the space of real Lebesgue measurable functions on R×Ω. The norm ‖ϕ(t,z)‖=∫ΩϕT(t,z)ϕ(t,z)dz1/2 is defined for any ϕ(t,z)∈L2(R×Ω,Rn), and ‖ϕ(t,z)‖ω=max⁡sup-ω≤t≤0⁡‖ϕ(t,z)‖ is defined on C1([-ω,0]×Ω,Rn), z=(z1,z2,…,zm)T∈Ω⊂Rm.


## 2. Problem Formulation and Preliminaries

We consider the following delayed GRNs presented in [3]:(1)m˙it=-aimit+bip1t-σt,p2t-σt,…,pnt-σtp˙it=-cipit+dimit-τt,i=1,2,…,n.mi(t),pi(t) are concentrations of mRNA and protein in the ith node at time t, respectively. ai and ci are the degradation rates of mRNA and protein, and di is the translation rate. τ(t) and σ(t) are the time-varying delays. bi(·) is the regulatory function of the ith gene with the form bi(p1(t),p2(t),…,pn(t))=∑j=1nbij(pj(t)), which is called SUM logic [36].(2)bijpjt=αijpjt/βjHj1+pjt/βjHjif  transcription  factorjis  an  activator  of  geneiαij11+pjt/βjHjif  transcription  factorjis  a  repressor  of  geneiThe function bij(pj(t)) is a monotonic function of the Hill form. Hj is the Hill coefficient, βj is a positive constant, and the bounded constant αij is the dimensionless transcriptional rate of transcription factor j to gene i. Therefore, (1) can be rewritten into the following form:(3)m˙it=-aimit+∑j=1nwijhjpjt-σt+uip˙it=-cipit+dimit-τt,i=1,2,…,n.hj(x)=(x/βj)Hj/(1+(x/βj)Hj), ui=∑j∈Iiαij is defined as a basal rate, and Ii is the set of all the j which are repressors of gene i.Rewriting system (3) into compact matrix form, we can obtain(4)m˙t=-Amt+Whpt-σt+up˙t=-Cpt+Dmt-τtwhere (5)A=diag⁡a1,a2,…,an,C=diag⁡c1,c2,…,cn,D=diag⁡d1,d2,…,dn,u=col⁡u1,u2,…,un,mt=col⁡m1t,m2t,…,mnt,pt=col⁡p1t,p2t,…,pnt,hpt=col⁡h1p1t,h2p2t,…,hnpnt,W=wij∈Rn×nwithwij=αijif  transcription  factorjis  an  activator  of  genei0if  there  is  no  link  from  nodejto  nodei-αijif  transcription  factorjis  a  repressor  of  geneiLet(m∗,p∗) be an equilibrium point of system (4). In the following, we will always shift the equilibrium point (m∗,p∗) to the origin by letting x(t)=m(t)-m∗,y(t)=p(t)-p∗,fj(yj(t))=hj(yj(t)+pj∗)-hj(pj∗). Hence, system (4) can be transformed into the following form:(6)x˙t=-Axt+Wfyt-σty˙t=-Cyt+Dxt-τtwhere(7)xt=col⁡x1t,x2t,…,xnt∈Rn,yt=col⁡y1t,y2t,…,ynt∈Rn,fyt=col⁡f1y1t,f2y2t,…,fnynt∈Rn.In this paper, we take parameter uncertainties and reaction-diffusion terms into account, and then the GRNs model (6) can be expressed as follows:(8)∂xt,z∂t=∑k=1m∂∂zkBk∂xt,z∂zk-A+ΔAtxt,z+W+ΔWtfyt-σt,z∂yt,z∂t=∑k=1m∂∂zkGk∂yt,z∂zk-C+ΔCtyt,z+D+ΔDtxt-τt,zxs,z=φs,z,ys,z=ϕs,z,∀s∈-ω,0where z=(z1,z2,…,zm)T∈Ω⊂Rm, Ω={z∣zk≤lk,k=1,2,…,m} is the diffusion range of mRNAs and proteins, and lk is a constant. Bk=diag⁡(b1k,b2k,…,bnk)>0 and Gk=diag⁡(g1k,g2k,…,gnk)>0 denote the transmission diffusion rate matrices of mRNA and protein, respectively. x(t,z)=col⁡x1t,z,x2t,z,…,xnt,z∈Rn,y(t,z)=col⁡{y1(t,z),y2(t,z),…,yn(t,z)}∈Rn. The initial functions φ(s,z),ϕ(s,z)∈C1-ω,0×Ω,Rn,ω=max⁡{τ2,σ2}.τ ( t ) and σ(t) are the time-varying delays satisfying 0≤τ1≤τ(t)≤τ2, τ1d≤τ˙(t)≤τd<∞; 0≤σ1≤σ(t)≤σ2, σ1d≤σ˙(t)≤σ2d<∞, where τ1,τ2,σ1,σ2,τ1d,τ2d,σ1d, and σ2d are constants.The parametric uncertaintiesΔA(t),ΔW(t),ΔC(t), and ΔD(t) satisfy ΔA(t)=EaF1(t)Ha, ΔW(t)=EwF2(t)Hw, ΔC(t)=EcF3(t)Hc, ΔD(t)=EdF4(t)Hd, where Ea,Ew,Ec,Ed,Ha,Hw,Hc, and Hd are some given constant matrices with appropriate dimensions, and Fi(t) satisfies FiT(t)Fi(t)≤I(i=1,2,3,4) for any t≥0.In order to conduct the stability analysis for the aforementioned systems, it is necessary to make the following definition, assumptions, and lemmas.Definition 1 (see [27]).
The uncertain GRNs (8) are said to be finite-time robustly stable with respect to positive real numbers ε1, ε2, and T, if(9)φt,zω2+ϕt,zω2≤ε1⇒xt,z2+yt,z2≤ε2,∀t∈0,T.Assumption 2.
The GRNs (8) satisfy the Dirichlet boundary conditions; that is,(10)xit,z=0,z∈∂Ω,i=1,2,…,n,yit,z=0,z∈∂Ω,i=1,2,…,n.Assumption 3 (see [6]).
For some constantslj- and lj+(j=1,2,…,n.), we assume that f(·) satisfies the sector condition: lj-≤fj(xj)/xj≤lj+, which implies(11)fjxj-lj-xjxj≥0,lj+xj-fjxjxj≥0. Let L-=diag⁡(l1-,l2-,…,ln-), L+=diag⁡(l1+,l2+,…,ln+), lj=max⁡{|lj-|,|lj+|}, L=diag⁡(l1,l2,…,ln).Remark 4.
