# Finite-Time Stability of Fractional-Order Time-Varying Delays Gene Regulatory Networks with Structured Uncertainties and Controllers

Authors: Zhaohua Wu; Zhiming Wang; Tiejun Zhou
Journal: Complexity
 (2020)

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## Abstract

In this paper, we investigate a class of fractional-order time-varying delays gene regulatory networks with structured uncertainties and controllers (DFGRNs). Our contributions lie in three aspects: first, a necessary and sufficient condition on the existence of the solution for the DFGRNs is given by using the properties of the Riemann–Liouville fractional derivative and Caputo’s fractional derivative; second, the unique solution of the DFGRNs is proved under given initial function and certain condition; third, some novel sufficient conditions on finite-time stability of the DFGRNs are established by using a generalized Gronwall inequality and norm technique, and some conclusions on the finite-time stability of the DFGRNs with memory state-feedback controllers are reached, and those conditions and conclusions depend on the fractional order of the DFGRNs. One of the most interesting findings is that the “estimated time” of the finite-time stability is indeed related to the structured uncertainties, state-feedback controllers, time delays, and the fractional order of the system.


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## Body

## 1. Introduction

Genetic regulatory networks (GRNs), which describe the interaction functions in gene expressions between DNAs, RNAs, proteins, and small molecules in an organism, are fundamental and important biological networks. The analysis and control of GRNs involve two aspects: first, understanding the widespread phenomena in living organisms and providing potential routes to prolong life span, cure cancer and diabetes, and so on; second, potential application of GRNs in the development of related disciplines, such as synthetic biology, network medicine, and personalized medicine [1–5].With the development of sequencing technology, more and more genes and their regulatory sites are discovered. The structures and functions of a large number of genes are confirmed through experimental techniques, and even the regulatory mechanisms of the gene expressions controlled by some proteins are also identified [2]. Some qualitative models were proposed for distinguishing these regulatory mechanisms, such as directed graphs, Boolean networks, generalized logical networks, and rule-based formalisms [2]. However, it is difficult to know the biochemical reaction mechanisms’ underlying regulatory interactions when qualitatively handling great quantities of experimental data. Therefore, studying GRNs needs more accurately quantitative model. Based on experimental data, some simple genetic networks have been constructed, for example, genetic repressilator network [5], negative feedback GRNs [6], and genetic switch network [7]. The results in these experiments show that quantitative mathematical modeling approaches on dynamical systems have been great tools in providing insights on the mechanisms underlying the structure and the behaviors of GRNs [8–12].Using integer-order differential equation to model GRNs is a classical method. However, the fractional-order differential equations are more suitable for modeling the gene regulatory mechanism. Ji et al. [13] applied the particle swarm optimization technique in modeling the fractional-order GRNs with eight real target genes. The experimental results confirmed that the fractional-order model has achieved much lower fitting error on test data than integer-order model. Other studies also revealed that the fractional-order systems have excellent performance in describing the memory and hereditary properties of various processes in GRNs, which could be far better than the integer-order ones [8–11].Due to slow biochemical processes such as gene transcription, translation, and transportation (the synthesis of mRNAs and proteins at nucleus and cytoplasm, respectively, in eukaryotic cells), time delays are omnipresent in GRNs [14]. Many nonlinear differential equations with time delays have been proposed to model general GRNs, and the important role of time delays in dynamics of GRNs is now widely accepted [15–18]. Actually, time delays often degrade the system performance or destabilize the system [1, 19, 20]; even GRN models without time delay may generate wrong predictions [21]. As time delays often change with time and their precise measurement is difficult in real GRNs, the dynamics of fractional-order linear and nonlinear systems with time-varying delays has attracted increasing interest, and the results show that it is naturally of better practical significance than those with constant delays [21–25].In addition, in order to avoid undesirable states associated with disease, the control of GRNs is often regarded as developing therapeutic intervention strategies for some diseases [26, 27]. And many literatures focus on the research of control in the dynamic system [28–33]. In [32], the authors obtained some stabilization results for neural networks with leakage delay by designing state-feedback controller. Ebihara et al. [33] discovered that exact robust control is indeed attained for discrete-time linear systems by designing periodically time-varying memory state-feedback controller. Therefore, it is necessary to consider the controller for the DFGRNs.Since the modeling of GRNs is underlined with the real-world gene expression time-series data, some limitations of the current experimental techniques in GRNs make the modeling errors and parameter fluctuations unavoidable. Moreover, some point out that the system parameters identified with the experimental data may construct an unknown but bounded time-varying function, and this unknown nature is referred to as the structural uncertainty or the parametric uncertainty, also known as variation or fluctuation [34]. As is known, the structural uncertainties in GRNs may lead to the poor performance or even instability in real genetic networks [28, 34–37]. In [28], the authors studied the robust stabilization and state-feedback controller design for a class of integer-order GRNs with time-varying delays (DGRNs) and structured uncertainties and established some delay-dependent stability results by using some matrix techniques. Therefore, taking into account the structural uncertainties while investigating the dynamical behaviors of DFGRNs is essential.Since the expression of gene and mRNA-translated protein is accomplished in a much relatively short period, in recent decades, some scholars have paid more attention to the finite-time stability of GRNs [4, 38]. For example, Wu et al. [4] investigated the finite-time stability associated with a class of integer-order GRNs by designing adaptive controllers. Wang et al. [38] established some new sufficient conditions of the finite-time stability for a class of integer-order uncertain GRNs with time-varying delays. Lazarević [22] investigated the finite-time stability for fractional-order nonlinear differential equation with time-varying delays by using generalized Gronwall inequality and the classical Bellman–Gronwall inequality, respectively. Phat and Thanh [23] established some new sufficient conditions of robust finite-time stability for a class of nonlinear fractional-order differential systems with time-varying delays. Wang et al. [39] considered a class of nonlinear fractional-order systems with constant delays and studied the existence and uniqueness of the solution for this kind of systems by using relevant properties of the fractional derivative.However, the discussions on the existence and uniqueness of the solutions and the finite-time stability results for the fractional-order uncertain GRNs with time-varying delays and controllers seem rare.From above discussions, we focus on the existence and uniqueness of the solution and the finite-time stability for a class of DFGRNs with structured uncertainties and controllers. The remainder of this paper is organized as follows. In Section2, we give the model description, some definitions, and related properties on fractional calculus. In Section 3, we discuss the existence and uniqueness of the solution and give some sufficient criteria on the finite-time stability for the DFGRNs. In Section 4, we perform some numerical simulations, which support our findings. In Section 5, we briefly review and summarize the main results.


