Research ArticleFinite-Time Stability of Fractional-Order Time-Varying DelaysGene Regulatory Networks with Structured Uncertaintiesand Controllers

Zhaohua Wu12 Zhiming Wang 12 and Tiejun Zhou 1

1College of Information and Intelligence Science Hunan Agricultural University Changsha 410128 China2Hunan Engineering Research Center for Information Technology in Agriculture Hunan Agricultural UniversityChangsha 410128 China

Correspondence should be addressed to Zhiming Wang wzmhunaueducn and Tiejun Zhou hntjzhou126com

Received 27 April 2020 Revised 1 July 2020 Accepted 17 July 2020 Published 31 August 2020

Academic Editor Xianming Zhang

Copyright copy 2020 ZhaohuaWu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we investigate a class of fractional-order time-varying delays gene regulatory networks with structureduncertainties and controllers (DFGRNs) Our contributions lie in three aspects first a necessary and sufficient condition onthe existence of the solution for the DFGRNs is given by using the properties of the RiemannndashLiouville fractional derivativeand Caputorsquos fractional derivative second the unique solution of the DFGRNs is proved under given initial function andcertain condition third some novel sufficient conditions on finite-time stability of the DFGRNs are established by usinga generalized Gronwall inequality and norm technique and some conclusions on the finite-time stability of the DFGRNswith memory state-feedback controllers are reached and those conditions and conclusions depend on the fractional order ofthe DFGRNs One of the most interesting findings is that the ldquoestimated timerdquo of the finite-time stability is indeed related tothe structured uncertainties state-feedback controllers time delays and the fractional order of the system

1 Introduction

Genetic regulatory networks (GRNs) which describe theinteraction functions in gene expressions between DNAsRNAs proteins and small molecules in an organism arefundamental and important biological networks e anal-ysis and control of GRNs involve two aspects first un-derstanding the widespread phenomena in living organismsand providing potential routes to prolong life span curecancer and diabetes and so on second potential applicationof GRNs in the development of related disciplines such assynthetic biology network medicine and personalizedmedicine [1ndash5]

With the development of sequencing technology moreandmore genes and their regulatory sites are discoveredestructures and functions of a large number of genes are

confirmed through experimental techniques and even theregulatorymechanisms of the gene expressions controlled bysome proteins are also identified [2] Some qualitativemodels were proposed for distinguishing these regulatorymechanisms such as directed graphs Boolean networksgeneralized logical networks and rule-based formalisms [2]However it is difficult to know the biochemical reactionmechanismsrsquo underlying regulatory interactions whenqualitatively handling great quantities of experimental dataerefore studying GRNs needs more accurately quanti-tative model Based on experimental data some simplegenetic networks have been constructed for example ge-netic repressilator network [5] negative feedback GRNs [6]and genetic switch network [7] e results in these ex-periments show that quantitative mathematical modelingapproaches on dynamical systems have been great tools in

HindawiComplexityVolume 2020 Article ID 2315272 19 pageshttpsdoiorg10115520202315272

providing insights on the mechanisms underlying thestructure and the behaviors of GRNs [8ndash12]

Using integer-order differential equation tomodel GRNsis a classical method However the fractional-order differ-ential equations are more suitable for modeling the generegulatory mechanism Ji et al [13] applied the particleswarm optimization technique in modeling the fractional-order GRNs with eight real target genes e experimentalresults confirmed that the fractional-order model hasachieved 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 thememory and hereditary properties of various processes inGRNs which could be far better than the integer-order ones[8ndash11]

Due to slow biochemical processes such as gene tran-scription translation and transportation (the synthesis ofmRNAs and proteins at nucleus and cytoplasm re-spectively in eukaryotic cells) time delays are omnipresentin GRNs [14] Many nonlinear differential equations withtime delays have been proposed to model general GRNsand the important role of time delays in dynamics of GRNsis now widely accepted [15ndash18] Actually time delays oftendegrade the system performance or destabilize the system[1 19 20] even GRN models without time delay maygenerate wrong predictions [21] As time delays oftenchange with time and their precise measurement is difficultin real GRNs the dynamics of fractional-order linear andnonlinear systems with time-varying delays has attractedincreasing interest and the results show that it is naturallyof better practical significance than those with constantdelays [21ndash25]

In addition in order to avoid undesirable states asso-ciated with disease the control of GRNs is often regarded asdeveloping therapeutic intervention strategies for somediseases [26 27] And many literatures focus on the researchof control in the dynamic system [28ndash33] In [32] the au-thors obtained some stabilization results for neural networkswith leakage delay by designing state-feedback controllerEbihara et al [33] discovered that exact robust control isindeed attained for discrete-time linear systems by designingperiodically time-varying memory state-feedback controllererefore it is necessary to consider the controller for theDFGRNs

Since the modeling of GRNs is underlined with the real-world gene expression time-series data some limitations ofthe current experimental techniques in GRNs make themodeling errors and parameter fluctuations unavoidableMoreover some point out that the system parametersidentified with the experimental data may construct anunknown but bounded time-varying function and thisunknown nature is referred to as the structural uncertaintyor the parametric uncertainty also known as variation orfluctuation [34] As is known the structural uncertainties inGRNs may lead to the poor performance or even instabilityin real genetic networks [28 34ndash37] In [28] the authors

studied the robust stabilization and state-feedback controllerdesign for a class of integer-order GRNs with time-varyingdelays (DGRNs) and structured uncertainties and estab-lished some delay-dependent stability results by using somematrix techniques erefore taking into account thestructural uncertainties while investigating the dynamicalbehaviors of DFGRNs is essential

Since the expression of gene and mRNA-translatedprotein is accomplished in a much relatively short periodin recent decades some scholars have paid more attentionto the finite-time stability of GRNs [4 38] For exampleWu et al [4] investigated the finite-time stability associatedwith a class of integer-order GRNs by designing adaptivecontrollers Wang et al [38] established some new sufficientconditions of the finite-time stability for a class of integer-order uncertain GRNs with time-varying delays Lazarevic[22] investigated the finite-time stability for fractional-order nonlinear differential equation with time-varyingdelays by using generalized Gronwall inequality and theclassical BellmanndashGronwall inequality respectively Phatand anh [23] established some new sufficient conditionsof robust finite-time stability for a class of nonlinearfractional-order differential systems with time-varyingdelays Wang et al [39] considered a class of nonlinearfractional-order systems with constant delays and studiedthe existence and uniqueness of the solution for this kind ofsystems by using relevant properties of the fractionalderivative

However the discussions on the existence anduniqueness of the solutions and the finite-time stabilityresults for the fractional-order uncertain GRNs with time-varying delays and controllers seem rare

From above discussions we focus on the existence anduniqueness of the solution and the finite-time stability fora class of DFGRNs with structured uncertainties and con-trollers e remainder of this paper is organized as followsIn Section 2 we give the model description some defini-tions and related properties on fractional calculus InSection 3 we discuss the existence and uniqueness of thesolution and give some sufficient criteria on the finite-timestability for the DFGRNs In Section 4 we perform somenumerical simulations which support our findings InSection 5 we briefly review and summarize the main results

2 Problem Description and Preliminaries

For any vector x(t) isin Rn and matrix A (aij)ntimesn isin Rntimesndenote

x(t) 1113944n

i1xi(t)

11138681113868111386811138681113868111386811138681113868

A max1lejlen 1113944

n

i1aij

11138681113868111386811138681113868

11138681113868111386811138681113868

(1)

Let σ(middot) be the largest singular value of matrix

2 Complexity

η1 max A + μ1 C + μ41113864 1113865

η2 max W + μ2( 1113857L1 D + μ51113864 1113865

η3 max K + μ3( 1113857L2 H + μ61113864 1113865

η4 max σ(A) + μ1 + σ(D) + μ5 σ(W) + μ2( 1113857L1 + σ(C) + μ41113864 1113865

η5 max σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5 σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113864 1113865

η6 max σ Q1( 1113857 + μ7 σ Q2( 1113857 + μ81113864 1113865

ζ1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6

ζ2 B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

ζ3 η5 + σ(K) + μ3( 1113857L2 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857 + σ(H) + μ6( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857

(2)

where μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8 L1 L2 are positive con-stants that satisfy the later assumptions (I) and (II)respectively

We will focus on a class of DFGRNs with structureduncertainties and controllers which is established as follows

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t))

+ (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t)

+ (H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(3)

where

m(t) m1(t) m2(t) mn(t)1113858 1113859T

p(t) p1(t) p2(t) pn(t)1113858 1113859T

F(p(t)) f1 p1(t)( 1113857 f2 p2(t)( 1113857 fn pn(t)( 11138571113858 1113859T

B B1 B2 Bn1113858 1113859T

A diag a1 a2 an1113864 1113865

C diag c1 c2 cn1113864 1113865

D diag d1 d2 dn1113864 1113865

H diag e1 e2 en1113864 1113865

G p t minus τ1(t)( 1113857( 1113857 g1 p1 t minus τ1(t)( 1113857( 1113857 g2 p2 t minus τ1(t)( 1113857( 1113857 gn pn t minus τ1(t)( 1113857( 11138571113858 1113859T

(4)

in which CDq

t represents Caputorsquos fractional derivative andq isin (0 1) mi(t) pi(t) isin R are the concentrations of mRNAand protein of the ith node respectively e parameters ai gt 0and ci gt 0 are the decay rates ofmRNA and protein respectivelydi gt 0 are the translation rates ei ge 0 are the translation ratesBoth fj(pj(t)) and gj(pj(t minus τ1(t))) represent the feedbackregulation of the protein on the transcription Generally eachone of the two functions is a nonlinear function but has a formofmonotonicity with its variable As a monotonic increasing or

decreasing regulatory function fj and gj are usually of theMichaelisndashMenten or Hill forms [21] Bi 1113936jisinIi

bij + 1113936jisinIibij

where bij and bij are bounded constants which are respectivelythe dimensionless transcriptional rates of transcription factor j

to i at time t and t minus τ1(t) and Ii Ii respectively are the set ofall the j where the transcription factor j is a repressor of gene i attime t and t minus τ1(t) W (wij) isin Rntimesn K (kij) isin Rntimesn arethe coupling matrices of the gene network which are defined asfollows