In [27, 28, 30–34], f(·) is a monotonically increasing function satisfying 0≤fj(xj)/xj≤lj. The constants lj- and lj+ in Assumption 3 are allowed to be positive, negative, or zero. Obviously, it relaxes the restriction on this condition.Lemma 5 (see [29]).
For the compact setΩ={z∣|zk|≤lk,k=1,2,…,m}, if ui(t,z)∈C1(Ω) satisfies ui(t,z)∂Ω=0, that is, it vanishes on the boundary ∂Ω of Ω, then(12)∫Ωuit,z∂2uit,z∂zk2dz≤-π2lk2∫Ωui2t,zdz.Lemma 6 (Jensen’s integral inequality [37]).
For any positive definite matrixM∈Rn×n, scalars h2>h1≥0, vector function w:[h1,h2]↦Rn such that the integrations concerned are well defined, the following inequality holds:(13)-h2-h1∫t-h2t-h1wTsMwsds≤-∫t-h2t-h1wsdsTM∫t-h2t-h1wsds.Lemma 7 (Wirtinger-type integral inequality [38]).
For a given matrixQ>0, the following inequality holds for all continuously differentiable function w(t) in w:[a,b]↦Rn:(14)∫abw˙TsQw˙sds≥1b-awb-waTQwb-wa+3b-aηTQη=1b-awb-waηTQ003Qwb-waη, where η=w(b)+w(a)-2/b-a∫abw(u)du.Lemma 8 (reciprocally convex combination inequality [39]).
For all vectorξ∈Rn, the function Θ(α,Q) is given by(15)Θα,Q=1αξTW1TQW1ξ+11-αξTW2TQW2ξ, where α∈(0,1), W1, W2 and Q are matrices with appropriate dimensions. Then, the following inequality holds,(16)minα∈0,1⁡Θα,Q≥ξTW1W2TQX∗QW1W2ξ,if there exists a matrix X such that QX∗Q>0.Lemma 9 (see [30]).
For real diagonal matricesP3>0 and P4>0, system (8) satisfies(17)∫Ω∂xTt,z∂tP3∑k=1m∂∂zkBk∂xt,z∂zkdz=∫ΩxTt,zP3∑k=1m∂∂zkBk∂∂zk∂xt,z∂tdz.(18)∫Ω∂yTt,z∂tP4∑k=1m∂∂zkGk∂yt,z∂zkdz=∫ΩyTt,zP4∑k=1m∂∂zkGk∂∂zk∂yt,z∂tdz.Lemma 10 (Gronwall inequality [40]).
If the functionsu(t) and b(·,t) are nonnegative and integrable over t∈[0,T], a≥0,∂b/∂t(·,t)≥0 and b(t,t) exists, then(19)ut≤a+∫0tbs,tusds implies that(20)ut≤ae∫0tbs,tds.


## 3. Finite-Time Robust Stability Analysis for Genetic Regulatory Networks

In this section, we firstly provide a finite-time robust stability criterion for GRNs (8) under Dirichlet boundary conditions and then derive three corollaries based on Theorem 11.Theorem 11.
For given constantsτ1, τ2, τ1d, τ2d, σ1, σ2, σ1d, σ2d and positive constant γ>0, the GRNs (8) are finite-time robust stable if there exist positive definite matrices Qi(i=1,2,…,8), Rj(j=1,2,…,9), diagonal matrices Pk>0(k=1,2,3,4), Ti=diag⁡(t1i,t2i,…,tni)>0(i=1,2) and any matrices S1,X=X11X12X21X22, Y=Y11Y12Y21Y22 with appropriate dimension, any constants ρj≥0(j=1,2,3,4), such that R3S1∗R9≥0, Q^8=Q~8X∗Q~8≥0, R^8=R~8Y∗R~8≥0 and the following LMIs (21) hold: (21)Ωj<0,j=1,2,3,4.Ω1, Ω2, Ω3, and Ω4 are the particular matrices of matrix Φ when α=0, β=0; α=0, β=1; α=1, β=0 and α=1, β=1, respectively. (22)Φ=∑i=19ΦiΦ1=e1-π22P1BL-2P1Ae1T+e5-π22P2GL-2P2Ce5T+e1P1We10T-e1P1Eae19T+e1P1Ewe20T+e3DP2e5T-e5P2Ece21T+e5P2Ede22T+e1P1We10T-e1P1Eae19T+e1P1Ewe20T+e3DP2e5T-e5P2Ece21T+e5P2Ede22TTΦ2=e1Q1e1T+e2Q2-Q1e2T-e4Q2e4T+e5R1e5T+e6R2-R1e6T-e8R2e8TΦ3=e1Q3e1T+e31-τ1dQ4-1-τ2dQ3e3T-e4Q4e4T+e5e9R3S1∗R9e5e9T-1-σ2de7e10R3S1∗R9e7e10T+e71-σ1dR4e7T-e8R4e8TΦ4=e1τ1Q5e1T+e5σ1R5e5T-e13τ1Q5e13T-e16σ1R5e16TΦ5=e1τ12Q6e1T+e5σ12R6e5T-ατ12e15Q6e15T-1-ατ12e14Q6e14T-βσ12e18R6e18T-1-βσ12e17R6e17TΦ6=e11τ12Q7e11T+e12σ12R7e12T-e1-e2Q7e1-e2T-3e1+e2-2e13Q7e1+e2-2e13T-e5-e6R7e5-e6T-3e5+e6-2e16R7e5+e6-2e16TΦ7=e11τ122Q8e11T+e12σ122R8e12T-Γ1Γ2Q^8Γ1Γ2T-Γ3Γ4R^8Γ3Γ4TΦ8=-e112P3e11T-e122P4e12T-e1AP3e11T+e10WTP3e11T-e11P3Eae19T+e11P3Ewe20T-e5CP4e12T+e3DTP4e12T-e12P4Ece21T+e12P4Ede22T+-e1AP3e11T+e10WTP3e11T-e11P3Eae19T+e11P3Ewe20T-e5CP4e12T+e3DTP4e12T-e12P4Ece21T+e12P4Ede22TTΦ9=-2e9T1e9T+e5T1L-+L+e9T+e9L-+L+T1e5T+e5-L-T1L+-L+T1L-e5T-2e10T2e10T+e7T2L-+L+e10T+e10L-+L+T2e7T+e7-L-T2L+-L+T2L-e7T+e1ρ1HaTHae1T+e3ρ4HdTHde3T+e5ρ3HcTHce5T+e10ρ2HwTHwe10T-ρ1e19e19T-ρ2e20e20T-ρ3e21e21T-ρ4e22e22T-e1γP1e1T-e5γP2e5TQ~8=diag⁡Q8,3Q8,R~8=diag⁡R8,3R8,Γ1=e3-e4e3+e4-2e15,Γ2=e2-e3e2+e3-2e14,Γ3=e7-e8e7+e8-2e18,Γ4=e6-e7e6+e7-2e17.τ12=τ2-τ1,σ12=σ2-σ1,α=τ2-τtτ12,β=σ2-σtσ12.BL=diag⁡∑k=1mb1klk2,∑k=1mb2klk2,…,∑k=1mbnklk2,GL=diag⁡∑k=1mg1klk2,∑k=1mg2klk2,…,∑k=1mgnklk2.
e i   ( i = 1,2 , … , 22 ) are elementary matrices; for example, e3T=[00I019].Proof.
