# Anr-Order Finite-Time State Observer for Reaction-Diffusion Genetic Regulatory Networks with Time-Varying Delays

Authors: Xiaofei Fan; Yantao Wang; Ligang Wu; Xian Zhang
Journal: Complexity
 (2018)

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## Abstract

It will be settled out for the open problem of designing anr-order finite-time (F-T) state observer for reaction-diffusion genetic regulatory networks (RDGRNs) with time-varying delays. By assuming the Dirichlet boundary conditions, aiming to estimate the mRNA and protein concentrations via available network measurements. Firstly, sufficient F-T stability conditions for the filtering error system have been investigated via constructing an appropriate Lyapunov–Krasovskii functional (LKF) and using several integral inequalities and (reciprocally) convex technique simultaneously. These conditions are delay-dependent and reaction-diffusion-dependent and can be checked by MATLAB toolbox. Furthermore, a method is proposed to design an r-order F-T state observer, and the explicit expressions of observer gains are given. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Recently, due to a great many applications in the real world, genetic regulatory networks have become one of the hot topics in many fields. Much significant results (see [1–10] and the references therein) have been obtained. Usually, owing to the highly complexity of genetic regulatory networks, spatial homogeneity (i.e., the mRNA and protein concentrations are independent on their space positions) is assumed in the process of modelling genetic regulatory networks. However, this assumption is sometimes unreasonable, for example, the concentrations of proteins in ribosomal gathering are higher than ones of other parts of the cell in the process of translation. So, bringing the diffusing phenomenon into the models of genetic regulatory networks is urgent and necessary, which results in RDGRNs. Generally, the models of genetic regulatory networks are divided into discrete-time models and continuous-time ones [11]. A continuous-time model has wide applications in studying the complex features and the nonlinear behaviors of genetic regulatory networks. Moreover, due to the slow processes of transcription and translation, time delays should be considered in the continuous-time models of RDGRNs. It should be emphasized that time delays may lead to poor network performance, even instability. To the best our knowledge, those works in [12–17] have researched the problem of stability analysis of delayed RDGRNs. The asymptotic stability analysis of delayed RDGRNs have been involved in [13–15] by constructing an appropriate LKF and applying some inequality techniques. In [16], a sufficient condition of F-T stability for delayed RDGRNs has been given by constructing an LKF including quad-slope integrations and applying the Gronwall inequality and Wirtinger-type integral inequality. Related research on uncertain stochastic time-delay RDGRNs and impulsive stochastic time-delay RDGRNs can be found in [12, 17], respectively.Generally speaking, with the change of environment, not all mRNA and protein concentrations are measurable. So, it is important and necessary to estimate the mRNA and protein concentrations. Currently, one of effective approaches to estimate system states is to design observers based on the available measurement. As we all know, the problem of estimating the states of delayed RDGRNs is only studied in [18, 19], although some scholars have addressed the diffusion-free case (see [20–22] and the references therein). A full-order observer based on available measurement has been designed in [18] by introducing an LKF and employing several integral inequalities, convex approach and Green’s second identity. The existence condition and design method of a full-order F-T observer have been given in [19]. It is worth emphasizing that all these results are about the full-order observers, and all approaches proposed in these literatures are not available for designing reduced-order observers. But, the design of a reduced-order observer is necessary, since it can be more easily realized than the full-order one in engineering practise.The above discussion motivates us to design anr-order (i.e., reduced-order) F-T state observer for delayed RDGRNs. By constructing a novel LKF and employing several integral inequalities and (reciprocally) convex technique to estimate its derivative, a F-T stability criterion in the form of linear matrix inequalities (LMIs) is established for the resulting error system. In addition, we propose a method for designing an r-order F-T state observer for RDGRNs with time-varying delays, and the observer gains are parameterized by the solutions of these LMIs. Furthermore, the method proposed in this paper is explained by a numerical example.It is worth emphasizing that the method proposed in this paper has the following advantages:(i)
Ther-order observer is delay-dependent and reaction-diffusion-dependent, which is more practical.
(ii)
Compared with the full-order observers, the designedr-order one can save the cost in the engineering.
(iii)
For delayed genetic regulatory networks without reaction-diffusion items, the method is still keeping available by removing the corresponding parts ofV1.
(iv)
The method can also extend to some other time-delay models, including Markov jump neural networks [23–25] and stochastic delayed systems [26].Notation 1.
For givenn×n matrices X and Y, we say X>Y and X≥Y, if X−Y is real symmetric positive definite and semidefinite, respectively. The n×n identity matrix is defined by In, and the m×n zero matrix by 0m×n. AT and SymA stand for the transpose matrix of A and the sum of A and its transpose, respectively. For an arbitrary but fixed positive integer l, we denote by l the set 1,2,…,l. Let Ω=x∈ℝl:xk≤Lk,k∈l with Li>0, i=1,2,…,l. The set of all functions f:X→ℝn having the continuous second derivatives is defined by C2X,ℝn. · and ·d represent the norms on C2−d,0×Ω,ℝn and are defined by
(1)yt,x=∫ΩyTt,xyt,xdx1/2and
(2)ht,xd=maxsup−d≤t≤0ht,x,sup−d≤t≤0∂ht,x∂t,max1≤k≤nsup−d≤t≤0∂ht,x∂xk.
The symbolcolA1,…,Am refers to A1T⋯AmTT.