Complexity 3

wij kij1113872 1113873

bij bij1113872 1113873 if transcription factor j is an activator of gene i

minus bij minus bij1113872 1113873 if transcription factor j is a repressor of gene i

0 if there is no link fromnode j to i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(5)

e transcriptional delay τ1(t) and translational delayτ2(t) are bounded continuous functions on R with0le τi(t)le τlowast(i 1 2) here τlowast is a positive constantu1(t) (u11(t) u12(t) u1n(t))T u2(t) (u21(t)

u22(t) u2n(t))T are controller vectors Qi(i 1 2) are n

dimensions coefficient matrices ΔA(t) ΔD(t)ΔC(t)ΔW(t) ΔK(t) ΔH(t) ΔQ1(t)ΔQ2(t) are norm-boundedunknown matrices with time-varying structureduncertainties

e initial conditions for DFGRN (3) are as followsm(θ) ϕ1(θ) θ isin minus τlowast 0[ ]

p(θ) ϕ2(θ) θ isin minus τlowast 0[ ]1113896 (6)

where ϕi(t) isin C([minus τlowast 0] Rn)(i 1 2) is the given initialfunction with ϕic supminus τlowastleθ le0ϕi(θ)(i 1 2) andϕ0 ϕ1c + ϕ2c

(i) Assumption (I) the norm-bounded unknown ma-trices satisfy the following inequalities

ΔA(t)le μ1

ΔW(t)le μ2

ΔK(t)le μ3

ΔC(t)le μ4

ΔD(t)le μ5

ΔH(t)le μ6

ΔQ1(t)

le μ7

ΔQ2(t)

le μ8

(7)

where μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8 are positive constants

(ii) Assumption (II) the functions F G satisfy the fol-lowing inequalities

F(x) minus F(y)le L1x minus y

G(x) minus G(y)le L2x minus y x y isin Rn

(8)

where L1 L2 are positive constantsNext we give some definitions and lemmas

Definition 1 (see [40]) e fractional integral of order q fora function f(t) is defined as

aIqt f(t)

1Γ(q)

1113946t

a(t minus τ)

qminus 1f(τ)dτ (9)

where tge a a isin R qgt 0e gamma function Γ(q) is definedby the integral Γ(q) 1113938

infin0 tqminus 1eminus tdt

Definition 2 (see [40]) Caputorsquos fractional derivative oforder q for a function f is defined by

Ca D

q

t f(t) 1Γ(n minus q)

1113946t

a

1(t minus τ)qminus n+1f

(n)(τ)dτ (10)

where tge a and n is a positive integer such that n minus 1lt qlt n

Definition 3 (see [40]) e RiemannndashLiouville fractionalderivative of order q for a function f is defined as

RLa D

q

t f(t) 1Γ(n minus q)

dn

dtn1113946

t

a(t minus s)

nminus qminus 1f(s)ds (11)

where tge a and n is a positive integer such that n minus 1lt qlt nFor convenience we choose the notation I

qt 0I

qt

CDq

t C0 D

q

t RLD

q

t RL0 D

q

t

Definition 4 A mild solution of DFGRN (3) with initialcondition (6) is a vector (m(t) p(t))T composed of con-tinuous functions

m(t) p(t) minus τlowastinfin1113858 1113857⟶ Rn (12)

satisfying

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

m(θ) ϕ1(θ) θ isin minus τlowast 0[ ]

p(θ) ϕ2(θ) θ isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

4 Complexity

Definition 5 (see [22]) e system given by (3) (whenQi 0ΔQi 0 i 1 2) satisfying the initial condition (6) isfinite-time stable with respect to δ ε t0 J1113864 1113865 δ lt ε if and onlyif ϕ0 lt δ imply m(t) + p(t)lt εforallt isin J J sub R

Definition 6 (see [22]) e system given by (3) satisfying theinitial condition (6) is finite-time stable with respect toδ ε α1 t0 J1113864 1113865 δ lt ε if and only if ϕ0lt δ and

u1(t) + u2(t)lt α1forallt isin Jimply m(t) + p(t)lt εforallt isin J J sub R where α1 is a positive constant

Lemma 1 (see [40]) If f(t) isin Cn([0infin)) and n minus 1lt αltn isin Z+ then

(i) Iqt [CD

q

t f(t)] f(t) minus 1113936nminus 1k0f

(k)(0)(tkk)(ii) RLD

q

t [Iqt f(t)] f(t)

(iii) RLDq

t f(t) CDq

t f(t) + 1113936nminus 1k0(tkminus q

Γ(k + 1 minus q))f(k)(0)

Lemma 2 (see [41]) Suppose βgt 0 if 0le tltT (someTle +infin) a(t) is a locally integrable nonnegative function v(t)

is a nonnegative and nondecreasing continuous function v(t)

leM (constant) and u(t) is a nonnegative and locally integrablefunction with u(t)le a(t) + v(t) 1113938

t

0 (tminus s)βminus 1u(s)ds then

u(t)le a(t) + 1113946t

01113944

infin

n1

(v(t)Γ(β))n

Γ(nβ)(t minus s)

nβminus 1a(s)⎡⎣ ⎤⎦ds forall0le tleT

(14)

In addition if a(t) is a nondecreasing function thenu(t)le a(t)Eβ(v(t)Γ(β)tβ) where Eβ is the Mittag-Lefflerfunction defined by Eβ(z) 1113936

infink0(zkΓ(kβ + 1))

3 Main Results

31 e Existence and Uniqueness of the Mild Solution ofDFGRNs

Theorem 1 Continuously differentiable functionsm(t) p(t) [minus τlowast T]⟶ Rn(Tlt +infin) form a mild solution(m(t) p(t))T to DFGRN (3) with initial condition (6) if andonly if

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds for t isin [0 T]

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds for t isin [0 T]

m(t) ϕ1(t) for t isin minus τlowast 0[ ]

p(t) ϕ2(t) for t isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

Proof We firstly give the sufficient condition of the exis-tence of the mild solution to DFGRN (3)

When minus τlowast le tle 0 (m(t) p(t))T (ϕ1(t) ϕ2(t))T isobvious For 0le tleT according to (15) applying RLD

q

t andproperty (ii) of Lemma 1 we obtain

RLDq

t m(t) ϕ1(0)tminus q

Γ(1 minus q)minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

RLDq

t p(t) ϕ2(0)tminus q

Γ(1 minus q)minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Complexity 5

According to the property (iii) of Lemma 1 and 0lt qlt 1we get

RLDq

t m(t) CDq

t m(t) + ϕ1(0)tminus q

Γ(1 minus q)

RLDq

t p(t) CDq

t p(t) + ϕ2(0)tminus q

Γ(1 minus q)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

From (16) and (17) we haveCD

q

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

We secondly give the necessary condition of the exis-tence of the mild solution to DFGRN (3)

When t isin [minus τlowast 0] the solution of DFGRN (3) is

m(t) ϕ1(t)

p(t) ϕ2(t)

t isin minus τlowast 01113858 1113859

(19)

If 0le tleT from DFGRN (3) we have

Iqt

CDq

t m(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

Iqt

CDq

t p(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(20)

In the case of 0lt qlt 1 from property (i) of Lemma 1 wecan obtain

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

e proof is completed

Theorem 2 If assumptions (I) and (II) hold then DFGRN(3) with initial condition (6) has a unique mild solution

Proof Let (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T be any twodifferent solutions to DFGRN (3) with initial condition (6)

denote x(t) m(t) minus 1113957m(t) y(t) p(t) minus 1113957p(t) z(t)

x(t) + y(t) t isin [minus τlowast T] According to eorem 1 weknow that both (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T satisfycondition (15)

If 0le tleT then

x(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))x(s) +(W + ΔW(s))(F(p(s)) minus F(1113957p(s))) +(K + ΔK(s)) G p s minus τ1(s)( 1113857( 1113857 minus G 1113957p s minus τ1(s)( 1113857( 1113857( 1113857( 1113857ds

y(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))y(s) +(D + ΔD(s))x(s) +(H + ΔH(s))x s minus τ2(s)( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(22)

6 Complexity

From (22) by using the norm (middot) and assumptions (I)and (II) we can obtain

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s) + K + μ3( 1113857 middot L2 middot y s minus τ1(s)( 1113857

1113960 1113961ds t isin [0 T]

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s) + H + μ6( 1113857 middot x s minus τ2(s)( 1113857

1113960 1113961ds t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(23)

First when t isin [minus τlowast 0] x(t) ϕ1(θ) minus ϕ1(θ) 0

y(t) ϕ2(θ) minus ϕ2(θ) 0 So m(t) 1113957m(t) p(t) 1113957p(t) fort isin [minus τlowast 0]

Second when t isin (0 τlowast] from (23) we have

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s)1113858 1113859ds

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s)1113858 1113859ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

In accordance with (24) we can obtain

z(t) x(t) +y(t)

le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s)( 1113857ds

η1 + η2Γ(q)

1113946t

0(t minus s)

qminus 1z(s)ds

(25)

From Lemma 2 we can get

z(t)le 0 middot Eq

η1 + η2Γ(q)

middot Γ(q)tq

1113890 1113891 t isin 0 τlowast( 1113859 (26)

us z(t)le 0 t isin (0 τlowast] at is to say x(t)+

y(t)le 0 t isin (0 τlowast] So m(t) 1113957m(t) p(t) 1113957p(t) fort isin (0 τlowast]

ird when t isin (τlowast T] according to (23) we have

z(t) x(t) +y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s) + η3 x s minus τ2(s)( 1113857

+ y s minus τ1(s)( 1113857

1113872 11138731113960 1113961ds t isin τlowast T( 1113859 (27)

Let zlowast(t) supθ1113957θisin[minus τlowast 0]

[x(t + θ) + y(t + 1113957θ)] enwe obtain

zlowast(t)le

η1 + η2 + η3Γ(q)

1113946t

0(t minus s)

qminus 1zlowast(s)ds t isin τlowast T( 1113859

(28)

From Lemma 2 we can get

z(t)le zlowast(t)le 0 middot Eq

η1 + η2 + η3Γ(q)

middot Γ(q) middot tq

1113888 1113889 t isin τlowast T( 1113859

(29)

en we have x(t) + y(t) le 0 t isin (τlowast T] So m(t)

1113957m(t) p(t) 1113957p(t) for t isin (τlowast T]