Define vectors:(23)ϑ1t,z=F1tHaxt,z,ϑ2t,z=F2tHwfyt-σt,z,ϑ3t,z=F3tHcyt,z,ϑ4t,z=F4tHdxt-τt,zThen, the uncertain delayed GRNs (8) can be transformed into the following form:(24)∂xt,z∂t=∑k=1m∂∂zkBk∂xt,z∂zk-Axt,z+Wfyt-σt,z-Eaϑ1t,z+Ewϑ2t,z∂yt,z∂t=∑k=1m∂∂zkGk∂yt,z∂zk-Cyt,z+Dxt-τt,z-Ecϑ3t,z+Edϑ4t,zxs,z=φs,z,ys,z=ϕs,z,∀s∈-ω,0
For systems (24), we construct the following Lyapunov-Krasovskii functional:(25)Vt,x,y=∑i=17Vit,x,ywhere(26)V1t,x,y=∫ΩxTt,zP1xt,zdz+∫ΩyTt,zP2yt,zdz+∑k=1m∫Ω∂xTt,z∂zkP3Bk∂xt,z∂zkdz+∑k=1m∫Ω∂yTt,z∂zkP4Gk∂yt,z∂zkdzV2t,x,y=∫Ω∫t-τ1txTs,zQ1xs,zdsdz+∫Ω∫t-τ2t-τ1xTs,zQ2xs,zdsdz+∫Ω∫t-σ1tyTs,zR1ys,zdsdz+∫Ω∫t-σ2t-σ1yTs,zR2ys,zdsdzV3t,x,y=∫Ω∫t-τttxTs,zQ3xs,zdsdz+∫Ω∫t-τ2t-τtxTs,zQ4xs,zdsdz+∫Ω∫t-σttys,zfys,zTR3S1∗R9ys,zfys,zdsdz+∫Ω∫t-σ2t-σtyTs,zR4ys,zdsdzV4t,x,y=∫Ω∫-τ10∫t+θtxTs,zQ5xs,zdsdθdz+∫Ω∫-σ10∫t+θtyTs,zR5ys,zdsdθdzV5t,x,y=∫Ω∫-τ2-τ1∫t+θtxTs,zQ6xs,zdsdθdz+∫Ω∫-σ2-σ1∫t+θtyTs,zR6ys,zdsdθdzV6t,x,y=τ1∫Ω∫-τ10∫t+θt∂xTs,z∂sQ7∂xs,z∂sdsdθdz+σ1∫Ω∫-σ10∫t+θt∂yTs,z∂sR7∂ys,z∂sdsdθdzV7t,x,y=τ12∫Ω∫-τ2-τ1∫t+θt∂xTs,z∂sQ8∂xs,z∂sdsdθdz+σ12∫Ω∫-σ2-σ1∫t+θt∂yTs,z∂sR8∂ys,z∂sdsdθdzAs a convenience, we define(27)ξTt,z=1σ2-σt∫t-σ2t-σtys,zdsTxTt,zxTt-τ1,zxTt-τt,zxTt-τ2,zyTt,zyTt-σ1,zyTt-σt,zyTt-σ2,zfTyt,zfTyt-σt,z∂xTt,z∂t∂yTt,z∂t1τ1∫t-τ1txs,zdsT1τt-τ1∫t-τtt-τ1xs,zdsT1τ2-τt∫t-τ2t-τtxs,zdsT1σ1∫t-σ1tys,zdsT1σt-σ1∫t-σtt-σ1ys,zdsT1σ2-σt∫t-σ2t-σtys,zdsTϑ1Ttϑ2Ttϑ3Ttϑ4TtThen, taking derivative of Vt,x,y along the trajectories of systems (24), we can obtain the following differential:(28)∂∂tVt,x,y=∑i=17∂∂tVit,x,ywith(29)
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,
z
d
z
-
π
2
2
∫
Ω
y
T
t
,
z
P
2
G
L
y
t
,
z
d
z
+
2
∫
Ω
x
T
t
,
z
P
1
×
-
A
x
t
,
z
+
W
f
y
t
-
σ
t
,
z
-
E
a
ϑ
1
t
,
z
+
E
w
ϑ
2
t
,
z
d
z
+
2
∫
Ω
y
T
t
,
z
P
2
×
-
C
y
t
,
z
+
D
x
t
-
τ
t
,
z
-
E
c
ϑ
3
t
,
z
+
E
d
ϑ
4
t
,
z
d
z
+
2
∑
k
=
1
m
∫
Ω
∂
x
T
t
,
z
∂
z
k
×
P
3
B
k
∂
∂
z
k
∂
x
t
,
z
∂
t
d
z
+
2
∑
k
=
1
m
∫
Ω
∂
y
T
t
,
z
∂
z
k
P
4
G
k
∂
∂
z
k
∂
y
t
,
z
∂
t
d
z
=
∫
Ω
ξ
T
t
,
z
Φ
1
ξ
t
,
z
d
z
+
2
∑
k
=
1
m
∫
Ω
∂
x
T
t
,
z
∂
z
k
P
3
B
k
∂
∂
z
k
∂
x
t
,
z
∂
t
d
z
+
2
∑
k
=
1
m
∫
Ω
∂
y
T
t
,
z
∂
z
k
P
4
G
k
∂
∂
z
k
∂
y
t
,
z
∂
t
d
z
(30)
∂
∂
t
V
2
t
,
x
,
y
=
∫
Ω
x
T
t
,
z
Q
1
x
t
,
z
+
x
T
t
-
τ
1
,
z
Q
2
-
Q
1
x
t
-
τ
1
,
z
-
x
T
t
-
τ
2
,
z
Q
2
×
x
t
-
τ
2
,
z
d
z
+
∫
Ω
y
T
t
,
z
R
1
y
t
,
z
+
y
T
t
-
σ
1
,
z
R
2
-
R
1
y
t
-
σ
1
,
z
-
y
T
t
-
σ
2
,
z
R
2
y
t
-
σ
2
,
z
d
z
=
∫
Ω
ξ
T
t
,
z
Φ
2
ξ
t
,
z
d
z
(31)
∂
∂
t
V
3
t
,
x
,
y
≤
∫
Ω
x
T
t
,
z
Q
3
x
t
,
z
-
1
-
τ
2
d
x
T
t
-
τ
t
,
z
Q
3
x
t
-
τ
t
,
z
d
z
+
∫
Ω
1
-
τ
1
d
×
x
T
t
-
τ
t
,
z
Q
4
x
t
-
τ
t
,
z
-
x
T
t
-
τ
2
,
z
Q
4
x
t
-
τ
2
,
z
d
z
+
∫
Ω
y
t
,
z
f
y
t
,
z
T
×
R
3
S
1
∗
R
9
y
t
,
z
f
y
t
,
z
d
z
-
1
-
σ
2
d
y
t
-
σ
t
,
z
f
y
t
-
σ
t
,
z
T
R
3
S
1