## 2. Problem Formulation

Consider the following delayed RDGRN [15]:
(3)∂m~t,x∂t=∑k=1lDk∂2m~t,x∂xk2−Am~t,x+Wgp~t−κt,x+q,∂p~t,x∂t=∑k=1lDk∗∂2p~t,x∂xk2−Cp~t,x+Bm~t−ρt,x,where
(4)A=diaga1,a2,…,an,B=diagb1,b2,…,bn,C=diagc1,c2,…,cn,W=wij∈ℝn×n,q=colq1,q2,…,qn,m~t,x=colm~1t,x,…,m~nt,x,gp~t,x=colg1p~1t,x,g2p~2t,x,…,gnp~nt,x,p~t,x=colp~1t,x,p~2t,x,…,p~nt,x,x=colx1,x2,…,xl∈Ω⊂ℝl, m~it,x, and p~it,x stand for the concentrations of mRNAs and proteins, respectively; ai, ci, and bi are the rate constants; Dk>0 and Dk∗>0 denote the diagonal diffusion rate matrices; W represents the coupling matrix with elements defined as in [15]; gj:s→sH/1+sH is the Hill function, qi is the sum of dimensionless transcriptional rates which repress gene i, κt and ρt are delays subject to
(5)0≤ρt≤ρ¯,ρ̇t≤μρ,0≤κt≤κ¯,κ̇t≤μκ,where ρ¯, κ¯, μρ, and μκ are nonnegative real numbers.The expression ofgi indicates that
(6)0≤giy−gizy−z≤ξi,y,z∈ℝ,y≠zfor some common scalar ξi>0.Assume thatm∗x,p∗x is the unique equilibrium solution of (3). Set
(7)mt,x,pt,x=m~t,x−m∗x,p~t,x−p∗x.Then the delayed RDGRN (3) turns into
(8)∂mt,x∂t=∑k=1lDk∂2mt,x∂xk2−Amt,x+Wfpt−κt,x,∂pt,x∂t=∑k=1lDk∗∂2pt,x∂xk2−Cpt,x+Bmt−ρt,x,where
(9)fps,x=colf1p1s,x,…,fnpns,x,fipis,x=gipis,x+pi∗−gipi∗,i∈n.Next, it is assumed that the initial conditions and the Dirichlet boundary conditions of (8) are as follows:
(10)mt,x=ψt,x,pt,x=ψ∗t,x,x∈Ω,t∈−d,0,mt,x=0,pt,x=0,x∈∂Ω,t∈−d,+∞,where d=maxκ¯,ρ¯, and ψt,x and ψ∗t,x are functions in C2−d,0×Ω,ℝn. Furthermore, let the network outputs be
(11)zpt,x=Nppt,x,zmt,x=Nmmt,xwith the full-row-rank constant matrices Nm and Np. Here, zmt,x and zpt,x represent the expression levels of mRNAs and proteins at time t, respectively.This paper aims at designing anr-order F-T state observer for the delayed RDGRN (8), which is described as:
(12)∂m^t,x∂t=A^m^t,x+M^mzmt,x,∂p^t,x∂t=−C^p^t,x+M^pzpt,x,m^t,x=ψ^t,x,p^t,x=ψ^∗t,x,x∈Ω,t∈−d,0,m^t,x=0,p^t,x=0,x∈∂Ω,t∈−d,+∞.Here,m^t,x and p^t,x are the r-order observer states, and A^, C^, M^m, and M^p are the observer gains.Remark 1.
Clearly, whenr=n the observer (12) is of full-order. So, our method is also available to establish full-order observers for the delayed RDGRN (8).Define the augmented vectors(13)emt,x=mt,xm^t,x,ept,x=pt,xp^t,x.According to (8), (11), and (12), one can obtain the resulting error system as follows:
(14)∂emt,x∂t=∑k=1lD¯k∂2emt,x∂xk2−A¯emt,x+W¯fpt−κt,x,x∈Ω,t∈−d,+∞,∂ept,x∂t=∑k=1lD¯k∗∂2ept,x∂xk2−C¯ept,x+B¯mt−ρt,x,x∈Ω,t∈−d,+∞,emt,x=ψ¯t,x,ept,x=ψ¯∗t,x,x∈Ω,t∈−d,0,emt,x=0,ept,x=0,x∈∂Ω,t∈−d,+∞,where
(15)D¯k=diagDk,0,W¯=colW,0,D¯k∗=diagDk∗,0,B¯=colB,0,A¯=A0−M^mNm−A^,C¯=C0−M^pNp−C^,ψ¯t,x=colψt,x,ψ^t,x,ψ¯∗t,x=colψ∗t,x,ψ^∗t,x.Definition 1 (see [12]).
The trivial solution of system (14) is called F-T stable with respect to positive scalars c1, c2, and T, if
(16)ψ¯t,xd2+ψ¯∗t,xd2≤c1⇒emt,x2+ept,x2≤c2,∀t∈0,T.
We say that system (14) is F-T stable with respect to positive scalars c1, c2, and T, if so, it is its trivial solution.
To achieve our aim, one requires to seekr-order observer gains A^, C^, M^m, and M^p such that system (14) is F-T stable with respect to positive scalars c1, c2, and T.


## 3. Preliminaries

The following two lemmas are needed to design anr-order F-T state observer.Lemma 1 (Jensen’s inequality) [27].
For given scalarsa<b, an integral function χ:a,b→ℝn and a matrix MT=M>0, there holds the following inequality:
(17)b3−a36∫ba∫θ0∫λ0χTsMχsdsdλdθ≥∫ba∫θ0∫λ0χTsdsdλdθM∫ba∫θ0∫λ0χsdsdλdθ.Lemma 2 (Wirtinger-type integral inequalities) [28].