## 2. Problem Description and Preliminaries

For any vectorxt∈Rn and matrix A¯=a¯ijn×n∈Rn×n, denote(1)xt=∑i=1nxit,A¯=max1≤j≤n∑i=1na¯ij.Letσ⋅ be the largest singular value of matrix,(2)η1=maxA+μ1,C+μ4,η2=maxW+μ2L1,D+μ5,η3=maxK+μ3L2,H+μ6,η4=maxσA+μ1+σD+μ5,σW+μ2L1+σC+μ4,η5=maxσA+μ1+σQ1+μ7σγ1+σD+μ5,σW+μ2L1+σC+μ4+σQ2+μ8σγ2,η6=maxσQ1+μ7,σQ2+μ8,ζ1=η4+σK+μ3L2+σH+μ6,ζ2=B+σW+μ2F0+σK+μ3G0,ζ3=η5+σK+μ3L2+σQ1+μ7σγ3+σH+μ6+σQ2+μ8σγ4,where μ1,μ2,μ3,μ4,μ5,μ6,μ7,μ8,L1,L2 are positive constants that satisfy the later assumptions (I) and (II), respectively.We will focus on a class of DFGRNs with structured uncertainties and controllers, which is established as follows:(3)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tu1t,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tu2t,where(4)mt=m1t,m2t,…,mntT,pt=p1t,p2t,…,pntT,Fpt=f1p1t,f2p2t,…,fnpntT,B=B1,B2,…,BnT,A=diaga1,a2,…,an,C=diagc1,c2,…,cn,D=diagd1,d2,…,dn,H=diage1,e2,…,en,Gpt−τ1t=g1p1t−τ1t,g2p2t−τ1t,…,gnpnt−τ1tT,in which DCtqrepresents Caputo’s fractional derivative and q∈0,1,mit,pit∈R are the concentrations of mRNA and protein of the ith node, respectively. The parameters ai>0 and ci>0 are the decay rates of mRNA and protein, respectively; di>0 are the translation rates; ei≥0 are the translation rates. Both fjpjt and gjpjt−τ1t represent the feedback regulation of the protein on the transcription. Generally, each one of the two functions is a nonlinear function but has a form of monotonicity with its variable. As a monotonic increasing or decreasing regulatory function, fj and gj are usually of the Michaelis–Menten or Hill forms [21]. Bi=∑j∈Iibij+∑j∈I¯ib¯ij, where bij and b¯ij are bounded constants which are, respectively, the dimensionless transcriptional rates of transcription factor j to i at time t and t−τ1t, and Ii,I¯i, respectively, are the set of all the j where the transcription factor j is a repressor of gene i at time t and t−τ1t. W=wij∈Rn×n,K=kij∈Rn×n are the coupling matrices of the gene network, which are defined as follows:(5)wijkij=bijb¯ij,if transcription factorjis an activator of genei,−bij−b¯ij,if transcription factorjis a repressor of genei,0,if there is no link from nodejtoi.The transcriptional delayτ1t and translational delay τ2t are bounded continuous functions on R with 0≤τit≤τ∗i=1,2; here τ∗ is a positive constant. u1t=u11t,u12t,…,u1ntT, u2t=u21t,u22t,…,u2ntT are controller vectors, Qii=1,2 are n dimensions coefficient matrices. ΔAt, ΔDt,ΔCt, ΔWt, ΔKt, ΔHt, ΔQ1t,ΔQ2t are norm-bounded unknown matrices with time-varying structured uncertainties.The initial conditions for DFGRN (3) are as follows:(6)mθ=ϕ1θ,θ∈−τ∗,0,pθ=ϕ2θ,θ∈−τ∗,0,where ϕit∈C−τ∗,0,Rni=1,2 is the given initial function with ϕic=sup−τ∗≤θ≤0ϕiθi=1,2 and ϕ0=ϕ1c+ϕ2c.(i)
Assumption (I): the norm-bounded unknown matrices satisfy the following inequalities:(7)ΔAt≤μ1,ΔWt≤μ2,ΔKt≤μ3,ΔCt≤μ4,ΔDt≤μ5,ΔHt≤μ6,ΔQ1t≤μ7,ΔQ2t≤μ8,where μ1,μ2,μ3,μ4,μ5,μ6,μ7,μ8 are positive constants.(ii)
Assumption (II): the functionsF,G satisfy the following inequalities:(8)Fx−Fy≤L1x−y,Gx−Gy≤L2x−y,x,y∈Rn,where L1,L2 are positive constants.Next, we give some definitions and lemmas.Definition 1 (see [40]).
The fractional integral of orderq for a function ft is defined as(9)Itqaft=1Γq∫att−τq−1fτdτ,where t≥a,a∈R,q>0. The gamma function Γq is defined by the integral Γq=∫0∞tq−1e−tdt.Definition 2 (see [40]).
Caputo’s fractional derivative of orderq for a function f is defined by(10)DaCtqft=1Γn−q∫at1t−τq−n+1fnτdτ,where t≥a and n is a positive integer such that n−1<q<n.Definition 3 (see [40]).
The Riemann–Liouville fractional derivative of orderq for a function f is defined as(11)DaRLtqft=1Γn−qdndtn∫att−sn−q−1fsds,where t≥a and n is a positive integer such that n−1<q<n.
For convenience, we choose the notationItq=Itq0, DCtq=D0Ctq, DRLtq=D0RLtq.Definition 4.
A mild solution of DFGRN (3) with initial condition (6) is a vector mt,ptT composed of continuous functions(12)mt,pt:−τ∗,∞⟶Rn,satisfying(13)mt=m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt=p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds,mθ=ϕ1θ,θ∈−τ∗,0,pθ=ϕ2θ,θ∈−τ∗,0.Definition 5 (see [22]).
The system given by (3) (when Qi=0,ΔQi=0,i=1,2) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,t0,J, δ<ε if and only if ϕ0<δ imply mt+pt<ε,∀t∈J,J⊂R.Definition 6 (see [22]).
The system given by (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,α1,t0,J, δ<ε if and only if ϕ0<δ and u1t+u2t<α1,∀t∈Jimply mt+pt<ε,∀t∈J,J⊂R, where α1 is a positive constant.Lemma 1 (see [40]).
Ifft∈Cn0,∞ and n−1<α<n∈Z+, then(i)
ItqDCtqft=ft−∑k=0n−1fk0tk/k!.(ii)
DRLtqItqft=ft.(iii)
DRLtqft=DCtqft+∑k=0n−1tk−q/Γk+1−qfk0.Lemma 2 (see [41]).
Supposeβ>0; if 0≤t<T (some T≤+∞), at is a locally integrable nonnegative function, vt is a nonnegative and nondecreasing continuous function, vt≤M (constant), and ut is a nonnegative and locally integrable function with ut≤at+vt∫0tt−sβ−1usds, then(14)ut≤at+∫0t∑n=1∞vtΓβnΓnβt−snβ−1asds,∀0≤t≤T.In addition, ifat is a nondecreasing function, then ut≤atEβvtΓβtβ, where Eβ is the Mittag-Leffler function defined by Eβz=∑k=0∞zk/Γkβ+1.