∗
R
9
y
t
-
σ
t
,
z
f
y
t
-
σ
t
,
z
d
z
+
∫
Ω
1
-
σ
1
d
y
T
t
-
σ
t
,
z
R
4
y
t
-
σ
t
,
z
-
y
T
t
-
σ
2
,
z
R
4
y
t
-
σ
2
,
z
d
z
=
∫
Ω
ξ
T
t
,
z
Φ
3
ξ
t
,
z
d
z
(32)
∂
∂
t
V
4
t
,
x
,
y
=
τ
1
∫
Ω
x
T
t
,
z
Q
5
x
t
,
z
d
z
-
∫
Ω
∫
t
-
τ
1
t
x
T
s
,
z
Q
5
x
s
,
z
d
s
d
z
+
σ
1
∫
Ω
y
T
t
,
z
R
5
y
t
,
z
d
z
-
∫
Ω
∫
t
-
σ
1
t
y
T
s
,
z
R
5
y
s
,
z
d
s
d
z
≤
τ
1
∫
Ω
x
T
t
,
z
Q
5
x
t
,
z
d
z
+
σ
1
∫
Ω
y
T
t
,
z
R
5
y
t
,
z
d
z
-
τ
1
∫
Ω
1
τ
1
∫
t
-
τ
1
t
x
s
,
z
d
s
T
Q
5
1
τ
1
∫
t
-
τ
1
t
x
s
,
z
d
s
d
z
-
σ
1
∫
Ω
1
σ
1
∫
t
-
σ
1
t
y
s
,
z
d
s
T
R
5
1
σ
1
∫
t
-
σ
1
t
y
s
,
z
d
s
d
z
=
∫
Ω
ξ
T
t
,
z
Φ
4
ξ
t
,
z
d
z
(33)
∂
∂
t
V
5
t
,
x
,
y
=
τ
12
∫
Ω
x
T
t
,
z
Q
6
x
t
,
z
d
z
-
∫
Ω
∫
t
-
τ
2
t
-
τ
1
x
T
s
,
z
Q
6
x
s
,
z
d
s
d
z
+
σ
12
∫
Ω
y
T
t
,
z
R
6
y
t
,
z
d
z
-
∫
Ω
∫
t
-
σ
2
t
-
σ
1
y
T
s
,
z
R
6
y
s
,
z
d
s
d
z
(34)
∂
∂
t
V
6
t
,
x
,
y
=
τ
1
2
∫
Ω
∂
x
T
t
,
z
∂
t
Q
7
∂
x
t
,
z
∂
t
d
z
-
τ
1
∫
Ω
∫
t
-
τ
1
t
∂
x
T
s
,
z
∂
s
Q
7
∂
x
s
,
z
∂
s
d
s
d
z
+
σ
1
2
∫
Ω
∂
y
T
t
,
z
∂
t
R
7
∂
y
t
,
z
∂
t
d
z
-
σ
1
∫
Ω
∫
t
-
σ
1
t
∂
y
T
s
,
z
∂
s
R
7
∂
y
s
,
z
∂
s
d
s
d
z
≤
τ
1
2
∫
Ω
∂
x
T
t
,
z
∂
t
Q
7
∂
x
t
,
z
∂
t
d
z
+
σ
1
2
∫
Ω
∂
y
T
t
,
z
∂
t
R
7
∂
y
t
,
z
∂
t
d
z
-
∫
Ω
ξ
T
t
,
z
e
1
-
e
2
Q
7
e
1
-
e
2
T
+
3
e
1
+
e
2
-
2
e
13
Q
7
e
1
+
e
2
-
2
e
13
T
ξ
t
,
z
d
z
-
∫
Ω
ξ
T
t
,
z
e
5
-
e
6
R
7
e
5
-
e
6
T
+
3
e
5
+
e
6
-
2
e
16
R
7
e
5
+
e
6
-
2
e
16
T
ξ
t
,
z
d
z
=
∫
Ω
ξ
T
t
,
z
Φ
6
ξ
t
,
z
d
z
(35)
∂
∂
t
V
7
t
,
x
,
y
=
τ
12
2
∫
Ω
∂
x
T
t
,
z
∂
s
Q
8
∂
x
t
,
z
∂
s
d
z
-
τ
12
∫
Ω
∫
t
-
τ
2
t
-
τ
1
∂
x
T
s
,
z
∂
s
Q
8
∂
x
s
,
z
∂
s
d
s
d
z
+
σ
12
2
∫
Ω
∂
y
T
s
,
z
∂
s
R
8
∂
y
s
,
z
∂
s
d
z
-
σ
12
∫
Ω
∫
t
-
σ
2
t
-
σ
1
∂
y
T
s
,
z
∂
s
R
8
∂
y
s
,
z
∂
s
d
s
d
zNote that(36)-∫Ω∫t-τ2t-τ1xTs,zQ6xs,zdsdz=-∫Ω∫t-τ2t-τtxTs,zQ6xs,zdsdz-∫Ω∫t-τtt-τ1xTs,zQ6xs,zdsdzAccording to Lemma 6, we have(37)-∫Ω∫t-τ2t-τtxTs,zQ6xs,zdsdz≤-1τ2-τt∫Ω∫t-τ2t-τtxs,zdsTQ6∫t-τ2t-τtxs,zdsdz=-ατ12∫Ω1τ2-τt∫t-τ2t-τtxs,zdsTQ61τ2-τt∫t-τ2t-τtxs,zdsdz=∫ΩξTt,z-ατ12e15Q6e15Tξt,zdz(38)-∫Ω∫t-τtt-τ1xTs,zQ6xs,zdsdz≤-1τt-τ1∫Ω∫t-τtt-τ1xs,zdsTQ6∫t-τtt-τ1xs,zdsdz=-1-ατ12∫Ω1τt-τ1∫t-τtt-τ1xs,zdsTQ61τt-τ1∫t-τtt-τ1xs,zdsdz=∫ΩξTt,z-1-ατ12e14Q6e14Tξt,zdzThus,(39)-∫Ω∫t-τ2t-τ1xTs,zQ6xs,zdsdz≤∫ΩξTt,z-ατ12e15Q6e15T-1-ατ12e14Q6e14Tξt,zdzSimilarly,(40)-∫Ω∫t-σ2t-σ1yTs,zR6ys,zdsdz≤∫ΩξTt,z-βσ12e18R6e18T-1-βσ12e17R6e17Tξt,zdzCombining (33), (39), and (40), we can obtain(41)∂∂tV5t,x,y≤∫ΩξTt,zΦ5ξt,zdz