For given scalarsa<b, a function χ:a,b→ℝn which is derivative and a matrix MT=M>0, there hold the following inequalities:
(18)b−a∫abχTsMχsds≥Θ5TΘ6TM¯Θ5TΘ6TT,∫ab∫bbχ̇TsMχ̇sdsdα≥Θ3TΘ4TM~Θ3TΘ4TT,b−a∫abχ̇TsMχ̇sds≥Θ0TΘ1TΘ2TM^Θ0TΘ1TΘ2TT,where
(19)M¯=diagM,3M,M~=diag2M,4M,M^=diagM,3M,5M,Θ0=χb−χa,Θ1=χb+χa−2b−a−1∫abχsds,Θ2=Θ0+6b−a∫abχsds−12b−a2∫ab∫abχsdsdα,Θ3=χb−b−a−1∫abχsds,Θ5=∫abχsds,Θ4=χb+2b−a∫abχsds−6b−a2∫ab∫abχsdsdα,Θ6=∫abχsds−2b−a−1∫ab∫abχsdsdα.


## 4. Design Method of Observer

In this section, a method to design anr-order F-T state observer for the delayed RDGRN (8) is proposed, that is, determine the observer gains A^,C^,M^m, and M^p such that the error system (14) is F-T stable. For this end, we define
(20)E1=colIn+r,017n+3r×n+r,E2=col0n+r×n,In,016n+3r×n,E3=col02n+r×n,In,015n+3r×n,E4=col03n+r×n+r,In+r,014n+2r×n+r,E4+i=col0in+3n+2r×n,In,014n−in+2r×n,i∈4,E9=col08n+2r×n+r,In+r,09n+r×n+r,E10=col09n+3r×n+r,In+r,08n×n+r,E10+i=col09n+in+4r×n,In,08−in×n,i∈8,Π1=E3−E2E3+E2−2E11E3−E2+6E11−12E12,Π2=E1K¯T−E3E1K¯T+E3−2E13E1K¯T−E3+6E13−12E14,Π3=E6−E5E6+E5−2E15E6−E5+6E15−12E16,Π4=E4K¯T−E6E4K¯T+E6−2E17E4K¯T−E6+6E17−12E18,Π5=E11E11−2E12,Π6=E13E13−2E14,Π7=E15E15−2E16,Π8=E17E17−2E18,Υ1=E1K¯T−E13E1K¯T+2E13−6E14,Υ2=E3−E11E3+2E11−6E12,Υ3=E4K¯T−E17E4K¯T+2E17−6E18,Υ4=E6−E15E6+2E15−6E16,Ψ1=Ψ11+Ψ12,Ψ11=Sym−π24E1U1DLE1T−E9U1E9T−π24E4U2DL∗E4T−E10U2E10T+E9U1W¯E8T+E4U2B¯E3T+E10U2B¯E3T+E1U1W¯E8T,Ψ12=Sym−E1U11A0−MmNm−AmE1T−E4U21C0−MpNp−CpE4T−E1ATU11−NmTMmT0−AmTE9T−E4CTU21−NpTMpT0−CpTE10T,Ψ2=E1K¯TV1+V2K¯E1T−E2V2E2T+μρ−1E3V1E3T+E4K¯TV3+V4K¯E4T−E5V4E5T+μκ−1E6V3E6T,Ψ3=μκ−1E8V5E8T+E7V5E7T,Ψ4ρ,κ=Ψ40+Ψ41+Ψ42+Ψ43ρ+Ψ44κ,Ψ40=ρ¯2E9K¯TW1K¯E9T+E1K¯TW3K¯E1T+κ¯2E10K¯TW2K¯E10T+E4K¯TW4K¯E4T,Ψ41=−Π1Π2W^1H^1H^1TW^1Π1Π2T,Ψ42=−Π3Π4W^2H^2H^2TW^2Π3Π4T,Ψ43ρ=ρ¯ρ¯−ρΠ5W~3Π5T−ρ¯ρΠ6W~3Π6T,Ψ44κ=−κ¯κ¯−κΠ7W~4Π7T−κ¯κΠ8W~4Π8T,Ψ5ρ,κ=Ψ50+Ψ51ρ+Ψ52κ,Ψ50=ρ¯22E9K¯TX1K¯E9T+κ¯22E10K¯TX2K¯E10T,Ψ51ρ=−Υ1X~1Υ1T−Υ2X~1Υ2T−ρ¯−ρρΠ2X^1Π2T,Ψ52κ=−Υ3X~2Υ3T−Υ4X~2Υ4T−κ¯−κκΠ4X^2Π4T,Ψ6ρ,κ=Ψ60+Ψ61ρ+Ψ62κ−ρ¯−ρΥ1Y~1Υ1T−κ¯−κΥ3Y~2Υ3T,Ψ60=ρ¯36E9K¯TY1K¯E9T+κ¯36E10K¯TY2K¯E10T,Ψ61ρ=−32ρE1K¯T−2E14Y1E1K¯T−2E14T−32ρ¯−ρE3−2E12Y1E3−2E12T,Ψ62κ=−32κE4K¯T−2E18Y2E4K¯T−2E18T−32κ¯−κE6−2E16Y2E6−2E16T,Ψ01=−2E7Z1E7T+E4K¯TKZ1E7T+E7Z1KK¯E4T,Ψ02=−2E8Z2E8T+E6KZ2E8T+E8Z2KE6T,X~i=diag2Xi,4Xi,W^i=diagWi,3Wi,5Wi,X^i=diagXi,3Xi,5Xi,Y~i=diag2Yi,4Yii=1,2,W~j=diagWj,3Wj,j=3,4,U=diagU1,U2,Uk=diagUk1,Uk2,k=1,2,λ11=λmaxU1+ρ¯λmaxV1+ρ¯λmaxV2+16ρ¯3maxX1+∑k=1lλmaxU1λmaxDk¯+12ρ¯3maxW1+W3+124ρ¯4λmaxY1,λ12=λmaxU2+κ¯λmaxV3+κ¯λmaxV4+κ¯λmaxV5λmaxKTK+16κ¯3λmaxX2+∑k=1lλmaxU2λmaxDk∗+12κ¯3λmaxW2+W4+124κ¯4λmaxY2,ηt,x=colemt,x,mt−ρ¯,x,mt−ρt,x,ept,x,pt−κ¯,x,pt−κt,x,fpt,x,fpt−κt,x,∂emt,x∂t,∂ept,x∂t,1ρ¯−ρt∫t−ρ¯t−ρtms,xds,1ρ¯−ρt2∫t−ρ¯t−ρt∫αt−ρtms,xdsdα,1ρt∫t−ρttms,xds,1ρ2t∫t−ρtt∫αtms,xdsdα,1κ¯−κt∫t−κt−κtps,xds,1κ¯−κt2∫t−κ¯t−κt∫αt−κtps,xdsdα,1κt∫t−κttps,xds,1κ2t∫t−κtt∫αtps,xdsdα,DL=∑k=1lD¯kLk2,DL∗=∑k=1lD¯k∗Lk2,K¯=I0,where Lk,D¯k, and D¯k∗ are the same with previous ones.Now we can provide an approach to design anr-order F-T state observer for the delayed RDGRN (8).Theorem 1.