Summarizing the above three cases we can obtain thatm(t) 1113957m(t) p(t) 1113957p(t) for t isin [minus τlowast T] Due to the ar-bitrary nature of the solution (m(t) p(t))T and( 1113957m(t) 1113957p(t))T of DFGRN (3) and in accordance with Def-inition 4 we can conclude that DFGRN (3) has a uniquemild solution e proof is completed

32 Finite-Time Stability of DFGRNs withStructured Uncertainties

Theorem 3 If assumptions (I) and (II) and [1 + ((ζ1+ζ4)tqΓ(q + 1))] middot Eq(ζ1tq)le (εδ)forallt isin J0 [0 T] holdthen the uncertain DFGRNs with controllers given by (3) withinitial condition (6) are finite-time stable with respect toδ ε α1 J01113864 1113865 δ lt ε where ζ4 ≔ ((η6α1 + ζ2)δ)

Complexity 7

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

η1 max A + μ1 C + μ41113864 1113865

η2 max W + μ2( 1113857L1 D + μ51113864 1113865

η3 max K + μ3( 1113857L2 H + μ61113864 1113865

η4 max σ(A) + μ1 + σ(D) + μ5 σ(W) + μ2( 1113857L1 + σ(C) + μ41113864 1113865

η5 max σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5 σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113864 1113865

η6 max σ Q1( 1113857 + μ7 σ Q2( 1113857 + μ81113864 1113865

ζ1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6

ζ2 B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

ζ3 η5 + σ(K) + μ3( 1113857L2 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857 + σ(H) + μ6( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857

(2)

where μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8 L1 L2 are positive con-stants that satisfy the later assumptions (I) and (II)respectively

We will focus on a class of DFGRNs with structureduncertainties and controllers which is established as follows

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t))

+ (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t)

+ (H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(3)

where

m(t) m1(t) m2(t) mn(t)1113858 1113859T

p(t) p1(t) p2(t) pn(t)1113858 1113859T

F(p(t)) f1 p1(t)( 1113857 f2 p2(t)( 1113857 fn pn(t)( 11138571113858 1113859T

B B1 B2 Bn1113858 1113859T

A diag a1 a2 an1113864 1113865

C diag c1 c2 cn1113864 1113865

D diag d1 d2 dn1113864 1113865

H diag e1 e2 en1113864 1113865

G p t minus τ1(t)( 1113857( 1113857 g1 p1 t minus τ1(t)( 1113857( 1113857 g2 p2 t minus τ1(t)( 1113857( 1113857 gn pn t minus τ1(t)( 1113857( 11138571113858 1113859T

(4)

in which CDq

t represents Caputorsquos fractional derivative andq isin (0 1) mi(t) pi(t) isin R are the concentrations of mRNAand protein of the ith node respectively e parameters ai gt 0and ci gt 0 are the decay rates ofmRNA and protein respectivelydi gt 0 are the translation rates ei ge 0 are the translation ratesBoth fj(pj(t)) and gj(pj(t minus τ1(t))) represent the feedbackregulation of the protein on the transcription Generally eachone of the two functions is a nonlinear function but has a formofmonotonicity with its variable As a monotonic increasing or

decreasing regulatory function fj and gj are usually of theMichaelisndashMenten or Hill forms [21] Bi 1113936jisinIi

bij + 1113936jisinIibij

where bij and bij are bounded constants which are respectivelythe dimensionless transcriptional rates of transcription factor j

to i at time t and t minus τ1(t) and Ii Ii respectively are the set ofall the j where the transcription factor j is a repressor of gene i attime t and t minus τ1(t) W (wij) isin Rntimesn K (kij) isin Rntimesn arethe coupling matrices of the gene network which are defined asfollows

Complexity 3

wij kij1113872 1113873

bij bij1113872 1113873 if transcription factor j is an activator of gene i

minus bij minus bij1113872 1113873 if transcription factor j is a repressor of gene i

0 if there is no link fromnode j to i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(5)

e transcriptional delay τ1(t) and translational delayτ2(t) are bounded continuous functions on R with0le τi(t)le τlowast(i 1 2) here τlowast is a positive constantu1(t) (u11(t) u12(t) u1n(t))T u2(t) (u21(t)

u22(t) u2n(t))T are controller vectors Qi(i 1 2) are n

dimensions coefficient matrices ΔA(t) ΔD(t)ΔC(t)ΔW(t) ΔK(t) ΔH(t) ΔQ1(t)ΔQ2(t) are norm-boundedunknown matrices with time-varying structureduncertainties

e initial conditions for DFGRN (3) are as followsm(θ) ϕ1(θ) θ isin minus τlowast 0[ ]

p(θ) ϕ2(θ) θ isin minus τlowast 0[ ]1113896 (6)

where ϕi(t) isin C([minus τlowast 0] Rn)(i 1 2) is the given initialfunction with ϕic supminus τlowastleθ le0ϕi(θ)(i 1 2) andϕ0 ϕ1c + ϕ2c

(i) Assumption (I) the norm-bounded unknown ma-trices satisfy the following inequalities

ΔA(t)le μ1

ΔW(t)le μ2

ΔK(t)le μ3

ΔC(t)le μ4

ΔD(t)le μ5

ΔH(t)le μ6

ΔQ1(t)

le μ7

ΔQ2(t)

le μ8

(7)

where μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8 are positive constants

(ii) Assumption (II) the functions F G satisfy the fol-lowing inequalities

F(x) minus F(y)le L1x minus y

G(x) minus G(y)le L2x minus y x y isin Rn

(8)

where L1 L2 are positive constantsNext we give some definitions and lemmas

Definition 1 (see [40]) e fractional integral of order q fora function f(t) is defined as

aIqt f(t)

1Γ(q)

1113946t

a(t minus τ)

qminus 1f(τ)dτ (9)

where tge a a isin R qgt 0e gamma function Γ(q) is definedby the integral Γ(q) 1113938

infin0 tqminus 1eminus tdt

Definition 2 (see [40]) Caputorsquos fractional derivative oforder q for a function f is defined by

Ca D

q

t f(t) 1Γ(n minus q)

1113946t

a

1(t minus τ)qminus n+1f

(n)(τ)dτ (10)

where tge a and n is a positive integer such that n minus 1lt qlt n

Definition 3 (see [40]) e RiemannndashLiouville fractionalderivative of order q for a function f is defined as

RLa D

q

t f(t) 1Γ(n minus q)

dn

dtn1113946

t

a(t minus s)

nminus qminus 1f(s)ds (11)

where tge a and n is a positive integer such that n minus 1lt qlt nFor convenience we choose the notation I

qt 0I

qt

CDq

t C0 D

q

t RLD

q

t RL0 D

q

t

Definition 4 A mild solution of DFGRN (3) with initialcondition (6) is a vector (m(t) p(t))T composed of con-tinuous functions

m(t) p(t) minus τlowastinfin1113858 1113857⟶ Rn (12)

satisfying

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

m(θ) ϕ1(θ) θ isin minus τlowast 0[ ]

p(θ) ϕ2(θ) θ isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

4 Complexity

Definition 5 (see [22]) e system given by (3) (whenQi 0ΔQi 0 i 1 2) satisfying the initial condition (6) isfinite-time stable with respect to δ ε t0 J1113864 1113865 δ lt ε if and onlyif ϕ0 lt δ imply m(t) + p(t)lt εforallt isin J J sub R

Definition 6 (see [22]) e system given by (3) satisfying theinitial condition (6) is finite-time stable with respect toδ ε α1 t0 J1113864 1113865 δ lt ε if and only if ϕ0lt δ and

u1(t) + u2(t)lt α1forallt isin Jimply m(t) + p(t)lt εforallt isin J J sub R where α1 is a positive constant

Lemma 1 (see [40]) If f(t) isin Cn([0infin)) and n minus 1lt αltn isin Z+ then

(i) Iqt [CD

q

t f(t)] f(t) minus 1113936nminus 1k0f

(k)(0)(tkk)(ii) RLD

q

t [Iqt f(t)] f(t)

(iii) RLDq

t f(t) CDq

t f(t) + 1113936nminus 1k0(tkminus q

Γ(k + 1 minus q))f(k)(0)

Lemma 2 (see [41]) Suppose βgt 0 if 0le tltT (someTle +infin) a(t) is a locally integrable nonnegative function v(t)

is a nonnegative and nondecreasing continuous function v(t)

leM (constant) and u(t) is a nonnegative and locally integrablefunction with u(t)le a(t) + v(t) 1113938

t

0 (tminus s)βminus 1u(s)ds then

u(t)le a(t) + 1113946t

01113944

infin

n1

(v(t)Γ(β))n

Γ(nβ)(t minus s)

nβminus 1a(s)⎡⎣ ⎤⎦ds forall0le tleT

(14)

In addition if a(t) is a nondecreasing function thenu(t)le a(t)Eβ(v(t)Γ(β)tβ) where Eβ is the Mittag-Lefflerfunction defined by Eβ(z) 1113936

infink0(zkΓ(kβ + 1))

3 Main Results

31 e Existence and Uniqueness of the Mild Solution ofDFGRNs

Theorem 1 Continuously differentiable functionsm(t) p(t) [minus τlowast T]⟶ Rn(Tlt +infin) form a mild solution(m(t) p(t))T to DFGRN (3) with initial condition (6) if andonly if

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds for t isin [0 T]

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds for t isin [0 T]

m(t) ϕ1(t) for t isin minus τlowast 0[ ]

p(t) ϕ2(t) for t isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

Proof We firstly give the sufficient condition of the exis-tence of the mild solution to DFGRN (3)

When minus τlowast le tle 0 (m(t) p(t))T (ϕ1(t) ϕ2(t))T isobvious For 0le tleT according to (15) applying RLD

q

t andproperty (ii) of Lemma 1 we obtain

RLDq

t m(t) ϕ1(0)tminus q

Γ(1 minus q)minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

RLDq

t p(t) ϕ2(0)tminus q

Γ(1 minus q)minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Complexity 5

According to the property (iii) of Lemma 1 and 0lt qlt 1we get

RLDq

t m(t) CDq

t m(t) + ϕ1(0)tminus q

Γ(1 minus q)

RLDq

t p(t) CDq

t p(t) + ϕ2(0)tminus q

Γ(1 minus q)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

From (16) and (17) we haveCD

q

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

We secondly give the necessary condition of the exis-tence of the mild solution to DFGRN (3)