Next, we deal with the integral terms in (35) by applying Lemmas 7 and 8,(42)-τ12∫Ω∫t-τ2t-τ1∂xTs,z∂sQ8∂xs,z∂sdsdz=-τ12∫Ω∫t-τ2t-τt∂xTs,z∂sQ8∂xs,z∂sdsdz-τ12∫Ω∫t-τtt-τ1∂xTs,z∂sQ8∂xs,z∂sdsdz≤-τ12τ2-τt∫ΩξTt,ze3-e4Q8e3-e4T+3e3+e4-2e15Q8e3+e4-2e15Tξt,zdz-τ12τt-τ1∫ΩξTt,ze2-e3Q8e2-e3T+3e2+e3-2e14Q8e2+e3-2e14Tξt,zdz=∫ΩξTt,z-τ12τ2-τtΓ1Q~8Γ1T-τ12τt-τ1Γ2Q~8Γ2Tξt,zdz≤-∫ΩξTt,zΓ1Γ2Q^8Γ1Γ2Tξt,zdzSimilarly,(43)-σ12∫Ω∫t-σ2t-σ1∂yTs,z∂sR8∂ys,z∂sdsdz≤-∫ΩξTt,zΓ3Γ4R^8Γ3Γ4Tξt,zdzThe combination of (35), (42), and (43) gives(44)∂∂tV7t,x,y≤∫ΩξTt,zΦ7ξt,zdzBesides, for diagonal matrices P3>0 and P4>0, the following equations are true from the model (24):(45)2∫Ω∂xTt,z∂tP3∑k=1m∂∂zkBk∂xt,z∂zk-Axt,z+Wfyt-σt,z-Eaϑ1t,z+Ewϑ2t,z-∂xt,z∂tdz=0(46)2∫Ω∂yTt,z∂tP4∑k=1m∂∂zkGk∂yt,z∂zk-Cyt,z+Dxt-τt,z-Ecϑ3t,z+Edϑ4t,z-∂yt,z∂tdz=0In accordance with Green formula, Lemma 9, and Dirichlet boundary conditions, we have(47)2∫Ω∂xTt,z∂tP3∑k=1m∂∂zkBk∂xt,z∂zkdz=2∫ΩxTt,zP3∑k=1m∂∂zkBk∂∂zk∂xt,z∂tdz=2∑k=1m∫Ω∂∂zkxTt,zP3Bk∂∂zk∂xt,z∂tdz-2∑k=1m∫Ω∂xTt,z∂zkP3Bk∂∂zk∂xt,z∂tdz=2∫∂ΩxTt,zP3Bk∂∂zk∂xt,z∂tk=1m·n¯dS-2∑k=1m∫Ω∂xTt,z∂zkP3Bk∂∂zk∂xt,z∂tdz=-2∑k=1m∫Ω∂xTt,z∂zkP3Bk∂∂zk∂xt,z∂tdzwhere n¯ is an outward pointing normal vector, dS is the element of ∂Ω,(48)xTt,zP3Bk∂∂zk∂xt,z∂tk=1m=xt,zP3B1∂∂z1∂xt,z∂t,…,xt,zP3Bm∂∂zm∂xt,z∂t. Analogously,(49)2∫Ω∂yTt,z∂tP4∑k=1m∂∂zkGk∂yt,z∂zkdz=-2∑k=1m∫Ω∂yTt,z∂zkP4Gk∂∂zk∂yt,z∂tdz
Furthermore, we can obtain the following inequalities from Assumption3 for any Ti= diag(t1i,t2i,…,tni)>0,i=1,2.:(50)-2fTyt,zT1fyt,z-yTt,zT1L-+L+fyt,z+yTt,zL-T1L+yt,z≥0(51)-2fTyt-σt,zT2fyt-σt,z-yTt-σt,zT2L-+L+fyt-σt,z+yTt-σt,zL-T2L+yt-σt,z≥0
Moreover, the following inequalities are true for any constantsρi≥0,i=1,2,3,4 according to (23) and FiT(t)Fi(t)≤I(i=1,2,3,4):(52)ρ1xTt,zHaTHaxt,z-ϑ1Tt,zϑ1t,z=ρ1xTt,zHaTHaxt,z-xTt,zHaTFTtFtHaxt,z≥0(53)ρ2fTyt-σt,zHwTHwfyt-σt,z-ϑ2Tt,zϑ2t,z≥0(54)ρ3yTt,zHcTHcyt,z-ϑ3Tt,zϑ3t,z≥0(55)ρ4xTt-τt,zHdTHdxt-τt,z-ϑ4Tt,zϑ4t,z≥0
Combining (29), (30), (31), (32), (33), (34), (35), (36), (37), (38), (39), (40), (41), (42), (43), (44), (45), (46), (47), (49), (50), (51), (52), (53), (54), and (55), we can obtain(56)∂∂tVt,x,y=∑i=17∂∂tVit,x,y≤∫ΩξTt,zΦξt,zdz+γ∫ΩxTt,zP1xt,z+yTt,zP2yt,zdz≤∫ΩξTt,zΦξt,zdz+γVt,x,y
According to the inequalities in (21) and the convex combination method, we have ∫ΩξT(t,z)Φξ(t,z)dz<0, which implies (57)∂∂tVt,x,y≤γVt,x,y
Integrating the two sides of inequality (57) from 0 to t,t∈[0,T], we have(58)Vt,x,y≤V0,x0,z,y0,z+∫0tγVs,x,ydsThen, we can obtain the following inequality by the well-known Gronwall inequality (Lemma 10),(59)VT,x,y≤eγTV0,x0,z,y0,zwhere(60)
V
0
,
x
0
,
z
,
y
0
,
z
=
∫
Ω
x
T
0
,
z
P
1
x
0
,
z
d
z
+
∫
Ω
y
T
0
,
z
P
2
y
0
,
z
d
z
+
∑
k
=
1
m
∫
Ω
∂
x
T
0
,
z
∂
z
k
P
3
B
k
∂
x
0
,
z
∂
z
k
d
z
+
∑
k
=
1
m
∫
Ω
∂
y
T
0
,
z
∂
z
k
P
4
G
k
∂
y
0
,
z
∂
z
k
d
z
+
∫
Ω
∫
-
τ
1
0
x
T
s
,
z
Q
1
x
s
,
z
d
s
d
z
+
∫
Ω
∫
-
τ
2
-
τ
1
x
T
s
,
z
Q
2
x
s
,
z
d
s
d
z
+
∫
Ω
∫
-
σ
1
0
y
T
s
,
z
R
1
y
s
,
z
d
s
d
z
+
∫
Ω
∫
-
σ
2
-
σ
1
y
T
s
,
z
R
2
y
s
,
z
d
s
d
z
+
∫
Ω
∫
-
τ
0
0
x
T
s
,
z
Q
3
x
s
,
z
d
s
d
z
+
∫