For given scalarsρ¯,κ¯,μρ, and μκ satisfying (5) and positive constants T,c1,c2, and α, system (14) is F-T stable with respect to c1,c2, and T, if there exist matrices 0<ViT=Vi∈ℝn×ni∈5,0<WjT=Wj∈ℝn×nj∈4,0<XkT=Xk∈ℝn×n, and 0<YkT=Yk∈ℝn×n, diagonal matrices 0<Uk∈ℝn+r×n+r and 0<Zk∈ℝn×n, and matrices H^k∈ℝ3n×3nk∈2,Am∈ℝr×r,Cp∈ℝr×r,Mm, and Mp of appropriate dimensions, such that the following inequalities are feasible for ρ∈0,ρ¯ and κ∈{0,κ¯}:
(21)W^kH^kH^kTW^k≥0,k∈2,(22)Ψρ,κ≔∑i=13Ψi+∑i=46Ψiρ,κ+∑i=12Ψ0i−αE1U1E1T−αE4U2E4T<0,(23)c1eαTλ11+λ12−c2λminU≤0,where K=diagξ1,ξ2,…,ξn>0, and A¯, B¯, C¯, and W¯ are defined previously.In addition, based on a feasible solution of (21), (22), and (23), an r-order F-T state observer can be represented by (12) with the following gains:
(24)A^M^mC^M^p=diagU12,U22−1AmMmCpMp.Proof 1.
Choose the following LKF functional:(25)Vt,em,ep=∑i=16Vit,em,ep,where
(26)V1t,em,ep=∫ΩemTt,xU1emt,xdx+∫ΩepTt,xU2ept,xdx+∑k=1l∫Ω∂emTt,x∂xkU1D¯k∂emt,x∂xkdx+∑k=1l∫Ω∂epTt,x∂xkU2D¯k∗∂ept,x∂xkdx,V2t,em,ep=∫Ω∫t−ρttemTs,xK¯TV1K¯ems,xdsdx+∫Ω∫t−ρ¯temTs,xK¯TV2K¯ems,xdsdx+∫Ω∫t−κttepTs,xK¯TV3K¯eps,xdsdx+∫Ω∫t−κ¯tepTs,xK¯TV4K¯eps,xdsdx,V3t,em,ep=∫Ω∫t−κttfTps,xV5fps,xdsdx,V4t,em,ep=ρ¯∫Ω∫−ρ¯0∫t+θt∂emTs,x∂sK¯TW1K¯∂ems,x∂sdsdθdx+κ¯∫Ω∫−κ¯0∫t+θt∂epTs,x∂sK¯TW2K¯∂eps,x∂sdsdθdx+ρ¯∫Ω∫−ρ¯0∫t+θtemTs,xK¯TW3K¯ems,xdsdθdx+κ¯∫Ω∫−κ¯0∫t+θtepTs,xK¯TW4K¯eps,xdsdθdx,V5t,em,ep=∫Ω∫−ρ¯0∫s0∫t+θt∂emTu,x∂uK¯TX1K¯∂emu,x∂ududθdsdx+∫Ω∫−κ¯0∫s0∫t+θt∂epTu,x∂uK¯TX2K¯∂epu,x∂ududθdsdx,V6t,em,ep=∫Ω∫−ρ¯0∫s0∫α0∫t+θt∂emTu,x∂uK¯TY1K¯∂emu,x∂ududθdαdsdx+∫Ω∫−κ¯0∫s0∫α0∫t+θt∂epTu,x∂uK¯TY2K¯∂epu,x∂ududθdαdsdx.Then, calculating the derivatives ofVit,em,ep i∈6 along on the solution of system (14), one can obtain that
(27)∂∂tV1t,em,ep=2∫ΩemTt,xU1−A¯emt,x+W¯fpt−κt,x+∑k=1lD¯k∂2emt,x∂xk2dx+2∫ΩepTt,xU2−C¯ept,x+B¯mt−ρt,x+∑k=1lDk∗∂2ept,x∂xk2dx+2∑k=1l∫Ω∂emTt,x∂xkU1D¯k∂∂xk∂emt,x∂tdx+2∑k=1l∫Ω∂epTt,x∂xkU2D¯k∗∂∂xk∂ept,x∂tdx,(28)∂∂tV2t,em,ep=∫ΩemTt,xK¯TV1+V2K¯emt,xdx−1−ρ̇t∫ΩmTt−ρt,xV1mt−ρt,xdx−∫ΩmTt−ρ¯,xV2mt−ρ¯,xdx+∫ΩepTt,xK¯TV3+V4K¯ept,xdx−1−κ̇t∫ΩpTt−κt,xV3pt−κt,xdx−∫ΩpTt−κ¯,xV4pt−κ¯,xdx≤∫ΩηTt,xΨ2ηt,xdx,(29)∂∂tV3t,em,ep=∫ΩfTpt,xV5fpt,xdx−1−κ̇t∫ΩfTpt−κt,xV5fpt−κt,xdx≤∫ΩηTt,xΨ3ηt,xdx,(30)∂∂tV4t,em,ep=−ρ¯∫Ω∫t−ρ¯t∂emTs,x∂sK¯TW1K¯∂ems,x∂sdsdx+ρ¯2∫Ω∂emTt,x∂tK¯TW1K¯∂emt,x∂tdx−κ¯∫Ω∫t−κ¯t∂epTs,x∂sK¯TW2K¯∂eps,x∂sdsdx+κ¯2∫Ω∂epTt,x∂tK¯TW2K¯∂ept,x∂tdx−ρ¯∫Ω∫t−ρ¯temTs,xK¯TW3K¯ems,xdsdx+ρ¯2∫ΩemTt,xK¯TW3K¯emt,xdx−κ¯∫Ω∫t−κ¯tepTs,xK¯TW4K¯eps,xdsdx+κ¯2∫ΩepTt,xK¯TW4K¯ept,xdx,(31)∂∂tV5t,em,ep=ρ¯22∫Ω∂emTt,x∂tK¯TX1K¯∂emt,x∂tdx−∫Ω∫−ρ¯0∫t+st∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx+κ¯22∫Ω∂epTt,x∂tK¯TX2K¯∂ept,x∂tdx−∫Ω∫−κ¯0∫t+st∂epTu,x∂uK¯TX2K¯∂epu,x∂ududsdx,(32)∂∂tV6t,em,ep=ρ¯36∫Ω∂emTt,x∂tK¯TY1K¯∂emt,x∂tdx−∫Ω∫−ρ¯0∫α0∫t+st∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdαdx+κ¯36∫Ω∂epTt,x∂tK¯TY2K¯∂ept,x∂tdx−∫Ω∫−κ¯0∫α0∫t+st∂epTu,x∂uK¯TY2K¯∂epu,x∂ududsdαdx.Firstly, it