When t isin [minus τlowast 0] the solution of DFGRN (3) is

m(t) ϕ1(t)

p(t) ϕ2(t)

t isin minus τlowast 01113858 1113859

(19)

If 0le tleT from DFGRN (3) we have

Iqt

CDq

t m(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

Iqt

CDq

t p(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(20)

In the case of 0lt qlt 1 from property (i) of Lemma 1 wecan obtain

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

e proof is completed

Theorem 2 If assumptions (I) and (II) hold then DFGRN(3) with initial condition (6) has a unique mild solution

Proof Let (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T be any twodifferent solutions to DFGRN (3) with initial condition (6)

denote x(t) m(t) minus 1113957m(t) y(t) p(t) minus 1113957p(t) z(t)

x(t) + y(t) t isin [minus τlowast T] According to eorem 1 weknow that both (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T satisfycondition (15)

If 0le tleT then

x(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))x(s) +(W + ΔW(s))(F(p(s)) minus F(1113957p(s))) +(K + ΔK(s)) G p s minus τ1(s)( 1113857( 1113857 minus G 1113957p s minus τ1(s)( 1113857( 1113857( 1113857( 1113857ds

y(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))y(s) +(D + ΔD(s))x(s) +(H + ΔH(s))x s minus τ2(s)( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(22)

6 Complexity

From (22) by using the norm (middot) and assumptions (I)and (II) we can obtain

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s) + K + μ3( 1113857 middot L2 middot y s minus τ1(s)( 1113857

1113960 1113961ds t isin [0 T]

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s) + H + μ6( 1113857 middot x s minus τ2(s)( 1113857

1113960 1113961ds t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(23)

First when t isin [minus τlowast 0] x(t) ϕ1(θ) minus ϕ1(θ) 0

y(t) ϕ2(θ) minus ϕ2(θ) 0 So m(t) 1113957m(t) p(t) 1113957p(t) fort isin [minus τlowast 0]

Second when t isin (0 τlowast] from (23) we have

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s)1113858 1113859ds

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s)1113858 1113859ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

In accordance with (24) we can obtain

z(t) x(t) +y(t)

le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s)( 1113857ds

η1 + η2Γ(q)

1113946t

0(t minus s)

qminus 1z(s)ds

(25)

From Lemma 2 we can get

z(t)le 0 middot Eq

η1 + η2Γ(q)

middot Γ(q)tq

1113890 1113891 t isin 0 τlowast( 1113859 (26)

us z(t)le 0 t isin (0 τlowast] at is to say x(t)+

y(t)le 0 t isin (0 τlowast] So m(t) 1113957m(t) p(t) 1113957p(t) fort isin (0 τlowast]

ird when t isin (τlowast T] according to (23) we have

z(t) x(t) +y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s) + η3 x s minus τ2(s)( 1113857

+ y s minus τ1(s)( 1113857

1113872 11138731113960 1113961ds t isin τlowast T( 1113859 (27)

Let zlowast(t) supθ1113957θisin[minus τlowast 0]

[x(t + θ) + y(t + 1113957θ)] enwe obtain

zlowast(t)le

η1 + η2 + η3Γ(q)

1113946t

0(t minus s)

qminus 1zlowast(s)ds t isin τlowast T( 1113859

(28)

From Lemma 2 we can get

z(t)le zlowast(t)le 0 middot Eq

η1 + η2 + η3Γ(q)

middot Γ(q) middot tq

1113888 1113889 t isin τlowast T( 1113859

(29)

en we have x(t) + y(t) le 0 t isin (τlowast T] So m(t)

1113957m(t) p(t) 1113957p(t) for t isin (τlowast T]

Summarizing the above three cases we can obtain thatm(t) 1113957m(t) p(t) 1113957p(t) for t isin [minus τlowast T] Due to the ar-bitrary nature of the solution (m(t) p(t))T and( 1113957m(t) 1113957p(t))T of DFGRN (3) and in accordance with Def-inition 4 we can conclude that DFGRN (3) has a uniquemild solution e proof is completed

32 Finite-Time Stability of DFGRNs withStructured Uncertainties

Theorem 3 If assumptions (I) and (II) and [1 + ((ζ1+ζ4)tqΓ(q + 1))] middot Eq(ζ1tq)le (εδ)forallt isin J0 [0 T] holdthen the uncertain DFGRNs with controllers given by (3) withinitial condition (6) are finite-time stable with respect toδ ε α1 J01113864 1113865 δ lt ε where ζ4 ≔ ((η6α1 + ζ2)δ)

Complexity 7

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

wij kij1113872 1113873

bij bij1113872 1113873 if transcription factor j is an activator of gene i

minus bij minus bij1113872 1113873 if transcription factor j is a repressor of gene i

0 if there is no link fromnode j to i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(5)

e transcriptional delay τ1(t) and translational delayτ2(t) are bounded continuous functions on R with0le τi(t)le τlowast(i 1 2) here τlowast is a positive constantu1(t) (u11(t) u12(t) u1n(t))T u2(t) (u21(t)

u22(t) u2n(t))T are controller vectors Qi(i 1 2) are n

dimensions coefficient matrices ΔA(t) ΔD(t)ΔC(t)ΔW(t) ΔK(t) ΔH(t) ΔQ1(t)ΔQ2(t) are norm-boundedunknown matrices with time-varying structureduncertainties

e initial conditions for DFGRN (3) are as followsm(θ) ϕ1(θ) θ isin minus τlowast 0[ ]

p(θ) ϕ2(θ) θ isin minus τlowast 0[ ]1113896 (6)

where ϕi(t) isin C([minus τlowast 0] Rn)(i 1 2) is the given initialfunction with ϕic supminus τlowastleθ le0ϕi(θ)(i 1 2) andϕ0 ϕ1c + ϕ2c

(i) Assumption (I) the norm-bounded unknown ma-trices satisfy the following inequalities

ΔA(t)le μ1

ΔW(t)le μ2

ΔK(t)le μ3

ΔC(t)le μ4

ΔD(t)le μ5

ΔH(t)le μ6

ΔQ1(t)

le μ7

ΔQ2(t)

le μ8

(7)

where μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8 are positive constants

(ii) Assumption (II) the functions F G satisfy the fol-lowing inequalities

F(x) minus F(y)le L1x minus y

G(x) minus G(y)le L2x minus y x y isin Rn

(8)

where L1 L2 are positive constantsNext we give some definitions and lemmas

Definition 1 (see [40]) e fractional integral of order q fora function f(t) is defined as

aIqt f(t)

1Γ(q)

1113946t

a(t minus τ)

qminus 1f(τ)dτ (9)

where tge a a isin R qgt 0e gamma function Γ(q) is definedby the integral Γ(q) 1113938

infin0 tqminus 1eminus tdt

Definition 2 (see [40]) Caputorsquos fractional derivative oforder q for a function f is defined by

Ca D

q

t f(t) 1Γ(n minus q)

1113946t

a

1(t minus τ)qminus n+1f

(n)(τ)dτ (10)

where tge a and n is a positive integer such that n minus 1lt qlt n

Definition 3 (see [40]) e RiemannndashLiouville fractionalderivative of order q for a function f is defined as

RLa D

q

t f(t) 1Γ(n minus q)

dn

dtn1113946

t

a(t minus s)

nminus qminus 1f(s)ds (11)

where tge a and n is a positive integer such that n minus 1lt qlt nFor convenience we choose the notation I

qt 0I

qt

CDq

t C0 D

q

t RLD

q

t RL0 D

q

t

Definition 4 A mild solution of DFGRN (3) with initialcondition (6) is a vector (m(t) p(t))T composed of con-tinuous functions

m(t) p(t) minus τlowastinfin1113858 1113857⟶ Rn (12)

satisfying

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

m(θ) ϕ1(θ) θ isin minus τlowast 0[ ]

p(θ) ϕ2(θ) θ isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

4 Complexity

Definition 5 (see [22]) e system given by (3) (whenQi 0ΔQi 0 i 1 2) satisfying the initial condition (6) isfinite-time stable with respect to δ ε t0 J1113864 1113865 δ lt ε if and onlyif ϕ0 lt δ imply m(t) + p(t)lt εforallt isin J J sub R

Definition 6 (see [22]) e system given by (3) satisfying theinitial condition (6) is finite-time stable with respect toδ ε α1 t0 J1113864 1113865 δ lt ε if and only if ϕ0lt δ and

u1(t) + u2(t)lt α1forallt isin Jimply m(t) + p(t)lt εforallt isin J J sub R where α1 is a positive constant

Lemma 1 (see [40]) If f(t) isin Cn([0infin)) and n minus 1lt αltn isin Z+ then

(i) Iqt [CD

q

t f(t)] f(t) minus 1113936nminus 1k0f

(k)(0)(tkk)(ii) RLD

q

t [Iqt f(t)] f(t)

(iii) RLDq

t f(t) CDq

t f(t) + 1113936nminus 1k0(tkminus q

Γ(k + 1 minus q))f(k)(0)

Lemma 2 (see [41]) Suppose βgt 0 if 0le tltT (someTle +infin) a(t) is a locally integrable nonnegative function v(t)

is a nonnegative and nondecreasing continuous function v(t)

leM (constant) and u(t) is a nonnegative and locally integrablefunction with u(t)le a(t) + v(t) 1113938

t

0 (tminus s)βminus 1u(s)ds then

u(t)le a(t) + 1113946t

01113944

infin

n1

(v(t)Γ(β))n

Γ(nβ)(t minus s)

nβminus 1a(s)⎡⎣ ⎤⎦ds forall0le tleT

(14)

In addition if a(t) is a nondecreasing function thenu(t)le a(t)Eβ(v(t)Γ(β)tβ) where Eβ is the Mittag-Lefflerfunction defined by Eβ(z) 1113936

infink0(zkΓ(kβ + 1))

3 Main Results

31 e Existence and Uniqueness of the Mild Solution ofDFGRNs

Theorem 1 Continuously differentiable functionsm(t) p(t) [minus τlowast T]⟶ Rn(Tlt +infin) form a mild solution(m(t) p(t))T to DFGRN (3) with initial condition (6) if andonly if

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds for t isin [0 T]

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds for t isin [0 T]

m(t) ϕ1(t) for t isin minus τlowast 0[ ]

p(t) ϕ2(t) for t isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

Proof We firstly give the sufficient condition of the exis-tence of the mild solution to DFGRN (3)