Ω
∫
-
τ
2
-
τ
0
x
T
s
,
z
Q
4
x
s
,
z
d
s
d
z
+
∫
Ω
∫
-
σ
0
0
y
s
,
z
f
y
s
,
z
T
R
3
S
1
∗
R
9
y
s
,
z
f
y
s
,
z
d
s
d
z
+
∫
Ω
∫
-
σ
2
-
σ
0
y
T
s
,
z
R
4
y
s
,
z
d
s
d
z
+
∫
Ω
∫
-
τ
1
0
∫
θ
0
x
T
s
,
z
Q
5
x
s
,
z
d
s
d
θ
d
z
+
∫
Ω
∫
-
σ
1
0
∫
θ
0
y
T
s
,
z
R
5
×
y
s
,
z
d
s
d
θ
d
z
+
∫
Ω
∫
-
τ
2
-
τ
1
∫
θ
0
x
T
s
,
z
Q
6
x
s
,
z
d
s
d
θ
d
z
+
∫
Ω
∫
-
σ
2
-
σ
1
∫
θ
0
y
T
s
,
z
R
6
y
s
,
z
d
s
d
θ
d
z
+
τ
1
∫
Ω
∫
-
τ
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Note thatφi(t,z),ϕi(t,z)∈C1[-ω,0]×Ω,R(i=1,2,…,n); that is, there exist nonnegative real numbers γ1, γ2, γ3, and γ4 such that(61)∂φit,z∂t≤γ1,∂ϕit,z∂t≤γ2,∂φit,z∂zk≤γ3,∂ϕit,z∂zk≤γ4. Sequentially, the following inequalities hold:(62)∑k=1m∫Ω∂xT0,z∂zkP3Bk∂x0,z∂zkdz≤∑k=1mγ32λmaxP3λmaxBkmes⁡Ω∑k=1m∫Ω∂yT0,z∂zkP4Gk∂y0,z∂zkdz≤∑k=1mγ42λmaxP4λmaxGkmes⁡Ωτ1∫Ω∫-τ10∫θ0∂xTs,z∂sQ7∂xs,z∂sdsdθdz≤τ1λmaxQ7γ12∫Ω∫-τ10∫θ0dsdθdz=12τ13γ12λmaxQ7mes⁡Ωσ1∫Ω∫-σ10∫θ0∂yTs,z∂sR7∂ys,z∂sdsdθdz≤12σ13γ22λmaxR7mes⁡Ωτ12∫Ω∫-τ2-τ1∫θ0∂xTs,z∂sQ8∂xs,z∂sdsdθdz≤12τ12τ22-τ12γ12λmaxQ8mes⁡Ωσ12∫Ω∫-σ2-σ1∫θ0∂yTs,z∂sR8∂ys,z∂sdsdθdz≤12σ12σ22-σ12γ22λmaxR8mes⁡ΩThus, there exist nonnegative real numbers Υ1 and Υ2 satisfying the following inequalities:(63)mes⁡Ω∑k=1mγ32λmaxP3λmaxBk+12τ13γ12λmaxQ7+12τ12τ22-τ12γ12λmaxQ8=Υ1φt,zw2(64)mes⁡Ω∑k=1mγ42λmaxP4λmaxGk+12σ13γ22λmaxR7+12σ12σ22-σ12γ22λmaxR8=Υ2ϕt,zw2Consequently, we have(65)VT,x,y≤eγTV0,x0,z,y0,z≤eγTΥ3φt,zw2+Υ4ϕt,zw2≤eγTΥ3+Υ4φt,zw2+ϕt,zw2where(66)Υ3=λmaxP1+τ1λmaxQ1+τ12λmaxQ2+τ2λmaxQ3+τ2λmaxQ4+12τ12λmaxQ5+12τ22-τ12λmaxQ6+Υ1Υ4=λmaxP2+σ1λmaxR1+σ12λmaxR2+σ2λmaxR3+2λmaxS1λmaxL+λmaxR9λmaxLL+λmaxR4+12σ12λmaxR5+12σ22-σ12λmaxR6+Υ2
On the other hand,(67)VT,x,y≥λminP1xt,z2+λminP2yt,z2≥minλminP1,λminP2xt,z2+yt,z2t∈0,T.Together with (65) and (67), we have(68)xt,z2+yt,z2≤eγTΥ3+Υ4min⁡λminP1,λminP2φt,zw2+ϕt,zw2According to Definition 1, it is easy to see that the trivial solution of GRNs (8) under Dirichlet boundary conditions is finite-time robust stable. This completes the proof.Remark 12.
The time-varying delaysτ(t) and σ(t) in [27, 28, 30–34] are supposed to meet the conditions 0≤τ(t)≤τ¯ and 0≤σ(t)≤σ¯, respectively. However, the lower bound of time-varying delay is not always zero. Therefore, we set 0≤τ1≤τ(t)≤τ2 and 0≤σ1≤σ(t)≤σ2 in this paper, where τ1,τ2,,σ1, and σ2 are constants.Remark 13.
In this study, we consider the delayed GRNs with reaction-diffusion and uncertain parameters. To handle with the uncertain parameters,ϑ1(t,z), ϑ2(t,z), ϑ3(t,z), and ϑ4(t,z) are defined as new variables, which can be seen as another way to deal with the uncertainties. In this way, the model (8) is rewritten as (24). And then, we present the finite-time robust stability criteria for this system in Theorem 11.Remark 14.