follows from Green formula that(33)2∫ΩemTt,xU1∑k=1lD¯k∂2emt,x∂xk2dx=2∑k=1l∫Ω∂∂xkemTt,xU1D¯k∂emt,x∂xkdx−2∑k=1l∫Ω∂emTt,x∂xkU1D¯k∂emt,x∂xkdx.Using Dirichlet boundary conditions, one can derive that(34)2∫ΩemTt,xU1∑k=1lD¯k∂2emt,x∂xk2dx=−2∑k=1l∫Ω∂emTt,x∂xkU1D¯k∂emt,x∂xkdx.This, together with the so-called Wirtinger’s inequality [29], implies that
(35)2∫ΩemTt,xU1∑k=1lD¯k∂2emt,x∂xk2dx≤−π22∫ΩemTt,xU1DLemt,xdx.In a similar way,(36)2∫ΩepTt,xU2∑k=1lD¯k∗∂2ept,x∂xk2dx≤−π22∫ΩepTt,xU2DL∗ept,xdx.From (14) we get
(37)2∫Ω∂emTt,x∂tU1−∂emt,x∂t+∑k=1lD¯k∂2emt,x∂xk2−A¯emt,x+W¯fpt−κt,xdx=0and
(38)2∫Ω∂epTt,x∂tU2−∂ept,x∂t+∑k=1lD¯k∗∂2ept,x∂xk2−C¯ept,x+B¯mt−ρt,xdx=0.By means of Dirichlet boundary conditions, Green formula and [15], Lemma 4, it yields that
(39)2∫Ω∂emTt,x∂tU1∑k=1lD¯k∂2emt,x∂xk2dx=2∫ΩemTt,xU1∑k=1l∂∂xkDk∂∂xk∂emt,x∂tdx=−2∑k=1l∫Ω∂emTt,x∂xkU1D¯k∂∂xk∂emTt,x∂tdx.Similarly,(40)2∫Ω∂epTt,x∂tU2∑k=1lD¯k∗∂2ept,x∂xk2dx=−2∑k=1l∫Ω∂epTt,x∂xkU2D¯k∗∂∂xk∂ept,x∂tdx.Combining (27) and (35), (36), (37), (38), (39), and (40), we get
(41)∂∂tV1t,em,ep≤2∫ΩemTt,xU1−A¯emt,x−π24DLemt,x+W¯fpt−κt,xdx+2∫ΩepTt,xU2−C¯ept,x−π24DL∗ept,x+B¯mt−ρt,xdx+2∫Ω∂emTt,x∂tU1−∂emt,x∂t−A¯emt,x+W¯fpt−κt,xdx+2∫Ω∂epTt,x∂tU2−∂ept,x∂t−C¯ept,x+B¯mt−ρt,xdx=∫ΩηTt,xΨ1ηt,xdx.Secondly, in view of (21), the reciprocally convex technique [30] and Lemma 2, it follows that
(42)−ρ¯∫Ω∫t−ρ¯t∂emTs,x∂sK¯TW1K¯∂ems,x∂sdsdx=−ρ¯∫Ω∫t−ρtt∂emTs,x∂sK¯TW1K¯∂ems,x∂sdsdx−ρ¯∫Ω∫t−ρ¯t−ρt∂emTs,x∂sK¯TW1K¯∂ems,x∂sdsdx≤−∫ΩηTt,xΨ41ηt,xdx.Similarly,(43)−κ¯∫Ω∫t−κ¯t∂epTs,x∂sK¯TW2K¯∂eps,x∂sdsdx≤−∫ΩηTt,xΨ42ηt,xdx.Again using Lemma2, one can obtain
(44)−ρ¯∫Ω∫t−ρ¯temTs,xK¯TW3K¯ems,xdsdx=−ρ¯∫Ω∫t−ρttemTs,xK¯TW3K¯ems,xdsdx−ρ¯∫Ω∫t−ρ¯t−ρtemTs,xK¯TW3K¯ems,xdsdx≤−ρ¯ρt∫ΩηTt,xΠ6W~3Π6Tηt,xdx−ρ¯ρ¯−ρt∫ΩηTt,xΠ5W~3Π5Tηt,xdx=∫ΩηTt,xΨ43ρtηt,xdxand
(45)−κ¯∫Ω∫t−ρ¯tepTs,xK¯TW4K¯eps,xdsdx≤−κ¯κt∫ΩηTt,xΠ8W~4Π8Tηt,xdx−κ¯κ¯−κt∫ΩηTt,xΠ7W~4Π7Tηt,xdx=∫ΩηTt,xΨ44κtηt,xdx.The combination of (30) and (42), (43), (44), and (45) gives
(46)∂∂tV4t,em,ep≤∫ΩηTt,xΨ4ρt,κtηt,xdx.Thirdly, it is clear that(47)−∫Ω∫−ρ¯0∫t+st∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx=−∫Ω∫−ρ¯−ρt∫t+st−ρt∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx−∫Ω∫−ρt0∫t+st∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx−ρ¯−ρt∫Ω∫t−ρtt∂emTu,x∂uK¯TX1K¯∂emu,x∂ududx.It follows from Lemma2 that
(48)−∫Ω∫−ρt0∫t+st∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx≤−∫ΩηTt,xΥ1X~1Υ1Tηt,xdx,−∫Ω∫−ρ¯−ρt∫t+st−ρt∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx≤−∫ΩηTt,xΥ2X~1Υ2Tηt,xdxand
(49)−ρ¯−ρt∫Ω∫t−ρtt∂emTu,x∂uK¯TX1K¯∂emu,x∂ududx≤−∫ΩηTt,xρ¯−ρtρ¯Π2X^1Π2Tηt,xdx.By (47), it implies that