## 3. Main Results

### 3.1. The Existence and Uniqueness of the Mild Solution of DFGRNs

Theorem 1.
Continuously differentiable functionsmt,pt:−τ∗,T⟶RnT<+∞ form a mild solution mt,ptT to DFGRN (3) with initial condition (6) if and only if(15)mt=ϕ10+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,fort∈0,T,pt=ϕ20+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds,fort∈0,T,mt=ϕ1t,fort∈−τ∗,0,pt=ϕ2t,fort∈−τ∗,0.Proof.
We firstly give the sufficient condition of the existence of the mild solution to DFGRN (3).
When−τ∗≤t≤0, mt,ptT=ϕ1t,ϕ2tT is obvious. For 0≤t≤T, according to (15), applying DRLtq and property (ii) of Lemma 1, we obtain(16)DRLtqmt=ϕ10t−qΓ1−q−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tu1t,DRLtqpt=ϕ20t−qΓ1−q−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tu2t.
According to the property (iii) of Lemma1 and 0<q<1, we get(17)DRLtqmt=DCtqmt+ϕ10t−qΓ1−q,DRLtqpt=DCtqpt+ϕ20t−qΓ1−q.
From (16) and (17), we have(18)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tu1t,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tu2t.
We secondly give the necessary condition of the existence of the mild solution to DFGRN (3).
Whent∈−τ∗,0, the solution of DFGRN (3) is(19)mt=ϕ1t,pt=ϕ2t,t∈−τ∗,0.
If0≤t≤T, from DFGRN (3), we have(20)ItqDCtqmt=1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,ItqDCtqpt=1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
In the case of0<q<1, from property (i) of Lemma 1, we can obtain(21)mt=ϕ10+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt=ϕ20+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
The proof is completed.Theorem 2.
If assumptions (I) and (II) hold, then DFGRN (3) with initial condition (6) has a unique mild solution.Proof.
Letmt,ptT and m˜t,p˜tT be any two different solutions to DFGRN (3) with initial condition (6); denote xt=mt−m˜t,yt=pt−p˜t, zt=xt+yt, t∈−τ∗,T. According to Theorem 1, we know that both mt,ptT and m˜t,p˜tT satisfy condition (15).
If0≤t≤T, then(22)xt=1Γq∫0tt−sq−1−A+ΔAsxs+W+ΔWsFps−Fp˜s+K+ΔKsGps−τ1s−Gp˜s−τ1sds,yt=1Γq∫0tt−sq−1−C+ΔCsys+D+ΔDsxs+H+ΔHsxs−τ2sds.
From (22), by using the norm ⋅ and assumptions (I) and (II), we can obtain(23)xt≤1Γq∫0tt−sq−1A+μ1⋅xs+W+μ2⋅L1⋅ys+K+μ3⋅L2⋅ys−τ1sds,t∈0,T,yt≤1Γq∫0tt−sq−1C+μ4⋅ys+D+μ5⋅xs+H+μ6⋅xs−τ2sds,t∈0,T.
First, whent∈−τ∗,0, xt=ϕ1θ−ϕ1θ=0,yt=ϕ2θ−ϕ2θ=0. So, mt=m˜t,pt=p˜t for t∈−τ∗,0.
Second, whent∈0,τ∗, from (23), we have(24)xt≤1Γq∫0tt−sq−1A+μ1⋅xs+W+μ2⋅L1⋅ysds,yt≤1Γq∫0tt−sq−1C+μ4⋅ys+D+μ5⋅xsds.
In accordance with (24), we can obtain(25)zt=xt+yt≤1Γq∫0tt−sq−1η1zs+η2zsds=η1+η2Γq∫0tt−sq−1zsds.
From Lemma2, we can get(26)zt≤0⋅Eqη1+η2Γq⋅Γqtq,t∈0,τ∗.
Thus,zt≤0,t∈0,τ∗. That is to say, xt+yt≤0,t∈0,τ∗. So, mt=m˜t,pt=p˜t for t∈0,τ∗.
Third, whent∈τ∗,T, according to (23), we have(27)zt=xt+yt≤1Γq∫0tt−sq−1η1zs+η2zs+η3xs−τ2s+ys−τ1sds,t∈τ∗,T.
Letz∗t=supθ,θ˜∈−τ∗,0xt+θ+yt+θ˜. Then, we obtain(28)z∗t≤η1+η2+η3Γq∫0tt−sq−1z∗sds,t∈τ∗,T.
From Lemma2, we can get(29)zt≤z∗t≤0⋅Eqη1+η2+η3Γq⋅Γq⋅tq,t∈τ∗,T.Then, we have xt+yt≤0,t∈τ∗,T. So, mt=m˜t,pt=p˜t for t∈τ∗,T.
Summarizing the above three cases, we can obtain thatmt=m˜t,pt=p˜t, for t∈−τ∗,T. Due to the arbitrary nature of the solution mt,ptT and m˜t,p˜tT of DFGRN (3) and in accordance with Definition 4, we can conclude that DFGRN (3) has a unique mild solution. The proof is completed.