When minus τlowast le tle 0 (m(t) p(t))T (ϕ1(t) ϕ2(t))T isobvious For 0le tleT according to (15) applying RLD

q

t andproperty (ii) of Lemma 1 we obtain

RLDq

t m(t) ϕ1(0)tminus q

Γ(1 minus q)minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

RLDq

t p(t) ϕ2(0)tminus q

Γ(1 minus q)minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Complexity 5

According to the property (iii) of Lemma 1 and 0lt qlt 1we get

RLDq

t m(t) CDq

t m(t) + ϕ1(0)tminus q

Γ(1 minus q)

RLDq

t p(t) CDq

t p(t) + ϕ2(0)tminus q

Γ(1 minus q)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

From (16) and (17) we haveCD

q

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

We secondly give the necessary condition of the exis-tence of the mild solution to DFGRN (3)

When t isin [minus τlowast 0] the solution of DFGRN (3) is

m(t) ϕ1(t)

p(t) ϕ2(t)

t isin minus τlowast 01113858 1113859

(19)

If 0le tleT from DFGRN (3) we have

Iqt

CDq

t m(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

Iqt

CDq

t p(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(20)

In the case of 0lt qlt 1 from property (i) of Lemma 1 wecan obtain

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

e proof is completed

Theorem 2 If assumptions (I) and (II) hold then DFGRN(3) with initial condition (6) has a unique mild solution

Proof Let (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T be any twodifferent solutions to DFGRN (3) with initial condition (6)

denote x(t) m(t) minus 1113957m(t) y(t) p(t) minus 1113957p(t) z(t)

x(t) + y(t) t isin [minus τlowast T] According to eorem 1 weknow that both (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T satisfycondition (15)

If 0le tleT then

x(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))x(s) +(W + ΔW(s))(F(p(s)) minus F(1113957p(s))) +(K + ΔK(s)) G p s minus τ1(s)( 1113857( 1113857 minus G 1113957p s minus τ1(s)( 1113857( 1113857( 1113857( 1113857ds

y(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))y(s) +(D + ΔD(s))x(s) +(H + ΔH(s))x s minus τ2(s)( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(22)

6 Complexity

From (22) by using the norm (middot) and assumptions (I)and (II) we can obtain

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s) + K + μ3( 1113857 middot L2 middot y s minus τ1(s)( 1113857

1113960 1113961ds t isin [0 T]

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s) + H + μ6( 1113857 middot x s minus τ2(s)( 1113857

1113960 1113961ds t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(23)

First when t isin [minus τlowast 0] x(t) ϕ1(θ) minus ϕ1(θ) 0

y(t) ϕ2(θ) minus ϕ2(θ) 0 So m(t) 1113957m(t) p(t) 1113957p(t) fort isin [minus τlowast 0]

Second when t isin (0 τlowast] from (23) we have

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s)1113858 1113859ds

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s)1113858 1113859ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

In accordance with (24) we can obtain

z(t) x(t) +y(t)

le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s)( 1113857ds

η1 + η2Γ(q)

1113946t

0(t minus s)

qminus 1z(s)ds

(25)

From Lemma 2 we can get

z(t)le 0 middot Eq

η1 + η2Γ(q)

middot Γ(q)tq

1113890 1113891 t isin 0 τlowast( 1113859 (26)

us z(t)le 0 t isin (0 τlowast] at is to say x(t)+

y(t)le 0 t isin (0 τlowast] So m(t) 1113957m(t) p(t) 1113957p(t) fort isin (0 τlowast]

ird when t isin (τlowast T] according to (23) we have

z(t) x(t) +y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s) + η3 x s minus τ2(s)( 1113857

+ y s minus τ1(s)( 1113857

1113872 11138731113960 1113961ds t isin τlowast T( 1113859 (27)

Let zlowast(t) supθ1113957θisin[minus τlowast 0]

[x(t + θ) + y(t + 1113957θ)] enwe obtain

zlowast(t)le

η1 + η2 + η3Γ(q)

1113946t

0(t minus s)

qminus 1zlowast(s)ds t isin τlowast T( 1113859

(28)

From Lemma 2 we can get

z(t)le zlowast(t)le 0 middot Eq

η1 + η2 + η3Γ(q)

middot Γ(q) middot tq

1113888 1113889 t isin τlowast T( 1113859

(29)

en we have x(t) + y(t) le 0 t isin (τlowast T] So m(t)

1113957m(t) p(t) 1113957p(t) for t isin (τlowast T]

Summarizing the above three cases we can obtain thatm(t) 1113957m(t) p(t) 1113957p(t) for t isin [minus τlowast T] Due to the ar-bitrary nature of the solution (m(t) p(t))T and( 1113957m(t) 1113957p(t))T of DFGRN (3) and in accordance with Def-inition 4 we can conclude that DFGRN (3) has a uniquemild solution e proof is completed

32 Finite-Time Stability of DFGRNs withStructured Uncertainties

Theorem 3 If assumptions (I) and (II) and [1 + ((ζ1+ζ4)tqΓ(q + 1))] middot Eq(ζ1tq)le (εδ)forallt isin J0 [0 T] holdthen the uncertain DFGRNs with controllers given by (3) withinitial condition (6) are finite-time stable with respect toδ ε α1 J01113864 1113865 δ lt ε where ζ4 ≔ ((η6α1 + ζ2)δ)

Complexity 7

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

Definition 5 (see [22]) e system given by (3) (whenQi 0ΔQi 0 i 1 2) satisfying the initial condition (6) isfinite-time stable with respect to δ ε t0 J1113864 1113865 δ lt ε if and onlyif ϕ0 lt δ imply m(t) + p(t)lt εforallt isin J J sub R

Definition 6 (see [22]) e system given by (3) satisfying theinitial condition (6) is finite-time stable with respect toδ ε α1 t0 J1113864 1113865 δ lt ε if and only if ϕ0lt δ and

u1(t) + u2(t)lt α1forallt isin Jimply m(t) + p(t)lt εforallt isin J J sub R where α1 is a positive constant

Lemma 1 (see [40]) If f(t) isin Cn([0infin)) and n minus 1lt αltn isin Z+ then

(i) Iqt [CD

q

t f(t)] f(t) minus 1113936nminus 1k0f

(k)(0)(tkk)(ii) RLD

q

t [Iqt f(t)] f(t)

(iii) RLDq

t f(t) CDq

t f(t) + 1113936nminus 1k0(tkminus q

Γ(k + 1 minus q))f(k)(0)

Lemma 2 (see [41]) Suppose βgt 0 if 0le tltT (someTle +infin) a(t) is a locally integrable nonnegative function v(t)

is a nonnegative and nondecreasing continuous function v(t)

leM (constant) and u(t) is a nonnegative and locally integrablefunction with u(t)le a(t) + v(t) 1113938

t

0 (tminus s)βminus 1u(s)ds then

u(t)le a(t) + 1113946t

01113944

infin

n1

(v(t)Γ(β))n

Γ(nβ)(t minus s)

nβminus 1a(s)⎡⎣ ⎤⎦ds forall0le tleT

(14)

In addition if a(t) is a nondecreasing function thenu(t)le a(t)Eβ(v(t)Γ(β)tβ) where Eβ is the Mittag-Lefflerfunction defined by Eβ(z) 1113936

infink0(zkΓ(kβ + 1))

3 Main Results

31 e Existence and Uniqueness of the Mild Solution ofDFGRNs

Theorem 1 Continuously differentiable functionsm(t) p(t) [minus τlowast T]⟶ Rn(Tlt +infin) form a mild solution(m(t) p(t))T to DFGRN (3) with initial condition (6) if andonly if

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds for t isin [0 T]

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds for t isin [0 T]

m(t) ϕ1(t) for t isin minus τlowast 0[ ]

p(t) ϕ2(t) for t isin minus τlowast 0[ ]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

Proof We firstly give the sufficient condition of the exis-tence of the mild solution to DFGRN (3)

When minus τlowast le tle 0 (m(t) p(t))T (ϕ1(t) ϕ2(t))T isobvious For 0le tleT according to (15) applying RLD

q

t andproperty (ii) of Lemma 1 we obtain

RLDq

t m(t) ϕ1(0)tminus q

Γ(1 minus q)minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

RLDq

t p(t) ϕ2(0)tminus q

Γ(1 minus q)minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Complexity 5

According to the property (iii) of Lemma 1 and 0lt qlt 1we get

RLDq

t m(t) CDq

t m(t) + ϕ1(0)tminus q

Γ(1 minus q)

RLDq

t p(t) CDq

t p(t) + ϕ2(0)tminus q

Γ(1 minus q)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

From (16) and (17) we haveCD

q

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

We secondly give the necessary condition of the exis-tence of the mild solution to DFGRN (3)

When t isin [minus τlowast 0] the solution of DFGRN (3) is

m(t) ϕ1(t)

p(t) ϕ2(t)

t isin minus τlowast 01113858 1113859

(19)

If 0le tleT from DFGRN (3) we have

Iqt

CDq

t m(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

Iqt

CDq

t p(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(20)

In the case of 0lt qlt 1 from property (i) of Lemma 1 wecan obtain

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

e proof is completed

Theorem 2 If assumptions (I) and (II) hold then DFGRN(3) with initial condition (6) has a unique mild solution

Proof Let (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T be any twodifferent solutions to DFGRN (3) with initial condition (6)

denote x(t) m(t) minus 1113957m(t) y(t) p(t) minus 1113957p(t) z(t)

x(t) + y(t) t isin [minus τlowast T] According to eorem 1 weknow that both (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T satisfycondition (15)

If 0le tleT then

x(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))x(s) +(W + ΔW(s))(F(p(s)) minus F(1113957p(s))) +(K + ΔK(s)) G p s minus τ1(s)( 1113857( 1113857 minus G 1113957p s minus τ1(s)( 1113857( 1113857( 1113857( 1113857ds

y(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))y(s) +(D + ΔD(s))x(s) +(H + ΔH(s))x s minus τ2(s)( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(22)

6 Complexity

From (22) by using the norm (middot) and assumptions (I)and (II) we can obtain

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s) + K + μ3( 1113857 middot L2 middot y s minus τ1(s)( 1113857

1113960 1113961ds t isin [0 T]