(50)−∫Ω∫−ρ¯0∫t+st∂emTu,x∂uK¯TX1K¯∂emu,x∂ududsdx≤∫ΩηTt,xΨ51ρtηt,xdx.Similarly,(51)−∫Ω∫−κ¯0∫t+st∂epTu,x∂uK¯TX2K¯∂epu,x∂ududsdx≤∫ΩηTt,xΨ52κtηt,xdx.Combining (31), (50), and (51), we have
(52)∂∂tV5t,em,ep≤∫ΩηTt,xΨ5ρt,κtηt,xdx.Fourthly, it is obvious that(53)−∫Ω∫−ρ¯0∫α0∫t+st∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdαdx≤−∫Ω∫−ρt0∫α0∫t+st∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdαdx−∫Ω∫−ρ¯−ρt∫α−ρt∫t+st−ρt∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdαdx−ρ¯−ρt∫Ω∫−ρt0∫t+st∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdx.By Lemmas1 and 2, one can obtain
(54)−∫Ω∫−ρ¯−ρt∫α−ρt∫t+st−ρt∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdαdx−∫Ω∫−ρt0∫α0∫t+st∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdαdx≤∫ΩηTt,xΨ61ρtηt,xdxand
(55)−ρ¯−ρt∫Ω∫−ρt0∫t+st∂emTu,x∂uK¯TY1K¯∂emu,x∂ududsdx≤−∫ΩηTt,xρ¯−ρtΥ1Y~1Υ1Tηt,xdx,respectively. Similarly,
(56)−∫Ω∫−κt0∫α0∫t+st∂epTu,x∂uK¯TY2K¯∂epu,x∂ududsdαdx−∫Ω∫−κ¯−κt∫α−κt∫t+st−κt∂epTu,x∂uK¯TY2K¯∂epu,x∂ududsdαdx≤∫ΩηTt,xΨ62κtηt,xdxand
(57)−κ¯−κt∫Ω∫−κt0∫t+st∂epTu,x∂uK¯TY2K¯∂epu,x∂ududsdx≤−∫ΩηTt,xκ¯−κtΥ3Y~2Υ3Tηt,xdx.By (32) and (53), (54), (55), (56), and (57), it is obtained that
(58)∂∂tV6t,em,ep≤∫ΩηTt,xΨ6ρt,κtηt,xdx.Finally, from (6) and the relationship between fi and gi, it is clear to say that f0=0 and
(59)fz−KzTZifz≤0,∀z∈ℝn,i=1,2,that is, for given diagonal matrices Z1>0 and Z2>0, we get
(60)ηTt,xΨ0iηt,x≥0,i=1,2.The combination of (22), (28), (29), (41), (46), (52), (39), and (60) yields
(61)∂∂tVt,em,ep=∑i=16∂∂tVit,em,ep≤α∫ΩemTt,xU1emt,x+epTt,xU2ept,xdx+∫ΩηTt,xΨρt,κtηt,xdx≤∫ΩηTt,xΨρt,κtηt,xdx+αVt,em,ep.SinceΨρt,κt depends affinely upon ρt and κt, one can derive from (22) that Ψρt,κt<0 for any 0≤κt≤κ¯ and 0≤ρt≤ρ¯. Utilizing (61), we can get
(62)∂∂tVt,em,ep≤αVt,em,ep.For anyt∈0,T, integrating the two sides of (62) from 0 to t, we derive
(63)Vt,em,ep≤V0,em0,x,ep0,x+∫0tαVs,em,epds.By means of the so-called Gronwall inequality, one can obtain(64)VT,em,ep≤eαTV0,em0,x,ep0,x.According to the expression ofVt,em,ep, it apparently will come
(65)V0,em0,x,ep0,x≤λ11+λ12ψ¯t,xd2+ψ¯∗t,xd2.Combination of (64) and (65) derives
(66)VT,em,ep≤eαTλ11+λ12ψ¯t,x+ψ¯∗t,xd2d2.In addition, whent∈0,T, one can easily derive that
(67)VT,em,ep≥λminUemt,x2+ept,x2,where λminU is the minimum eigenvalue of diagU1,U2. It can be seen from (66) and (67) that
(68)emt,x2+ept,x2≤eαTλ11+λ12ψ¯t,xd2+ψ¯∗t,xd2λminUBy Definition1 and (23), under Dirichlet boundary conditions, system (14) is F-T stable with respect to c1, c2, and T. The proof is completed.Following the proof of Theorem1, one can easily derive the following conclusion which gives a method to design an r-order observer for the delayed RDGRN (8).Corollary 1.