### 3.2. Finite-Time Stability of DFGRNs with Structured Uncertainties

Theorem 3.
If assumptions (I) and (II) and1+ζ1+ζ4tq/Γq+1⋅Eqζ1tq≤ε/δ,∀t∈J0=0,T hold, then the uncertain DFGRNs with controllers given by (3) with initial condition (6) are finite-time stable with respect to δ,ε,α1,J0,δ<ε, where ζ4≔η6α1+ζ2/δ.Proof.
According to Theorem1 and Theorem 2, we can know that DFGRN (3) has a mild solution and the solution satisfies the following integral equation:(30)mt=m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt=p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
Using the norm⋅, we can obtain the solution estimate of system (30):(31)mt≤m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt≤p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
By applying norm⋅ to DFGRN (3) and combining assumptions (I) and (II), we can get(32)DCtqmt≤σA+μ1mt+σW+μ2L1pt+F0+σK+μ3L2pt−τ1t+G0+B+σQ1+μ7u1t,DCtqpt≤σC+μ4pt+σD+μ5mt+σH+μ6mt−τ2t+σQ2+μ8u2t.
Letxt=mt+pt. According to (31), (3), and (32), if u1t+u2t<α1, we have(33)xt≤m0+p0+1Γq∫0tt−sq−1DCtqms+DCtqpsds≤m0+p0+1Γq∫0tt−sq−1σA+μ1+σD+μ5ms+σW+μ2L1+σC+μ4ps+σK+μ3L2ps−τ1s+σH+μ6⋅ms−τ2s+B+σW+μ2F0+σK+μ3G0+σQ1+μ7u1s+σQ2+μ8u2sds≤x0+1Γq∫0tt−sq−1η4xs+σK+μ3L2xs−τ1s+σH+μ6xs−τ2s+ζ2+σQ1+μ7u1s+σQ2+μ8u2sds,≤x0+1Γq∫0tt−sq−1η4+σK+μ3L2+σH+μ6sups−τ∗≤t∗≤sxt∗+ϕ0+ζ2+σQ1+μ7u1s+σQ2+μ8u2sds≤x0+1Γq∫0tt−sq−1ζ1sups−τ∗≤t∗≤sxt∗+ϕ0+ζ2+η6u1s+u2sds≤ϕ0+1Γqζ1∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ1Γq∫0tt−sq−1ϕ0ds+η6Γq∫0tt−sq−1u1s+u2sds+ζ2Γq∫0tt−sq−1ds≤ϕ0+ζ1Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ1Γqϕ0qtq+η6Γqq⋅α1⋅tq+ζ2tqΓqq≤ϕ01+ζ1tqΓq+1+ζ1Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+η6⋅α1+ζ2Γq+1⋅tq,t>0.
Let(34)ρt=ϕ01+ζ1tqΓq+1+η6⋅α1+ζ2Γq+1⋅tq,t>0.Then, we know that ρt is a nonnegative and nondecreasing function. By using Lemma 2 (the generalized Gronwall inequality), we have(35)xt≤supt−τ∗≤t∗≤txt∗≤ρtEqζ1ΓqΓqtq.If ϕ0<δ, we have(36)xt≤δ1+ζ1tqΓq+1+η6α1+ζ2Γq+1δtqEqζ1tq.
Because1+ζ1+ζ4tq/Γq+1Eqζ1tq≤ε/δ and ζ4=η6α1+ζ2/δ, then(37)xt<ε,∀t∈J0.
Hence,(38)mt+pt<ε,∀t∈J0.
The proof is completed.Remark 1.
If we adoptu1t≡0,u2t≡0,∀t∈J0 in DFGRN (3), we can obtain the following conclusion.
The uncertain DFGRN (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0,δ<ε, if assumptions (I) and (II) hold and the following condition is satisfied:(39)1+ζ1+ζ5Γq+1tq⋅Eqζ1tq≤εδ,∀t∈J0=0,T,where ζ5≔ζ2/δ.Remark 2.
In the proof of Theorem3, if we use the “classical” Bellman–Gronwall inequality instead of the generalized Gronwall inequality, we can get the following result.
The uncertain DFGRN with controllers given by (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,α1,J0,δ<ε, if assumptions (I) and (II) hold and the following condition is satisfied:(40)1+ζ1+ζ4Γq+1t−t0qexpζ1t−t0qΓq+1<εδ.Remark 3.
If we takeu1t≡0,u2t≡0,∀t∈J0 in system (3), the above results turn into the following conclusion.
The uncertain DFGRN (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0,δ<ε. If assumptions (I) and (II) hold, the following condition is satisfied:(41)1+ζ1+ζ5Γq+1t−t0qexpζ1t−t0qΓq+1<εδ.


### 3.3. Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers

We consider the following memory state-feedback controllers on DFGRN (3):(42)u1t=γ1mt+γ3pt−τ^1t,u2t=γ2pt+γ4mt−τ^2t,where γi,i=1,2,3,4 are the gain matrices of uit,0≤τ^1t≤τ∗,0≤τ^2t≤τ∗. Then, DFGRN (3) can be changed into(43)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tγ1mt+γ3pt−τ^1t,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tγ2pt+γ4mt−τ^2t.Theorem 4.
If assumptions (I) and (II) and(44)1+ζ3+ζ5Γq+1tqEqζ3tq≤εδ,hold, then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0, δ<ε.Proof.
Similar to Theorem1 and Theorem 2, it is easy to prove that DFGRN (43) has a mild solution satisfying the following integral equation:(45)mt=m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1sγ1ms+γ3ps−τ^1sds,pt=p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2sγ2ps+γ4ms−τ^2sds.
Using the norm⋅, we have(46)mt≤m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1sγ1ms+γ3ps−τ^1sds,pt≤p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2sγ2ps+γ4ms−τ^2sds.
From (43), by using assumptions (I) and (II), we have(47)DCtqmt≤σA+μ1mt+σW+μ2L1pt+F0+σK+μ3L2pt−τ1t+G0+B+σQ1+μ7σγ1mt+σγ3pt−τ^1t,DCtqpt≤σC+μ4pt+σD+μ5mt+σH+μ6mt−τ2t+σQ2+μ8σγ2pt+σγ4mt−τ^2t.
Letxt=mt+pt. From (46) and (47), we obtain(48)xt≤m0+p0+1Γq∫0tt−sq−1σA+μ1+σQ1+μ7σγ1+σD+μ5‖ms‖+B+σW+μ2F0+σK+μ3G0+σW+μ2L1+σC+μ4+σQ2+μ8σγ2ps+σK+μ3L2ps−τ1s+σQ1+μ7σγ3ps−τ^1s+σH+μ6ms−τ2s+σQ2+μ8σγ4ms−τ^2sds.
Hence,(49)xt≤x0+1Γq∫0tt−sq−1η5xs+ζ2+σK+μ3L2xs−τ1s+σQ1+μ7σγ3xs−τ^1s+σH+μ6xs−τ2s+σQ2+μ8σγ4xs−τ^2sds≤ϕ0+ζ3Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗+ϕ0ds+ζ2Γq∫0tt−sq−1ds=ϕ0+ζ3Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ2tqΓq+1+ζ3Γqϕ0∫0tt−sq−1ds≤ϕ01+ζ3Γq+1tq+ζ3Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ2tqΓq+1.
Let(50)ρt=ϕ01+ζ3tqΓq+1+ζ2tqΓq+1,t>0.Then, we know ρt is a nonnegative and nondecreasing function. From Lemma 2, we have(51)xt≤supt−τ∗≤t∗≤txt∗≤ρtEqζ3ΓqΓqtq.
Ifϕ0<δ, we obtain(52)xt≤δ1+ζ3+ζ2/δΓq+1tqEqζ3tq.
From the condition of1+ζ3+ζ5tq/Γq+1Eqζ3tq≤ε/δ and ζ5=ζ2/δ, we can get(53)xt<ε,∀t∈J0.
Therefore,(54)mt+pt<ε,∀t∈J0.
The proof is completed.Remark 4.
Similar to Remark2, we can get the following result.
The uncertain DFGRN (3) with memory state-feedback controller given by (43) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0,δ<ε, if assumptions (I) and (II) hold and the following condition is satisfied:(55)1+ζ3+ζ5Γq+1t−t0qexpζ3t−t0qΓq+1<εδ.Remark 5.
We can obtain the same conclusion as Theorem3 and Theorem 4 if the inequalities in assumption (II) are(56)Fx≤L1x,Gx≤L2x.Remark 6.
All the results in Remarks1–4 are still new.