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s) + H + μ6( 1113857 middot x s minus τ2(s)( 1113857

1113960 1113961ds t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(23)

First when t isin [minus τlowast 0] x(t) ϕ1(θ) minus ϕ1(θ) 0

y(t) ϕ2(θ) minus ϕ2(θ) 0 So m(t) 1113957m(t) p(t) 1113957p(t) fort isin [minus τlowast 0]

Second when t isin (0 τlowast] from (23) we have

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s)1113858 1113859ds

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s)1113858 1113859ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

In accordance with (24) we can obtain

z(t) x(t) +y(t)

le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s)( 1113857ds

η1 + η2Γ(q)

1113946t

0(t minus s)

qminus 1z(s)ds

(25)

From Lemma 2 we can get

z(t)le 0 middot Eq

η1 + η2Γ(q)

middot Γ(q)tq

1113890 1113891 t isin 0 τlowast( 1113859 (26)

us z(t)le 0 t isin (0 τlowast] at is to say x(t)+

y(t)le 0 t isin (0 τlowast] So m(t) 1113957m(t) p(t) 1113957p(t) fort isin (0 τlowast]

ird when t isin (τlowast T] according to (23) we have

z(t) x(t) +y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s) + η3 x s minus τ2(s)( 1113857

+ y s minus τ1(s)( 1113857

1113872 11138731113960 1113961ds t isin τlowast T( 1113859 (27)

Let zlowast(t) supθ1113957θisin[minus τlowast 0]

[x(t + θ) + y(t + 1113957θ)] enwe obtain

zlowast(t)le

η1 + η2 + η3Γ(q)

1113946t

0(t minus s)

qminus 1zlowast(s)ds t isin τlowast T( 1113859

(28)

From Lemma 2 we can get

z(t)le zlowast(t)le 0 middot Eq

η1 + η2 + η3Γ(q)

middot Γ(q) middot tq

1113888 1113889 t isin τlowast T( 1113859

(29)

en we have x(t) + y(t) le 0 t isin (τlowast T] So m(t)

1113957m(t) p(t) 1113957p(t) for t isin (τlowast T]

Summarizing the above three cases we can obtain thatm(t) 1113957m(t) p(t) 1113957p(t) for t isin [minus τlowast T] Due to the ar-bitrary nature of the solution (m(t) p(t))T and( 1113957m(t) 1113957p(t))T of DFGRN (3) and in accordance with Def-inition 4 we can conclude that DFGRN (3) has a uniquemild solution e proof is completed

32 Finite-Time Stability of DFGRNs withStructured Uncertainties

Theorem 3 If assumptions (I) and (II) and [1 + ((ζ1+ζ4)tqΓ(q + 1))] middot Eq(ζ1tq)le (εδ)forallt isin J0 [0 T] holdthen the uncertain DFGRNs with controllers given by (3) withinitial condition (6) are finite-time stable with respect toδ ε α1 J01113864 1113865 δ lt ε where ζ4 ≔ ((η6α1 + ζ2)δ)

Complexity 7

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

According to the property (iii) of Lemma 1 and 0lt qlt 1we get

RLDq

t m(t) CDq

t m(t) + ϕ1(0)tminus q

Γ(1 minus q)

RLDq

t p(t) CDq

t p(t) + ϕ2(0)tminus q

Γ(1 minus q)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

From (16) and (17) we haveCD

q

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

We secondly give the necessary condition of the exis-tence of the mild solution to DFGRN (3)

When t isin [minus τlowast 0] the solution of DFGRN (3) is

m(t) ϕ1(t)

p(t) ϕ2(t)

t isin minus τlowast 01113858 1113859

(19)

If 0le tleT from DFGRN (3) we have

Iqt

CDq

t m(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

Iqt

CDq

t p(t)1113960 1113961 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(20)

In the case of 0lt qlt 1 from property (i) of Lemma 1 wecan obtain

m(t) ϕ1(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) ϕ2(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

e proof is completed

Theorem 2 If assumptions (I) and (II) hold then DFGRN(3) with initial condition (6) has a unique mild solution

Proof Let (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T be any twodifferent solutions to DFGRN (3) with initial condition (6)

denote x(t) m(t) minus 1113957m(t) y(t) p(t) minus 1113957p(t) z(t)

x(t) + y(t) t isin [minus τlowast T] According to eorem 1 weknow that both (m(t) p(t))T and ( 1113957m(t) 1113957p(t))T satisfycondition (15)

If 0le tleT then

x(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))x(s) +(W + ΔW(s))(F(p(s)) minus F(1113957p(s))) +(K + ΔK(s)) G p s minus τ1(s)( 1113857( 1113857 minus G 1113957p s minus τ1(s)( 1113857( 1113857( 1113857( 1113857ds

y(t) 1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))y(s) +(D + ΔD(s))x(s) +(H + ΔH(s))x s minus τ2(s)( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(22)

6 Complexity

From (22) by using the norm (middot) and assumptions (I)and (II) we can obtain

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s) + K + μ3( 1113857 middot L2 middot y s minus τ1(s)( 1113857

1113960 1113961ds t isin [0 T]

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s) + H + μ6( 1113857 middot x s minus τ2(s)( 1113857

1113960 1113961ds t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(23)

First when t isin [minus τlowast 0] x(t) ϕ1(θ) minus ϕ1(θ) 0

y(t) ϕ2(θ) minus ϕ2(θ) 0 So m(t) 1113957m(t) p(t) 1113957p(t) fort isin [minus τlowast 0]

Second when t isin (0 τlowast] from (23) we have

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s)1113858 1113859ds

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s)1113858 1113859ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

In accordance with (24) we can obtain

z(t) x(t) +y(t)

le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s)( 1113857ds

η1 + η2Γ(q)

1113946t

0(t minus s)

qminus 1z(s)ds

(25)

From Lemma 2 we can get

z(t)le 0 middot Eq

η1 + η2Γ(q)

middot Γ(q)tq

1113890 1113891 t isin 0 τlowast( 1113859 (26)

us z(t)le 0 t isin (0 τlowast] at is to say x(t)+

y(t)le 0 t isin (0 τlowast] So m(t) 1113957m(t) p(t) 1113957p(t) fort isin (0 τlowast]

ird when t isin (τlowast T] according to (23) we have

z(t) x(t) +y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s) + η3 x s minus τ2(s)( 1113857

+ y s minus τ1(s)( 1113857

1113872 11138731113960 1113961ds t isin τlowast T( 1113859 (27)

Let zlowast(t) supθ1113957θisin[minus τlowast 0]

[x(t + θ) + y(t + 1113957θ)] enwe obtain

zlowast(t)le

η1 + η2 + η3Γ(q)

1113946t

0(t minus s)

qminus 1zlowast(s)ds t isin τlowast T( 1113859

(28)

From Lemma 2 we can get

z(t)le zlowast(t)le 0 middot Eq

η1 + η2 + η3Γ(q)

middot Γ(q) middot tq

1113888 1113889 t isin τlowast T( 1113859

(29)

en we have x(t) + y(t) le 0 t isin (τlowast T] So m(t)

1113957m(t) p(t) 1113957p(t) for t isin (τlowast T]

Summarizing the above three cases we can obtain thatm(t) 1113957m(t) p(t) 1113957p(t) for t isin [minus τlowast T] Due to the ar-bitrary nature of the solution (m(t) p(t))T and( 1113957m(t) 1113957p(t))T of DFGRN (3) and in accordance with Def-inition 4 we can conclude that DFGRN (3) has a uniquemild solution e proof is completed

32 Finite-Time Stability of DFGRNs withStructured Uncertainties

Theorem 3 If assumptions (I) and (II) and [1 + ((ζ1+ζ4)tqΓ(q + 1))] middot Eq(ζ1tq)le (εδ)forallt isin J0 [0 T] holdthen the uncertain DFGRNs with controllers given by (3) withinitial condition (6) are finite-time stable with respect toδ ε α1 J01113864 1113865 δ lt ε where ζ4 ≔ ((η6α1 + ζ2)δ)

Complexity 7

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

From (22) by using the norm (middot) and assumptions (I)and (II) we can obtain

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s) + K + μ3( 1113857 middot L2 middot y s minus τ1(s)( 1113857

1113960 1113961ds t isin [0 T]

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s) + H + μ6( 1113857 middot x s minus τ2(s)( 1113857

1113960 1113961ds t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(23)

First when t isin [minus τlowast 0] x(t) ϕ1(θ) minus ϕ1(θ) 0

y(t) ϕ2(θ) minus ϕ2(θ) 0 So m(t) 1113957m(t) p(t) 1113957p(t) fort isin [minus τlowast 0]

Second when t isin (0 τlowast] from (23) we have

x(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1A + μ1( 1113857 middot x(s) + W + μ2( 1113857 middot L1 middot y(s)1113858 1113859ds

y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1C + μ4( 1113857 middot y(s) + D + μ5( 1113857 middot x(s)1113858 1113859ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

In accordance with (24) we can obtain

z(t) x(t) +y(t)

le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s)( 1113857ds

η1 + η2Γ(q)

1113946t

0(t minus s)

qminus 1z(s)ds

(25)

From Lemma 2 we can get

z(t)le 0 middot Eq

η1 + η2Γ(q)

middot Γ(q)tq

1113890 1113891 t isin 0 τlowast( 1113859 (26)

us z(t)le 0 t isin (0 τlowast] at is to say x(t)+

y(t)le 0 t isin (0 τlowast] So m(t) 1113957m(t) p(t) 1113957p(t) fort isin (0 τlowast]

ird when t isin (τlowast T] according to (23) we have

z(t) x(t) +y(t)le1Γ(q)

1113946t

0(t minus s)

qminus 1 η1z(s) + η2z(s) + η3 x s minus τ2(s)( 1113857

+ y s minus τ1(s)( 1113857

1113872 11138731113960 1113961ds t isin τlowast T( 1113859 (27)

Let zlowast(t) supθ1113957θisin[minus τlowast 0]

[x(t + θ) + y(t + 1113957θ)] enwe obtain

zlowast(t)le

η1 + η2 + η3Γ(q)

1113946t

0(t minus s)

qminus 1zlowast(s)ds t isin τlowast T( 1113859

(28)