For given scalarsρ¯, κ¯, μρ, and μκ subject to (5), system (14) is asymptotically stable under Dirichlet boundary conditions, if there are matrices 0<ViT=Vi∈ℝn×n i∈5, 0<WjT=Wj∈ℝn×n j∈4, 0<XkT=Xk∈ℝn×n, and 0<YkT=Yk∈ℝn×n, diagonal matrices 0<Uk∈ℝn+r×n+r and 0<Zk∈ℝn×n, and matrices H^k∈ℝ3n×3n k∈2, Am∈ℝr×r, Cp∈ℝr×r, Mm, and Mp of appropriate dimensions, such that LMIs (21) and (22) with α=0 are feasible for ρ∈0,ρ¯ and κ∈0,κ¯. In addition, the desired r-order state observer is described by (12) and (24).Since inequality (23) is not an LMI, the Toolbox YALMIP of MATLAB is not applicable. Now we will do the following transformation:Theorem 2.
For given scalarsρ¯, κ¯, μρ, and μκ satisfying (5) and positive constants T,c1,c2, and α, system (14) is F-T stable with respect to c1, c2, and T under Dirichlet boundary conditions, if there are real numbers λvi>0i∈5, λwj>0 j∈4, λxk>0, λyk>0, λuk>0 k∈2, and λu>0, matrices 0<ViT=Vi∈ℝn×n i∈5, 0<WjT=Wj∈ℝn×n, j∈4, 0<XkT=Xk∈ℝn×n, and 0<YkT=Yk∈ℝn×n, diagonal matrices 0<Uk∈ℝn+r×n+r and 0<Zk∈ℝn×n, and matrices H^k∈ℝ3n×3n k∈2, Am∈ℝr×r, Cp∈ℝr×r, Mm, and Mp of appropriate dimensions, such that LMIs (21), (22) and the following (69), (70), (71), (72), (73), (74), and (75) are feasible for ρ∈0,ρ¯ and κ∈0,κ¯:
(69)0≤Vi≤λviI,i∈5,(70)0≤Wj≤λwjI,j∈4,(71)0≤Uk≤λukI,k∈2,(72)0≤Xk≤λxkI,k∈2,(73)0≤Yk≤λykI,k∈2,(74)λuI≤U,(75)c1eαTλu1+ρ¯λv1+ρ¯λv2+16ρ¯3λx1+12ρ¯3λw1+λw3+λu1∑k=1lλmaxD¯k+λu2+κ¯λv3+κ¯λv4+124ρ¯4λy1+κ¯λv5λmaxKTK+16κ¯3λx2+12κ¯3λw2+λw4+124κ¯4λy2+λu2∑k=1lλmaxD¯k∗≤c2λu.In addition, the desiredr-order F-T state observer is described by (12) and (24).Proof 2.
It follows from (69), (70), (71) and (72) that
(76)n1≔c1eαTλmaxU1+ρ¯λmaxV1+ρ¯λmaxV2+16ρ¯3λmaxX1+∑k=1lλmaxU1λmaxD¯k+12ρ¯3λmaxW1+12ρ¯3λmaxW3+124ρ¯4λmaxY1≤c1eαTλu1+ρ¯λv1+ρ¯λv2+16ρ¯3λx1+12ρ¯3λw1+12ρ¯3λw3+124ρ¯4λy1+λu1∑k=1lλmaxD¯kand
(77)n2≔c1eαTλmaxU2+κ¯λmaxV3+κ¯λmaxV4+κ¯λmaxV5λmaxKTK+12κ¯3λmaxW4+16κ¯3λmaxX2+∑k=1lλmaxU2λmaxD¯k∗+12κ¯3λmaxW2+124κ¯4λmaxY2≤c1eαTλu2+κ¯λv3+κ¯λv4+κ¯λv5λmaxKTK+16κ¯3λx2+12κ¯3λw2+12κ¯3λw4+124κ¯4λy2+λu2∑k=1lλmaxD¯k∗.This, together with (74) and (75), implies that
(78)n1+n2≤λuc2≤c2λminU,which shows that (23) holds. By Theorem 1, we complete the proof.Finally, we make several remarks on the method proposed in this paper.Remark 2.
Different from [18, 19], this paper gives an r-order F-T state observer (12) for the delayed RDGRN (8). It should be mentioned that, compared with the full-order observer, the reduced-order one is more practical. Particularly, a reduced-order observer can save the cost in the engineering applications.Remark 3.
In the proof of Theorem1, we employ the so-called Wirtinger’s inequality to obtain the equalities (18, 19), which claims the necessity of Dirichlet boundary conditions. However, by employing the technique used in [14, 15], one can deal with the cases of Robin boundary conditions and Neumann boundary conditions, which will make the LMI conditions corresponding to ones in Theorems 1 and 2 more conservative.Remark 4.