## 3.1. The Existence and Uniqueness of the Mild Solution of DFGRNs

Theorem 1.
Continuously differentiable functionsmt,pt:−τ∗,T⟶RnT<+∞ form a mild solution mt,ptT to DFGRN (3) with initial condition (6) if and only if(15)mt=ϕ10+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,fort∈0,T,pt=ϕ20+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds,fort∈0,T,mt=ϕ1t,fort∈−τ∗,0,pt=ϕ2t,fort∈−τ∗,0.Proof.
We firstly give the sufficient condition of the existence of the mild solution to DFGRN (3).
When−τ∗≤t≤0, mt,ptT=ϕ1t,ϕ2tT is obvious. For 0≤t≤T, according to (15), applying DRLtq and property (ii) of Lemma 1, we obtain(16)DRLtqmt=ϕ10t−qΓ1−q−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tu1t,DRLtqpt=ϕ20t−qΓ1−q−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tu2t.
According to the property (iii) of Lemma1 and 0<q<1, we get(17)DRLtqmt=DCtqmt+ϕ10t−qΓ1−q,DRLtqpt=DCtqpt+ϕ20t−qΓ1−q.
From (16) and (17), we have(18)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tu1t,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tu2t.
We secondly give the necessary condition of the existence of the mild solution to DFGRN (3).
Whent∈−τ∗,0, the solution of DFGRN (3) is(19)mt=ϕ1t,pt=ϕ2t,t∈−τ∗,0.
If0≤t≤T, from DFGRN (3), we have(20)ItqDCtqmt=1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,ItqDCtqpt=1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
In the case of0<q<1, from property (i) of Lemma 1, we can obtain(21)mt=ϕ10+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt=ϕ20+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
The proof is completed.Theorem 2.
If assumptions (I) and (II) hold, then DFGRN (3) with initial condition (6) has a unique mild solution.Proof.
Letmt,ptT and m˜t,p˜tT be any two different solutions to DFGRN (3) with initial condition (6); denote xt=mt−m˜t,yt=pt−p˜t, zt=xt+yt, t∈−τ∗,T. According to Theorem 1, we know that both mt,ptT and m˜t,p˜tT satisfy condition (15).
If0≤t≤T, then(22)xt=1Γq∫0tt−sq−1−A+ΔAsxs+W+ΔWsFps−Fp˜s+K+ΔKsGps−τ1s−Gp˜s−τ1sds,yt=1Γq∫0tt−sq−1−C+ΔCsys+D+ΔDsxs+H+ΔHsxs−τ2sds.
From (22), by using the norm ⋅ and assumptions (I) and (II), we can obtain(23)xt≤1Γq∫0tt−sq−1A+μ1⋅xs+W+μ2⋅L1⋅ys+K+μ3⋅L2⋅ys−τ1sds,t∈0,T,yt≤1Γq∫0tt−sq−1C+μ4⋅ys+D+μ5⋅xs+H+μ6⋅xs−τ2sds,t∈0,T.
First, whent∈−τ∗,0, xt=ϕ1θ−ϕ1θ=0,yt=ϕ2θ−ϕ2θ=0. So, mt=m˜t,pt=p˜t for t∈−τ∗,0.
Second, whent∈0,τ∗, from (23), we have(24)xt≤1Γq∫0tt−sq−1A+μ1⋅xs+W+μ2⋅L1⋅ysds,yt≤1Γq∫0tt−sq−1C+μ4⋅ys+D+μ5⋅xsds.
In accordance with (24), we can obtain(25)zt=xt+yt≤1Γq∫0tt−sq−1η1zs+η2zsds=η1+η2Γq∫0tt−sq−1zsds.
From Lemma2, we can get(26)zt≤0⋅Eqη1+η2Γq⋅Γqtq,t∈0,τ∗.
Thus,zt≤0,t∈0,τ∗. That is to say, xt+yt≤0,t∈0,τ∗. So, mt=m˜t,pt=p˜t for t∈0,τ∗.
Third, whent∈τ∗,T, according to (23), we have(27)zt=xt+yt≤1Γq∫0tt−sq−1η1zs+η2zs+η3xs−τ2s+ys−τ1sds,t∈τ∗,T.
Letz∗t=supθ,θ˜∈−τ∗,0xt+θ+yt+θ˜. Then, we obtain(28)z∗t≤η1+η2+η3Γq∫0tt−sq−1z∗sds,t∈τ∗,T.
From Lemma2, we can get(29)zt≤z∗t≤0⋅Eqη1+η2+η3Γq⋅Γq⋅tq,t∈τ∗,T.Then, we have xt+yt≤0,t∈τ∗,T. So, mt=m˜t,pt=p˜t for t∈τ∗,T.
Summarizing the above three cases, we can obtain thatmt=m˜t,pt=p˜t, for t∈−τ∗,T. Due to the arbitrary nature of the solution mt,ptT and m˜t,p˜tT of DFGRN (3) and in accordance with Definition 4, we can conclude that DFGRN (3) has a unique mild solution. The proof is completed.