In order to get less conservative stability conditions, Jensen’s inequality and Wirtinger-type integral inequality together with convex combination theory are employed to handle integral terms. For example, the processing method of integral term is-τ12∫Ω∫t-τ2t-τ1∂xT(s,z)/∂sQ8∂x(s,z)/∂sdsdz in ∂/∂tV7t,x,y. Firstly, the integral interval is divided into two parts: [t-τ2,t-τ(t)] and [t-τ(t),t-τ1]. Then, we use the Wirtinger-type integral inequality to estimate these two parts separately. Finally, they have been managed by convex combination method.When the time-varying delaysτ(t) and σ(t) are satisfying 0≤τ(t)≤τ2,τ˙(t)≤τ2d<∞,0≤σ(t)≤σ2, σ˙(t)≤σ2d<∞. τ2,σ2,τ2d, and σ2d are constants. The following corollary can be used to check the stability of GRNs (8).Corollary 15.
For given constantsτ2, τ2d, σ2, σ2d and positive constant γ>0, the GRNs (8) are finite-time asymptotic stable if there exist positive definite matrices Qi(i=2,3,6,8), Rj(j=2,3,6,8,9), diagonal matrices Pk>0(k=1,2,3,4), Ti=diag⁡(t1i,t2i,…,tni)>0(i=1,2) and any matrices S1,X=X11X12X21X22,Y=Y11Y12Y21Y22 with appropriate dimension, any constants ρj≥0(j=1,2,3,4), such that R3S1∗R9≥0,Q^8=Q~8X∗Q~8≥0, R^8=R~8Y∗R~8≥0 and the following LMIs (69) hold: (69)Ω~j<0,j=1,2,3,4.Ω~1, Ω~2, Ω~3, and Ω~4 are the particular matrices of matrix Φ~ when α=0, β=0; α=0, β=1; α=1, β=0 and α=1, β=1, respectively.(70)Φ~=∑i=17Φ~iΦ~1=e1-π22P1BL-2P1Ae1T+e4-π22P2GL-2P2Ce4T+e1P1We8T-e1P1Eae15T+e1P1Ewe16T+e2DP2e4T-e4P2Ece17T+e4P2Ede18T+e1P1We8T-e1P1Eae15T+e1P1Ewe16T+e2DP2e4T-e4P2Ece17T+e4P2Ede18TTΦ~2=e1Q2e1T-e3Q2e3T+e2R2e4T-e6R2e6TΦ~3=e1Q3e1T+e2-1-τ2dQ3e2T+e4e7R3S1∗R9e4e7T-1-σ2de5e8R3S1∗R9e5e8TΦ~4=e1τ12Q6e1T+e4σ12R6e4T-ατ12e12Q6e12T-1-ατ12e11Q6e11T-βσ12e14R6e14T-1-βσ12e13R6e13TΦ~5=e9τ122Q8e9T+e10σ122R8e10T-Γ~1Γ~2Q^8Γ~1Γ~2T-Γ~3Γ~4R^8Γ~3Γ~4TΦ~6=-e92P3e9T-e102P4e10T-e1AP3e9T+e8WTP3e9T-e9P3Eae15T+e9P3Ewe16T-e4CP4e10T+e2DTP4e10T-e10P4Ece17T+e10P4Ede18T+-e1AP3e9T+e8WTP3e9T-e9P3Eae15T+e9P3Ewe16T-e4CP4e10T+e2DTP4e10T-e10P4Ece17T+e10P4Ede18TTΦ~7=-2e7T1e7T+e4T1L-+L+e7T+e7L-+L+T1e4T+e4-L-T1L+-L+T1L-e4T-2e8T2e8T+e5T2L-+L+e8T+e8L-+L+T2e5T+e5-L-T2L+-L+T2L-e5T+e1ρ1HaTHae1T+e2ρ4HdTHde2T+e4ρ3HcTHce4T+e8ρ2HwTHwe8T-ρ1e15e15T-ρ2e16e16T-ρ3e17e17T-ρ4e18e18T-e1γP1e1T-e4γP2e4TQ~8=diag⁡Q8,3Q8,R~8=diag⁡R8,3R8,Γ~1=e2-e3e2+e3-2e12,Γ~2=e1-e2e1+e2-2e11,Γ~3=e5-e6e5+e6-2e14,Γ~4=e4-e5e4+e5-2e13.τ12=τ2-τ1,σ12=σ2-σ1,α=τ2-τtτ12,β=σ2-σtσ12.BL=diag⁡∑k=1mb1klk2,∑k=1mb2klk2,…,∑k=1mbnklk2,GL=diag⁡∑k=1mg1klk2,∑k=1mg2klk2,…,∑k=1mgnklk2.ei(i=1,2,…,18) are elementary matrices; for example, e3T=[00I015].Furthermore, we consider the GRNs (8) without uncertain parameters; that is,(71)∂xt,z∂t=∑k=1m∂∂zkBk∂xt,z∂zk-Axt,z+Wfyt-σt,z∂yt,z∂t=∑k=1m∂∂zkGk∂yt,z∂zk-Cyt,z+Dxt-τt,zxs,z=φs,z,ys,z=ϕs,z,∀s∈-ω,0The following corollary can be used to check the stability of GRNs (71).Corollary 16.
For given constantsτ2, τ2d, σ2, σ2d and positive constant γ>0, systems (71) are finite-time asymptotic stable if there exist positive definite matrices Qi(i=2,3,6,8), Rj(j=2,3,6,8,9), diagonal matrices Pk>0(k=1,2,3,4), Ti=diag⁡(t1i,t2i,…,tni)>0(i=1,2) and any matrices S1,X=X11X12X21X22, Y=Y11Y12Y21Y22 with appropriate dimension, such that R3S1∗R9≥0, Q^8=Q~8X∗Q~8≥0, R^8=R~8Y∗R~8≥0 and the following LMIs (72) hold: (72)Ωˇj<0,j=1,2,3,4.Ωˇ1, Ωˇ2, Ωˇ3, and Ωˇ4 are the particular matrices of matrix Φˇ when α=0, β=0; α=0, β=1; α=1, β=0 and α=1, β=1, respectively. (73)Φˇ=Φˇ1+∑i=25Φ~i+Φˇ6+Φˇ7Φˇ1=e1-π22P1BL-2P1Ae1T+e4-π22P2GL-2P2Ce4T+e1P1We8T+e2DP2e4T+e1P1We8T+e2DP2e4TTΦˇ6=-e92P3e9T-e102P4e10T-e1AP3e9T+e8WTP3e9T-e4CP4e10T+e2DTP4e10T+-e1AP3e9T+e8WTP3e9T-e4CP4e10T+e2DTP4e10TTΦˇ7=-2e7T1e7T+e4T1L-+L+e7T+e7L-+L+T1e4T+e4-L-T1L+-L+T1L-e4T-2e8T2e8T+e5T2L-+L+e8T+e8L-+L+T2e5T+e5-L-T2L+-L+T2L-e5T-e1γP1e1T-e4γP2e4T
e i   ( i = 1,2 , … , 14 ) are elementary matrices with appropriate dimension, and the expressions of Φ~i in Φˇ are equal to the corresponding terms in Corollary 15.Remark 17.