For delayed genetic regulatory networks without reaction-diffusion items, the method proposed in Theorems1 and 2 is still keeping available by removing the corresponding parts of V1t,em,ep.Remark 5.
For delayed genetic regulatory networks without reaction-diffusion items, Zhang et al. [22] proposed a method to design full- and reduced-order state observers. It should be pointed that this method cannot be applied to delayed RDGRNs, since the equivalent decompositions of output matrices are required.Remark 6.
Several techniques used in this paper may be available for some other time-delay models:(1)
Wirtinger-type integral inequality, instead of Jensen’s inequality, is applied to estimate some integral items in the derivative of LKFs;
(2)
The convex technique and reciprocally convex technique are organically combined;
(3)
The coefficients,1/ρt and 1/ρ¯−ρt, are introduced into the augmented vector ηt,x.


## 5. An Illustrative Example

In this section, we will give a numerical example to verify the availability of the proposed method to design ther-order F-T state observer.Example 1.
Consider the delayed RDGRN (8), where l=L1=1, fix=x2/1+x2,i∈3, D1=0.1I3, D1∗=0.2I3,and
(79)A=diag0.2,1.1,1.2,C=diag0.3,0.7,1.3,B=diag1.0,0.4,0.7,W=00−0.5−0.5000−0.50,Nm=0.5−0.600.30.8−0.2,Np=0.7−0.250.30.40.2−0.3.In order to save space, we only design the1-order F-T state observer. Let μρ=μκ=1.5, ρ¯=κ¯=1, c1=1.2, c2=5, α=0.002, and T=10. The LMIs in Theorem 2 is solved by means of the MATLAB’s toolbox. The solution matrices are listed as follows:
(80)U1=diag0.0024,0.0023,0.0023,0.0023,U2=diag0.0021,0.0022,0.0022,0.0022,V1=10−4∗0.10450.00250.00060.00250.42620.00080.00060.00080.3188,V2=10−3∗0.83490.0014−0.00010.00140.68630.0001−0.00010.00010.7273,V3=10−4∗0.23750.00070.01770.00070.37050.00040.01770.00040.2184,V4=10−3∗0.24590.0004−0.00260.00040.35300.0001−0.00260.00010.3982,V5=10−3∗0.07240.00010.00440.00010.10080.00010.00440.00010.0504,W1=10−3∗0.4365−0.0003−0.0010−0.00030.0602−0.0002−0.0010−0.00020.0979,W2=10−3∗0.0650−0.0000−0.0001−0.00000.0676−0.0000−0.0001−0.00000.1329,W3=0.00190.0000−0.00000.00000.0013−0.0000−0.0000−0.00000.0017,W4=10−3∗0.38590.0007−0.06570.00070.72480.0003−0.06570.00030.8332,X1=10−4∗0.1836−0.0002−0.0002−0.00020.20280.0000−0.00020.00000.1890,X2=10−4∗0.2099−0.00000.0037−0.00000.2086−0.00000.0037−0.00000.1899,Y1=10−3∗0.55250.0028−0.00850.00280.2215−0.0002−0.0085−0.00020.2354,Y2=10−3∗0.17490.00000.02910.00000.2371−0.00030.0291−0.00030.2511,Z1=diag0.0012,0.0083,0.0083,Z2=diag0.0007,0.0010,0.0015,λv1=1.0785e−004,λv2=8.6624e−004,λv3=1.0586e−004,λv4=4.3530e−004,λv3=1.0586e−004,λv4=4.3530e−004,λv5=2.5342e−004,λw1=4.9465e−004,λw2=2.4219e−004,λw3=0.0019,λw4=9.0637e−004,λx1=4.8617e−004,λx2=4.8702e−004,λu1=0.0024,λu2=0.0022,λy1=0.0020,λy2=0.0019,λu=0.0021,Am=−0.0037,Mm=0.0008−0.0011,Cp=−0036,Mp=0.0016−0.0024.Furthermore, we can obtain the corresponding observer gains as follows:(81)A^=−1.6087,M^m=0.3478−0.4783,C^=−1.6364,M^p=0.7273−1.0909.When the initial functionψt=0.20.20.2T and ψ∗t=0.20.20.2T for t∈−1,0, the state responses of RDGRN (8), 1-order observer (12), and error system (14) are presented in Figures 1–6. From which, it is seen that our approach is effective.Figure 1
The real trajectories of mRNA concentration (mt,x2).Figure 2
The real trajectories of protein concentration (pt,x2).Figure 3
The estimation mRNA by the1-order observer (m^t,x2).Figure 4
The estimation protein concentrations by the1-order observer (p^t,x2).Figure 5
The mRNA concentrations of error system (emt,x2).Figure 6
The protein concentrations of error system (ept,x2).


## 6. Conclusions

The design problem ofr-order F-T state observer of RDGRNs with time-varying delays has been researched under Dirichlet boundary conditions. Utilizing available measurement outputs, we proposed a method to design r-order F-T observer which can be used to estimate the mRNA and protein concentrations. Sufficient F-T stability conditions for error system have been investigated by constructing an appropriate LKF and employing several integral inequalities and (reciprocally) convex technique. Thereby, the concrete expression of r-order F-T state observer is given. A numerical example is presented to illustrate the validity of the proposed method. It is worth emphasizing that the reduced-order observer problem of delayed RDGRNs is studied at the first time.In literature, the problem of full-order state estimation for complex systems have been addressed (see, for example, [23–25]). However, all approaches proposed in these literature are not available for designing reduced-order observers. Therefore, extending the method presented in this paper to the other system models will be left for future work.



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