## 3.2. Finite-Time Stability of DFGRNs with Structured Uncertainties

Theorem 3.
If assumptions (I) and (II) and1+ζ1+ζ4tq/Γq+1⋅Eqζ1tq≤ε/δ,∀t∈J0=0,T hold, then the uncertain DFGRNs with controllers given by (3) with initial condition (6) are finite-time stable with respect to δ,ε,α1,J0,δ<ε, where ζ4≔η6α1+ζ2/δ.Proof.
According to Theorem1 and Theorem 2, we can know that DFGRN (3) has a mild solution and the solution satisfies the following integral equation:(30)mt=m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt=p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
Using the norm⋅, we can obtain the solution estimate of system (30):(31)mt≤m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1su1sds,pt≤p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2su2sds.
By applying norm⋅ to DFGRN (3) and combining assumptions (I) and (II), we can get(32)DCtqmt≤σA+μ1mt+σW+μ2L1pt+F0+σK+μ3L2pt−τ1t+G0+B+σQ1+μ7u1t,DCtqpt≤σC+μ4pt+σD+μ5mt+σH+μ6mt−τ2t+σQ2+μ8u2t.
Letxt=mt+pt. According to (31), (3), and (32), if u1t+u2t<α1, we have(33)xt≤m0+p0+1Γq∫0tt−sq−1DCtqms+DCtqpsds≤m0+p0+1Γq∫0tt−sq−1σA+μ1+σD+μ5ms+σW+μ2L1+σC+μ4ps+σK+μ3L2ps−τ1s+σH+μ6⋅ms−τ2s+B+σW+μ2F0+σK+μ3G0+σQ1+μ7u1s+σQ2+μ8u2sds≤x0+1Γq∫0tt−sq−1η4xs+σK+μ3L2xs−τ1s+σH+μ6xs−τ2s+ζ2+σQ1+μ7u1s+σQ2+μ8u2sds,≤x0+1Γq∫0tt−sq−1η4+σK+μ3L2+σH+μ6sups−τ∗≤t∗≤sxt∗+ϕ0+ζ2+σQ1+μ7u1s+σQ2+μ8u2sds≤x0+1Γq∫0tt−sq−1ζ1sups−τ∗≤t∗≤sxt∗+ϕ0+ζ2+η6u1s+u2sds≤ϕ0+1Γqζ1∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ1Γq∫0tt−sq−1ϕ0ds+η6Γq∫0tt−sq−1u1s+u2sds+ζ2Γq∫0tt−sq−1ds≤ϕ0+ζ1Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ1Γqϕ0qtq+η6Γqq⋅α1⋅tq+ζ2tqΓqq≤ϕ01+ζ1tqΓq+1+ζ1Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+η6⋅α1+ζ2Γq+1⋅tq,t>0.
Let(34)ρt=ϕ01+ζ1tqΓq+1+η6⋅α1+ζ2Γq+1⋅tq,t>0.Then, we know that ρt is a nonnegative and nondecreasing function. By using Lemma 2 (the generalized Gronwall inequality), we have(35)xt≤supt−τ∗≤t∗≤txt∗≤ρtEqζ1ΓqΓqtq.If ϕ0<δ, we have(36)xt≤δ1+ζ1tqΓq+1+η6α1+ζ2Γq+1δtqEqζ1tq.
Because1+ζ1+ζ4tq/Γq+1Eqζ1tq≤ε/δ and ζ4=η6α1+ζ2/δ, then(37)xt<ε,∀t∈J0.
Hence,(38)mt+pt<ε,∀t∈J0.
The proof is completed.Remark 1.
If we adoptu1t≡0,u2t≡0,∀t∈J0 in DFGRN (3), we can obtain the following conclusion.
The uncertain DFGRN (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0,δ<ε, if assumptions (I) and (II) hold and the following condition is satisfied:(39)1+ζ1+ζ5Γq+1tq⋅Eqζ1tq≤εδ,∀t∈J0=0,T,where ζ5≔ζ2/δ.Remark 2.
In the proof of Theorem3, if we use the “classical” Bellman–Gronwall inequality instead of the generalized Gronwall inequality, we can get the following result.
The uncertain DFGRN with controllers given by (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,α1,J0,δ<ε, if assumptions (I) and (II) hold and the following condition is satisfied:(40)1+ζ1+ζ4Γq+1t−t0qexpζ1t−t0qΓq+1<εδ.Remark 3.
If we takeu1t≡0,u2t≡0,∀t∈J0 in system (3), the above results turn into the following conclusion.
The uncertain DFGRN (3) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0,δ<ε. If assumptions (I) and (II) hold, the following condition is satisfied:(41)1+ζ1+ζ5Γq+1t−t0qexpζ1t−t0qΓq+1<εδ.


## 3.3. Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers

We consider the following memory state-feedback controllers on DFGRN (3):(42)u1t=γ1mt+γ3pt−τ^1t,u2t=γ2pt+γ4mt−τ^2t,where γi,i=1,2,3,4 are the gain matrices of uit,0≤τ^1t≤τ∗,0≤τ^2t≤τ∗. Then, DFGRN (3) can be changed into(43)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tγ1mt+γ3pt−τ^1t,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tγ2pt+γ4mt−τ^2t.Theorem 4.
If assumptions (I) and (II) and(44)1+ζ3+ζ5Γq+1tqEqζ3tq≤εδ,hold, then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0, δ<ε.Proof.
Similar to Theorem1 and Theorem 2, it is easy to prove that DFGRN (43) has a mild solution satisfying the following integral equation:(45)mt=m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1sγ1ms+γ3ps−τ^1sds,pt=p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2sγ2ps+γ4ms−τ^2sds.
Using the norm⋅, we have(46)mt≤m0+1Γq∫0tt−sq−1−A+ΔAsms+W+ΔWsFps+K+ΔKsGps−τ1s+B+Q1+ΔQ1sγ1ms+γ3ps−τ^1sds,pt≤p0+1Γq∫0tt−sq−1−C+ΔCsps+D+ΔDsms+H+ΔHsms−τ2s+Q2+ΔQ2sγ2ps+γ4ms−τ^2sds.
From (43), by using assumptions (I) and (II), we have(47)DCtqmt≤σA+μ1mt+σW+μ2L1pt+F0+σK+μ3L2pt−τ1t+G0+B+σQ1+μ7σγ1mt+σγ3pt−τ^1t,DCtqpt≤σC+μ4pt+σD+μ5mt+σH+μ6mt−τ2t+σQ2+μ8σγ2pt+σγ4mt−τ^2t.
Letxt=mt+pt. From (46) and (47), we obtain(48)xt≤m0+p0+1Γq∫0tt−sq−1σA+μ1+σQ1+μ7σγ1+σD+μ5‖ms‖+B+σW+μ2F0+σK+μ3G0+σW+μ2L1+σC+μ4+σQ2+μ8σγ2ps+σK+μ3L2ps−τ1s+σQ1+μ7σγ3ps−τ^1s+σH+μ6ms−τ2s+σQ2+μ8σγ4ms−τ^2sds.
Hence,(49)xt≤x0+1Γq∫0tt−sq−1η5xs+ζ2+σK+μ3L2xs−τ1s+σQ1+μ7σγ3xs−τ^1s+σH+μ6xs−τ2s+σQ2+μ8σγ4xs−τ^2sds≤ϕ0+ζ3Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗+ϕ0ds+ζ2Γq∫0tt−sq−1ds=ϕ0+ζ3Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ2tqΓq+1+ζ3Γqϕ0∫0tt−sq−1ds≤ϕ01+ζ3Γq+1tq+ζ3Γq∫0tt−sq−1sups−τ∗≤t∗≤sxt∗ds+ζ2tqΓq+1.
Let(50)ρt=ϕ01+ζ3tqΓq+1+ζ2tqΓq+1,t>0.Then, we know ρt is a nonnegative and nondecreasing function. From Lemma 2, we have(51)xt≤supt−τ∗≤t∗≤txt∗≤ρtEqζ3ΓqΓqtq.
Ifϕ0<δ, we obtain(52)xt≤δ1+ζ3+ζ2/δΓq+1tqEqζ3tq.
From the condition of1+ζ3+ζ5tq/Γq+1Eqζ3tq≤ε/δ and ζ5=ζ2/δ, we can get(53)xt<ε,∀t∈J0.
Therefore,(54)mt+pt<ε,∀t∈J0.
The proof is completed.Remark 4.
Similar to Remark2, we can get the following result.
The uncertain DFGRN (3) with memory state-feedback controller given by (43) satisfying the initial condition (6) is finite-time stable with respect to δ,ε,J0,δ<ε, if assumptions (I) and (II) hold and the following condition is satisfied:(55)1+ζ3+ζ5Γq+1t−t0qexpζ3t−t0qΓq+1<εδ.Remark 5.
We can obtain the same conclusion as Theorem3 and Theorem 4 if the inequalities in assumption (II) are(56)Fx≤L1x,Gx≤L2x.Remark 6.
All the results in Remarks1–4 are still new.