From Lemma 2 we can get

z(t)le zlowast(t)le 0 middot Eq

η1 + η2 + η3Γ(q)

middot Γ(q) middot tq

1113888 1113889 t isin τlowast T( 1113859

(29)

en we have x(t) + y(t) le 0 t isin (τlowast T] So m(t)

1113957m(t) p(t) 1113957p(t) for t isin (τlowast T]

Summarizing the above three cases we can obtain thatm(t) 1113957m(t) p(t) 1113957p(t) for t isin [minus τlowast T] Due to the ar-bitrary nature of the solution (m(t) p(t))T and( 1113957m(t) 1113957p(t))T of DFGRN (3) and in accordance with Def-inition 4 we can conclude that DFGRN (3) has a uniquemild solution e proof is completed

32 Finite-Time Stability of DFGRNs withStructured Uncertainties

Theorem 3 If assumptions (I) and (II) and [1 + ((ζ1+ζ4)tqΓ(q + 1))] middot Eq(ζ1tq)le (εδ)forallt isin J0 [0 T] holdthen the uncertain DFGRNs with controllers given by (3) withinitial condition (6) are finite-time stable with respect toδ ε α1 J01113864 1113865 δ lt ε where ζ4 ≔ ((η6α1 + ζ2)δ)

Complexity 7

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

Proof According to eorem 1 and eorem 2 we canknow that DFGRN (3) has a mild solution and the solutionsatisfies the following integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) +(W + ΔW(s))F(p(s)) +(K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)( 1113857ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) +(D + ΔD(s))m(s) +(H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

Using the norm (middot) we can obtain the solution estimateof system (30)

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857u1(s)

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857u2(s)

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

By applying norm (middot) to DFGRN (3) and combiningassumptions (I) and (II) we can get

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W) + μ2( 1113857 L1p(t) + F(0)( 1113857

+ σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873 + B + σ Q1( 1113857 + μ7( 1113857 u1(t)

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Let x(t) m(t) + p(t) According to (31) (3) and(32) if u1(t) + u2(t)lt α1 we have

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 CD

q

t m(s)

+C

Dq

t p(s)

1113874 1113875ds

le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ(D) + μ5( 1113857m(s) + σ(W) + μ2( 1113857L1 + σ(C) + μ4( 1113857p(s)(

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ(H) + μ6( 1113857 middot m s minus τ2(s)( 1113857

+ B + σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0)

+ σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

8 Complexity

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4x(s) + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ(H) + μ6( 1113857( 1113857x s minus τ2(s)( 1113857(

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113873ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η4 + σ(K) + μ3( 1113857L2 + σ(H) + μ6( 1113857 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889

+ ζ2 + σ Q1( 1113857 + μ7( 1113857 u1(s)

+ σ Q2( 1113857 + μ8( 1113857 u2(s)

1113889ds

lex(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 ζ1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889 + ζ2 + η6 u1(s)

+ u2(s)

1113872 11138731113888 1113889ds

leϕ0 +1Γ(q)

ζ1 1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1ϕ0ds

+η6Γ(q)

1113946t

0(t minus s)

qminus 1u1(s)

+ u2(s)

1113872 1113873ds +

ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

leϕ0 +ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ1Γ(q)

ϕ0q

tq

+η6Γ(q)q

middot α1 middot tq

+ ζ2tq

Γ(q)q

leϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

ζ1Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0

(33)

Let

ρ(t) ϕ0 1 +ζ1tq

Γ(q + 1)1113890 1113891 +

η6 middot α1 + ζ2Γ(q + 1)

middot tq tgt 0 (34)

en we know that ρ(t) is a nonnegative and nondecreasingfunction By using Lemma 2 (the generalized Gronwallinequality) we have

x(t)le suptminus τlowast le tlowast le t

x tlowast

( 1113857le ρ(t)Eq

ζ1Γ(q)Γ(q)t

q1113888 1113889 (35)

If ϕ0 lt δ we have

x(t)le δ 1 +ζ1tq

Γ(q + 1)+η6α1 + ζ2Γ(q + 1)δ

tq

1113890 1113891Eq ζ1tq

( 1113857 (36)

Because [1 + ((ζ1 + ζ4)tqΓ(q + 1))]Eq(ζ1tq)le (εδ) andζ4 ((η6α1 + ζ2)δ) then

x(t)lt εforallt isin J0 (37)

Hence

m(t) +p(t)lt ε forallt isin J0 (38)

e proof is completed

Remark 1 If we adopt u1(t) equiv 0 u2(t) equiv 0forallt isin J0 inDFGRN (3) we can obtain the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε if

assumptions (I) and (II) hold and the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

tq

1113890 1113891 middot Eq ζ1tq

( 1113857leεδ forallt isin J0 [0 T] (39)

where ζ5 ≔ (ζ2δ)

Remark 2 In the proof of eorem 3 if we use the ldquoclas-sicalrdquo BellmanndashGronwall inequality instead of the general-ized Gronwall inequality we can get the following result

e uncertain DFGRN with controllers given by (3)satisfying the initial condition (6) is finite-time stable withrespect to δ ε α1 J01113864 1113865 δ lt ε if assumptions (I) and (II) holdand the following condition is satisfied

1 +ζ1 + ζ4Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (40)

Remark 3 If we take u1(t) equiv 0 u2(t) equiv 0forallt isin J0 in system(3) the above results turn into the following conclusion

e uncertain DFGRN (3) satisfying the initial condition(6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε Ifassumptions (I) and (II) hold the following condition issatisfied

1 +ζ1 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ1 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (41)

Complexity 9

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

33 Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers We consider the following memorystate-feedback controllers on DFGRN (3)

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(42)

where ci i 1 2 3 4 are the gain matrices ofui(t) 0le 1113954τ1(t)le τlowast 0le 1113954τ2(t)le τlowast en DFGRN (3) can bechanged into

CDq

t m(t) minus (A + ΔA(t))m(t) + (W + ΔW(t))F(p(t)) + (K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

+ Q1 + ΔQ1(t)( 1113857 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus (C + ΔC(t))p(t) + (D + ΔD(t))m(t) + (H + ΔH(t))m t minus τ2(t)( 1113857

+ Q2 + ΔQ2(t)( 1113857 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

Theorem 4 If assumptions (I) and (II) and

1 +ζ3 + ζ5Γ(q + 1)

tq

1113890 1113891Eq ζ3tq

( 1113857leεδ (44)

hold then the uncertain DFGRN (3) with memory state-feedback controllers given by (43) satisfying the initial con-dition (6) is finite-time stable with respect to δ ε J01113864 1113865 δ lt ε

Proof Similar to eorem 1 and eorem 2 it is easy toprove that DFGRN (43) has a mild solution satisfying thefollowing integral equation

m(t) m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857( ds

p(t) p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857( 1113857ds

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Using the norm (middot) we have

m(t) le m(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (A + ΔA(s))m(s) + (W + ΔW(s))F(p(s)) + (K + ΔK(s))G p s minus τ1(s)( 1113857( 1113857 + B + Q1 + ΔQ1(s)( 1113857 c1m(s) + c3p s minus 1113954τ1(s)( 1113857( 1113857

ds

p(t) le p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1minus (C + ΔC(s))p(s) + (D + ΔD(s))m(s) + (H + ΔH(s))m s minus τ2(s)( 1113857 + Q2 + ΔQ2(s)( 1113857 c2p(s) + c4m s minus 1113954τ2(s)( 1113857( 1113857

ds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

10 Complexity

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

From (43) by using assumptions (I) and (II) we have

CDq

t m(t)

le σ(A) + μ1( 1113857m(t) + σ(W +μ2(( 1113857 L1p(t) + F(0)( 1113857 + σ(K) + μ3( 1113857 L2 p t minus τ1(t)( 1113857

+ G(0)1113872 1113873

+B + σ Q1( 1113857 + μ7( 1113857 σ c1( 1113857m(t) + σ c3( 1113857 p t minus 1113954τ1(t)( 1113857

1113872 1113873

CDq

t p(t)

le σ(C) + μ4( 1113857p(t) + σ(D) + μ5( 1113857m(t) + σ(H) + μ6( 1113857 m t minus τ2(t)( 1113857

+ σ Q2( 1113857 + μ8( 1113857 σ c2( 1113857p(t) + σ c4( 1113857 m t minus 1113954τ2(t)( 1113857

1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(47)

Let x(t) m(t) + p(t) From (46) and (47) weobtain

x(t)le m(0) + p(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 σ(A) + μ1 + σ Q1( 1113857 + μ7( 1113857σ c1( 1113857 + σ(D) + μ5( 1113857m(s) + B(

+ σ(W) + μ2( 1113857F(0) + σ(K) + μ3( 1113857G(0) + σ(W) + μ2( 1113857L1 + σ(C) + μ4 + σ Q2( 1113857 + μ8( 1113857σ c2( 11138571113858 1113859p(s)

+ σ(K) + μ3( 1113857L2 p s minus τ1(s)( 1113857

+ σ Q1( 1113857 + μ7( σ c3( 1113857 p s minus 1113954τ1(s)( 1113857

+ σ(H) + μ6( 1113857 m s minus τ2(s)( 1113857

+ σ Q2( 1113857 + μ8( 1113857σ c4( 1113857 m s minus 1113954τ2(s)( 1113857

1113873ds

(48)

Hence

x(t)le x(0) +1Γ(q)

1113946t

0(t minus s)

qminus 1 η5x(s) + ζ2 + σ(K) + μ3( 1113857L2x s minus τ1(s)( 1113857 + σ Q1( 1113857 + μ7( 1113857σ c3( 1113857x s minus 1113954τ1(s)( 1113857(

+ σ(H) + μ6( 1113857x s minus τ2(s)( 1113857 + σ Q2( 1113857 + μ8( 1113857σ c4( 1113857x s minus 1113954τ2(s)( 11138571113857ds

le ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857 + ϕ01113888 1113889ds +ζ2Γ(q)

1113946t

0(t minus s)

qminus 1ds

ϕ0 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)+

ζ3Γ(q)

ϕ0 1113946t

0(t minus s)

qminus 1ds

le ϕ0 1 +ζ3Γ(q + 1)

tq

1113888 1113889 +ζ3Γ(q)

1113946t

0(t minus s)

qminus 1 supsminus τlowastletlowastles

x tlowast

( 1113857ds +ζ2tq

Γ(q + 1)

(49)