Theorem11 and Corollary 15 are obtained based on the Dirichlet boundary conditions. When the boundary conditions are Neumann boundary conditions or Robin boundary conditions, we can also give the similar finite-time stability criteria by employing the method proposed in [28, 30, 31].When there are no reaction-diffusion terms, the GRNs (8) can be simplified as follows:(74)x˙t,z=-A+ΔAtxt,z+W+ΔWtfyt-σt,zy˙t,z=-C+ΔCtyt,z+D+ΔDtxt-τt,zxs,z=φs,z,ys,z=ϕs,z,∀s∈-ω,0The following corollary can be used to check the stability of GRNs (74).Corollary 18.
For given constantsτ1, τ2, τ1d, τ2d, σ1, σ2, σ1d, σ2d and positive constant γ>0, systems (74) are finite-time robust stable if there exist positive definite matrices P1,P2,Qi(i=1,2,…,8),Rj(j=1,2,…,9), diagonal matrices Ti=diag⁡(t1i,t2i,…,tni)>0(i=1,2), and any matrices S1,X=X11X12X21X22,Y=Y11Y12Y21Y22 with appropriate dimension, any constants ρj≥0(j=1,2,3,4), such that R3S1∗R9≥0,Q^8=Q~8X∗Q~8≥0,R^8=R~8Y∗R~8≥0, and the following LMIs (75) hold: (75)Ω^j<0,j=1,2,3,4.Ω^1, Ω^2, Ω^3, and Ω^4 are the particular matrices of matrix Φ^ when α=0,β=0;α=0,β=1;α=1,β=0, and α=1,β=1, respectively.(76)Φ^=Φ^1+∑i=29Φi<0Φ^1=e1-P1A-AP1e1T+e5-P2C-CP2e5T+e1P1We10T-e1P1Eae19T+e1P1Ewe20T+e3DP2e5T-e5P2Ece21T+e5P2Ede22T+e1P1We10T-e1P1Eae19T+e1P1Ewe20T+e3DP2e5T-e5P2Ece21T+e5P2Ede22TTThe other elements in Φ^ are equal to the corresponding terms in Theorem 11.


## 4. Numerical Simulations

In this section, the following numerical simulation example is given to demonstrate the effectiveness and correctness of the stability criteria obtained above.Example 1.
Consider the uncertain reaction-diffusion GRNs (8) with Dirichlet boundary conditions and the following parameters [28]:(77)A=diag⁡1.2,1.8,2.5,0.9,C=diag⁡0.8,1.7,1.4,1.1,D=diag⁡1.6,0.5,1.9,1.0,W=0-101100001010-100B1=B2=B3=diag⁡0.1,0.1,0.1,0.1,G1=G2=G3=diag⁡0.2,0.2,0.2,0.2,l1=l2=l3=1,Ea=Ew=Ec=Ed=diag⁡0.2,0.2,0.2,0.2,Ha=diag⁡0.3,0.2,0.1,0.5,Hw=diag⁡0.1,0.2,0.2,0.3,Hc=diag⁡0.6,0.2,0.1,0.5,Hd=diag⁡0.1,0.4,0.7,0.3,fiyi=yi2yi2+1,L-=0,L+=diag⁡0.65,0.65,0.65,0.65,τ1=σ1=0,τ2=1.5,σ2=1.1.In [28], the derivatives of time delays must be less than one. However, such strict constraint is removed in this paper. For example, when τ2d=1.5, σ2d=1.5, we can obtain the following feasible solutions for LMIs (69) in Corollary 15 (to save space, we only list parts of our feasible solutions):(78)Q2=0.02180.02270.0177-0.00690.02270.02420.0149-0.00740.01770.01490.0383-0.0076-0.0069-0.0074-0.00760.0261,Q6=0.0003-0.00110.0003-0.0005-0.00110.0065-0.00210.00290.0003-0.00210.0008-0.0010-0.00050.0029-0.00100.0026,R2=0.0028-0.00010.0004-0.0007-0.00010.0540-0.0007-0.00010.0004-0.00070.0058-0.0010-0.0007-0.0001-0.00100.0307,R6=0.00040.00020.0002-0.00020.00020.0034-0.00020.00020.0002-0.00020.0019-0.0004-0.00020.0002-0.00040.0017,P1=0.021800000.022700000.024200000.0177,S1=-0.0053-0.0001-0.00010.0000-0.0004-0.00080.0000-0.0001-0.00010.0000-0.00760.00000.0002-0.00000.0001-0.0006,X11=-0.0075-0.0003-0.0025-0.0006-0.00870.0027-0.0008-0.00060.00050.0058-0.0177-0.00170.00040.00050.0010-0.0070,Y11=-0.00240.0000-0.00010.00040.0012-0.0322-0.00040.0006-0.00020.0003-0.00310.0005-0.00040.0003-0.0005-0.0209,ρ1=0.0288,ρ2=0.0195,ρ3=0.0317,ρ4=0.0107.Figures1 and 2 show the trajectories of variables x(t) (mRNA concentrations) and y(t) (protein concentrations) for GRNs (8) with the above parameters and Dirichlet boundary conditions.Figure 1
The trajectories of mRNA concentrationsx(t,z) for the genetic regulatory networks (8).Figure 2
The trajectories of protein concentrationsy(t,z) for the genetic regulatory networks (8).


## 5. Conclusion

This study has investigated the finite-time robust stability problem of uncertain GRNs with interval time-varying delays and reaction-diffusion terms. The time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. Besides, the approach for conducting uncertain parameters is to define parts of them as some new variables. In addition, we employ an appropriate Lyapunov-Krasovskii functional to derive some less conservative stability criteria for GRNs under Dirichlet boundary conditions, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. Moreover, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed to handle with the integral terms in the derivative of Lyapunov-Krasovskii functional. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the obtained stability criteria. The foregoing results obtained in this study hold the potential to be extended to the other problems of delayed GRNs, including the decentralized event-triggered exponential stability, state estimation, sampled-data control, and so on. Meanwhile, the approaches can be used for the other time-delay systems, such as complex networks, neural networks, and multiagent systems.



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