## 4. Numerical Examples

In this section, some numerical examples are given to illustrate the effectiveness of above theoretical results. In the following examples, the functionsfj and gj are taken as the Hill form. And in the Adams–Bashforth–Moulton predictor-corrector scheme [42], the step length is h=0.1.Example 1.
Consider the following DFGRNs of three mRNA and protein nodes with structured uncertainties and memory state-feedback controllers:(57)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B+Q1+ΔQ1tu1t,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t+Q2+ΔQ2tu2t.
Let(58)A=300030003,C=2.50002.50002.5,D=100010001,H=0.30000.30000.3,W=0.8147−0.91340.27850.90580.6324−0.5469−0.12700.09750.9575,K=0.289470.28716−0.042570.04728−0.145620.12654−0.291180.240090.27471,ΔAt=0.1cost−0.07sint0.02cost−0.05sint0.04cost−0.06sint0.1cost+0.01sint0.02cost0.04cost+0.03sint0.05cost+0.03sint0.01cost+0.01sint0.02cost+0.06sint,ΔCt=0.04cost−0.08sint0.04cost−0.03sint0.02cost−0.01sint0.04cost+0.04sint0.04cost+0.04sint0.02cost+0.03sint0.02cost+0.08sint0.02cost+0.07sint0.01cost+0.05sint,ΔDt=0.01sint0.04cost−0.09sint0.02sint0.04cost−0.03sint0.03sint0.02cost−0.03sint,ΔHt=0.06cost+0.01sint0.02cost−0.04sint0.04cost−0.02sint0.06cost+0.02sint0.02cost+0.02sint0.04cost+0.01sint0.03cost+0.03sint0.01cost+0.04sint0.02cost+0.02sint,ΔKt=0.02cost−0.01sint0.06cost−0.05sint0.04cost−0.02sint0.02cost+0.03sint0.06cost0.04cost+0.01sint0.01cost+0.05sint0.03cost+0.01sint0.02cost+0.02sint,ΔWt=0.04cost−0.1sint0.06cost−0.02sint0.08cost+0.01sint0.04cost0.06cost+0.01sint0.08cost+0.02sint0.02cost+0.02sint0.03cost+0.02sint0.04cost+0.03sint,ΔQ1t=0.02cost−0.02sint0.04cost−0.05sint0.04cost−0.08sint0.02cost+0.01sint0.04cost0.04cost−0.01sint0.01cost+0.02sint0.02cost+0.01sint0.02cost,ΔQ2t=0.02cost+0.02sint0.02cost−0.04sint0.04cost−0.08sint0.02cost+0.04sint0.02cost+0.02sint0.04cost−0.01sint0.01cost+0.06sint0.01cost+0.04sint0.02cost.
The memory state-feedback controllers are defined as follows:(59)u1t=γ1mt+γ3pt−τ^1t,u2t=γ2pt+γ4mt−τ^2t,where(60)γ1=0.04650.0457−0.0358−0.0342−0.0015−0.00780.04710.03000.0416,γ2=0.0195−0.04660.0266−0.0183−0.00610.02950.0450−0.0118−0.0313,γ3=−0.00100.02090.0180−0.00540.02550.01550.0146−0.0224−0.0337,γ4=−0.0381−0.01600.0251−0.00020.0085−0.02450.0460−0.02760.0006.
LetQ1=Q2=diag2,2,2,q=0.95,δ=1,ε=50,τ1t=τ2t=cost+1/4,τ∗=1/2, ϕ1t,ϕ2tT=0.1392,0.2734,0.4788,0.4824,0.0788,0.4853T−τ∗≤t≤0, L1=L2=1, Fx=Gx=x2/1+x2. According to the notations in Section 2, we obtain ϕ0=0.9641<1, σA=3,σD=1,σW=1.3710,σC=2.5,σH=0.3,σK=0.4793,σQ1=2, σQ2=2,η5=4.5081,ζ3=5.8292,ζ5=7.0183. When t<0.3339, simple computation reveals that(61)1+ζ3+ζ5tqΓq+1Eqζ3tq<1+5.8292+7.0183×0.33390.95Γ0.95+1Eq5.8292×0.33390.95<εδ=501.
From Theorem4, system (57) is finite-time stable with respect to 1,50,0,0.3339. Denote Te≈0.3339 as the “estimated time” of finite-time stability. The transient states of the variable mit and piti=1,2,3 of DFGRN (57) with q=0.95 and q=0.6 are shown in Figures 1(a) and 1(b), respectively.Figure 1
Transient states of DFGRN (57) with (a) q=0.95 and (b) q=0.6.
(a)(b)Example 2.
Consider the following DFGRNs of three mRNA and protein nodes with structured uncertainties and without controller:(62)DCtqmt=−A+ΔAtmt+W+ΔWtFpt+K+ΔKtGpt−τ1t+B,DCtqpt=−C+ΔCtpt+D+ΔDtmt+H+ΔHtmt−τ2t.
Using the same parameters in Example 1, we similarly getη4=4.3172,ζ1=5.3845,ζ5=7.0183. When t<0.3585, we have(63)1+ζ1+ζ5tqΓq+1Eqζ1tq<1+5.3845+7.0183×0.35850.95Γ0.95+1Eq5.3845×0.35850.95<εδ=501.
From Remark1, system (62) is finite-time stable with respect to 1,50,0,0.3585; then, the “estimated time” of finite-time stability Te≈0.3585. The transient states of the variables mit and piti=1,2,3 of DFGRN (62) with q=0.95 and q=0.6 are shown in Figures 2(a) and 2(b), respectively.
In Example2, when t⟶+∞, the case of infinite time, DFGRN (62) with structured uncertainties is unstable. The numerical simulations of the variables mit and piti=1,2,3 of DFGRN (62) with q=0.95 and q=0.6 are shown in Figures 3(a) and 3(b), respectively.Figure 2
Transient states of DFGRN (62) with (a) q=0.95 and (b) q=0.6.
(a)(b)Figure 3
Numerical simulations of DFGRN (62) with (a) q=0.95 and (b) q=0.6.
(a)(b)Remark 7.
It is worthy to note that in a special case of DFGRN (62) without structured uncertainties, it is proved that in the sense of infinite stability, (62) is globally asymptotically stable [16].Example 3.
Consider the following DFGRNs of three mRNA and protein nodes with memory state-feedback controllers and without structured uncertainties:(64)DCtqmt=−Amt+WFpt+KGpt−τ1t+B+Q1γ1mt+γ3pt−τ^1t,DCtqpt=−Cpt+Dmt+Hmt−τ2t+Q2γ2pt+γ4mt−τ^2t.
Using the same parameters in Example 1, we similarly obtainη5=4.1799,ζ3=5.2009,ζ5=7.0183. When t<0.3697, we can get(65)1+ζ3+ζ5tqΓq+1Eqζ3tq<1+5.2009+7.0183×0.36970.95Γ0.95+1Eq5.2009×0.36970.95<εδ=501.
From Theorem4, system (64) is finite-time stable with respect to 1,50,0,0.3697; then, the “estimated time” of finite-time stability Te≈0.3697. The transient states of the variables mit and piti=1,2,3 of DFGRN (64) with q=0.95 and q=0.6 are shown in Figures 4(a) and 4(b), respectively.Figure 4
Transient states of DFGRN (64) with (a) q=0.95 and (b) q=0.6.
(a)(b)Example 4.
Consider the following DFGRNs of three mRNA and protein nodes without structured uncertainties or controller:(66)DCtqmt=−Amt+WFpt+KGpt−τ1t+B,DCtqpt=−Cpt+Dmt+Hmt−τ2t.
Using the same parameters in Example1, we also obtain the “estimated time” of finite-time stability for system (66) as Te≈0.3984. The transient states of the variables mit and piti=1,2,3 of DFGRN (66) with q=0.95 and q=0.6 are shown in Figures 5(a) and 5(b), respectively.
If we adopt constant time-delayτ1t=τ2t=2 and q=0.4 in DFGRN (66), then system (66) is finite-time stable, and the “estimated time” of finite-time stability is 0.0315. The transient states of the variables mit and piti=1,2,3 of DFGRN (66) with q=0.4 are shown in Figure 6.Figure 5
Transient states of DFGRN (66) with (a) q=0.95 and (b) q=0.6.
(a)(b)Figure 6
Transient states of DFGRN (66) with q=0.40 and τ1t=τ2t=2.Remark 8.
Ifτ1t=τ2t=2 and q=0.4 in DFGRN (66), then system (66) converts to system (4.1) in [16]. When t⟶+∞, it is proved that system (4.1) is unstable in the sense of infinite-time stability [16], which means that the finite-time stability is different from the infinite-time stability of DFGRNs.
If we takeK=ΔKt=H=ΔHt=0 and γ3=γ4=0 in DFGRNs (57), (62), (64), and (66) systems (57), (62), (64), and (66) convert to the corresponding fractional-order gene regulatory networks without time delays (FGRNs).
In order to investigate the effects of structured uncertainties, controllers and time delays on the stability of the DFGRNs, we calculate the “estimated time”Te of finite-time stability for above four examples and the corresponding FGRNs with different fractional-order q; the results are shown in Tables 1 and 2, respectively.
From Table1 or Table 2, we have the following conclusions:(i)
The effect of the controllers: comparing column 2 with 3 (or column 4 with 5), we can know that the controllers can shorten the “estimated time” of finite-time stability under the same conditions of fractional-orderq and structured uncertainties.(ii)
The effect of the structured uncertainties: comparing column 3 with 5, we can know that the structured uncertainties can shorten the “estimated time” of finite-time stability under the same fractional-order itq.(iii)
The difference between the structured uncertainties and the controllers: comparing column 3 with 4, we can know that the size of “estimated time” of finite-time stability for DFGRN (62) with structured uncertainties is longer than DFGRN (64) with controllers under the same fractional-orderq.(iv)
The effect of the fractional-orderq: in the same column, we can know that decreasing the fractional-order q will be useful to decrease the “estimated time” of finite-time stability for DFGRNs or FGRNs.(v)
The effect of time delays: comparing Table1 with Table 2, we can know that the “estimated time” of finite-time stability is reduced under the same fractional-order q when considering time delays.Table 1
The “estimated time”Te of finite-time stability with different fractional-order q.
qDFGRN (57) TeDFGRN (62) TeDFGRN (64) TeDFGRN (66) Te0.950.33390.35850.36970.39840.850.26070.28240.29330.31920.750.19100.20930.21920.24160.600.09930.11160.11870.13440.500.05190.05980.06460.07510.450.03380.03960.04310.0511Table 2
The “estimated time”Te of finite-time stability without time delays.
qFGRN (57) TeFGRN (62) TeFGRN (64) TeFGRN (66) Te0.950.42040.43650.44770.46500.850.33820.35290.36440.38040.750.25750.27030.28130.29560.600.14520.15450.16320.17400.500.08240.08890.09520.10290.450.05660.06160.06660.0726


## 5. Concluding Remarks

This paper deals with the existence and uniqueness of the solution and the finite-time stability for a class of DFGRNs with structured uncertainties and controllers. In particular, we design the memory state-feedback controllers for DFGRNs with structured uncertainties and give the sufficient conditions for the system to achieve the finite-time stability.It should be pointed out that the conditions of finite-time stability in the present paper are dependent on the fractional-orderq, which is more different from the previous stability results for the case of integer order, i.e., the finite-time stability is independent of the integer order.In addition, from the numerical results, we find that all of the controllers, uncertain terms, fractional-orderq and time delays can affect the “estimated time” of finite-time stability. Particularly, (i) the size of “estimated time” of finite-time stability with controllers is shorter than the case without controller but only with structured uncertainties, which means that the controllers are more beneficial for controlling the “estimated time” than the structured uncertainties; (ii) the size of “estimated time” of finite-time stability with time delays is shorter than the case without time delays, which means that time delays degrade the GRN performance.If we takeΔAt=ΔWt=ΔKt=ΔCt=ΔDt=ΔHt=ΔQ1t=ΔQ2t=0 and controllers terms u1t=u2t=0, meanwhile, in the special case constant time delay, system (3) convert to (2.2) in [16], and we find that numerically: as t⟶+∞, DFGRN (62) in this paper is unstable; however, DFGRN (4.1) in [16] is globally asymptotically stable, which means that the structured uncertainty can change the stability of DFGRNs. Furthermore, from Remark 8, we know that DFGRN (66) is finite-time stable, while the corresponding system (4.1) in [16] is infinite-time unstable, which means that an infinite-time unstable system can change to a finite-time stable one under extra conditions. The analytical study on above questions is desirable in the future.



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