Let

ρ(t) ϕ0 1 +ζ3tq

Γ(q + 1)1113890 1113891 +

ζ2tq

Γ(q + 1) tgt 0 (50)

en we know ρ(t) is a nonnegative and nondecreasingfunction From Lemma 2 we have

x(t)le suptminus τlowastletlowastlet

x tlowast

( 1113857le ρ(t)Eq

ζ3Γ(q)Γ(q)t

q1113888 1113889 (51)

If ϕ0 lt δ we obtain

x(t)le δ 1 +ζ3 + ζ2δ( 1113857

Γ(q + 1)tq

1113890 1113891Eq ζ3tq

( 1113857 (52)

From the condition of [1 + (((ζ3 + ζ5)tq)Γ(q + 1))]Eq(ζ3tq)le (εδ) and ζ5 (ζ2δ) we can get

x(t)lt ε forallt isin J0 (53)

erefore

m(t) +p(t)lt ε forallt isin J0 (54)

e proof is completed

Remark 4 Similar to Remark 2 we can get the followingresult

Complexity 11

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

e uncertain DFGRN (3) with memory state-feedbackcontroller given by (43) satisfying the initial condition (6) isfinite-time stable with respect to δ ε J01113864 1113865 δ lt ε if assump-tions (I) and (II) hold and the following condition issatisfied

1 +ζ3 + ζ5Γ(q + 1)

t minus t0( 1113857q

1113890 1113891expζ3 t minus t0( 1113857

q

Γ(q + 1)1113890 1113891lt

εδ (55)

Remark 5 We can obtain the same conclusion aseorem 3and eorem 4 if the inequalities in assumption (II) are

F(x)leL1x

G(x)leL2x(56)

Remark 6 All the results in Remarks 1ndash4 are still new

4 Numerical Examples

In this section some numerical examples are given to il-lustrate the effectiveness of above theoretical results In thefollowing examples the functions fj and gj are taken as theHill form And in the AdamsndashBashforthndashMoulton pre-dictor-corrector scheme [42] the step length is h 01

Example 1 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andmemory state-feedback controllers

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t))

+(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B + Q1 + ΔQ1(t)( 1113857u1(t)

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t)

+(H + ΔH(t))m t minus τ2(t)( 1113857 + Q2 + ΔQ2(t)( 1113857u2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(57)

Let

A

3 0 0

0 3 0

0 0 3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

25 0 0

0 25 0

0 0 25

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

D

1 0 0

0 1 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

H

03 0 0

0 03 0

0 0 03

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

W

08147 minus 09134 02785

09058 06324 minus 05469

minus 01270 00975 09575

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K

028947 028716 minus 004257

004728 minus 014562 012654

minus 029118 024009 027471

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔA(t)

01 cos(t) minus 007 sin(t) 002 cos(t) minus 005 sin(t) 004 cos(t) minus 006 sin(t)

01 cos(t) + 001 sin(t) 002 cos(t) 004 cos(t) + 003 sin(t)

005 cos(t) + 003 sin(t) 001 cos(t) + 001 sin(t) 002 cos(t) + 006 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

12 Complexity

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

ΔC(t)

004 cos(t) minus 008 sin(t) 004 cos(t) minus 003 sin(t) 002 cos(t) minus 001 sin(t)

004 cos(t) + 004 sin(t) 004 cos(t) + 004 sin(t) 002 cos(t) + 003 sin(t)

002 cos(t) + 008 sin(t) 002 cos(t) + 007 sin(t) 001 cos(t) + 005 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔD(t)

001 sin(t) 004 cos(t) minus 009 sin(t)

002 sin(t) 004 cos(t) minus 003 sin(t)

003 sin(t) 002 cos(t) minus 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔH(t)

006 cos(t) + 001 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 002 sin(t)

006 cos(t) + 002 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) + 001 sin(t)

003 cos(t) + 003 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔK(t)

002 cos(t) minus 001 sin(t) 006 cos(t) minus 005 sin(t) 004 cos(t) minus 002 sin(t)

002 cos(t) + 003 sin(t) 006 cos(t) 004 cos(t) + 001 sin(t)

001 cos(t) + 005 sin(t) 003 cos(t) + 001 sin(t) 002 cos(t) + 002 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔW(t)

004 cos(t) minus 01 sin(t) 006 cos(t) minus 002 sin(t) 008 cos(t) + 001 sin(t)

004 cos(t) 006 cos(t) + 001 sin(t) 008 cos(t) + 002 sin(t)

002 cos(t) + 002 sin(t) 003 cos(t) + 002 sin(t) 004 cos(t) + 003 sin(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ1(t)

002 cos(t) minus 002 sin(t) 004 cos(t) minus 005 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 001 sin(t) 004 cos(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 002 sin(t) 002 cos(t) + 001 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ΔQ2(t)

002 cos(t) + 002 sin(t) 002 cos(t) minus 004 sin(t) 004 cos(t) minus 008 sin(t)

002 cos(t) + 004 sin(t) 002 cos(t) + 002 sin(t) 004 cos(t) minus 001 sin(t)

001 cos(t) + 006 sin(t) 001 cos(t) + 004 sin(t) 002 cos(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

e memory state-feedback controllers are defined asfollows

u1(t) c1m(t) + c3p t minus 1113954τ1(t)( 1113857

u2(t) c2p(t) + c4m t minus 1113954τ2(t)( 1113857(59)

where

c1

00465 00457 minus 00358

minus 00342 minus 00015 minus 00078

00471 00300 00416

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c2

00195 minus 00466 00266

minus 00183 minus 00061 00295

00450 minus 00118 minus 00313

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c3

minus 00010 00209 00180

minus 00054 00255 00155

00146 minus 00224 minus 00337

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

c4

minus 00381 minus 00160 00251

minus 00002 00085 minus 00245

00460 minus 00276 00006

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(60)

Let Q1 Q2 diag(222)q 095δ 1ε 50

τ1(t) τ2(t) ((|cos t| +1)4)τlowast (12) (ϕ1(t) ϕ2(t))T

(013920273404788048240078804853)T (minus τlowast letle0)L1 L2 1 F(x) G(x) x2(1+ x2) According to the

Complexity 13

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

notations in Section 2 we obtain ϕ0 09641lt1 σ(A)

3σ(D) 1σ (W) 13710 σ(C) 25σ(H) 03σ(K)

04793σ (Q1) 2 σ(Q2) 2

η5 45081ζ3 58292ζ5 70183 When tlt03339 simplecomputation reveals that

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(58292 + 70183) times 03339095

Γ(095 + 1)1113890 1113891Eq 58292 times 03339095

1113872 1113873ltεδ

501

(61)

From eorem 4 system (57) is finite-time stable withrespect to 1 50 [0 03339] Denote Te asymp 03339 as theldquoestimated timerdquo of finite-time stability e transient statesof the variable mi(t) and pi(t)(i 1 2 3) of DFGRN (57)with q 095 and q 06 are shown in Figures 1(a) and 1(b)respectively

Example 2 Consider the following DFGRNs of threemRNA and protein nodes with structured uncertainties andwithout controller

CDq

t m(t) minus (A + ΔA(t))m(t) +(W + ΔW(t))F(p(t)) +(K + ΔK(t))G p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus (C + ΔC(t))p(t) +(D + ΔD(t))m(t) +(H + ΔH(t))m t minus τ2(t)( 1113857

⎧⎨

⎩ (62)

Using the same parameters in Example 1 we similarlyget η4 43172 ζ1 53845 ζ5 70183 When tlt 03585we have

1 +ζ1 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ1t

q( 1113857lt 1 +

(53845 + 70183) times 03585095

Γ(095 + 1)1113890 1113891Eq 53845 times 03585095

1113872 1113873ltεδ

501

(63)

From Remark 1 system (62) is finite-time stable with re-spect to 1 50 [0 03585] then the ldquoestimated timerdquo of finite-time stability Te asymp 03585 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (62) with q 095 andq 06 are shown in Figures 2(a) and 2(b) respectively

In Example 2 when t⟶ +infin the case of infinite timeDFGRN (62) with structured uncertainties is unstable enumerical simulations of the variables mi(t) and pi(t)(i

1 2 3) of DFGRN (62) with q 095 and q 06 are shownin Figures 3(a) and 3(b) respectively

Remark 7 It is worthy to note that in a special case ofDFGRN (62) without structured uncertainties it is provedthat in the sense of infinite stability (62) is globally as-ymptotically stable [16]

Example 3 Consider the following DFGRNs of threemRNA and protein nodes with memory state-feedbackcontrollers and without structured uncertainties

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B + Q1 c1m(t) + c3p t minus 1113954τ1(t)( 1113857( 1113857

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857 + Q2 c2p(t) + c4m t minus 1113954τ2(t)( 1113857( 1113857

⎧⎨

⎩ (64)

Using the same parameters in Example 1 we similarlyobtain η5 41799 ζ3 52009 ζ5 70183 Whentlt 03697 we can get

1 +ζ3 + ζ5( 1113857tq

Γ(q + 1)1113890 1113891Eq ζ3t

q( 1113857lt 1 +

(52009 + 70183) times 03697095

Γ(095 + 1)1113890 1113891Eq 52009 times 03697095

1113872 1113873ltεδ

501

(65)

14 Complexity

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

From eorem 4 system (64) is finite-time stable with re-spect to 1 50 [0 03697] then the ldquoestimated timerdquo of finite-time stability Te asymp 03697 e transient states of the variablesmi(t) and pi(t)(i 1 2 3) of DFGRN (64) with q 095 andq 06 are shown in Figures 4(a) and 4(b) respectively

Example 4 Consider the following DFGRNs of threemRNA and protein nodes without structured uncertaintiesor controller

CDq

t m(t) minus Am(t) + WF(p(t)) + KG p t minus τ1(t)( 1113857( 1113857 + B

CDq

t p(t) minus Cp(t) + Dm(t) + Hm t minus τ2(t)( 1113857

⎧⎨

⎩

(66)

Using the same parameters in Example 1 we also obtainthe ldquoestimated timerdquo of finite-time stability for system (66) asTe asymp 03984 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 095 and q 06are shown in Figures 5(a) and 5(b) respectively

If we adopt constant time-delay τ1(t) τ2(t) 2 andq 04 in DFGRN (66) then system (66) is finite-timestable and the ldquoestimated timerdquo of finite-time stability is00315 e transient states of the variables mi(t) andpi(t)(i 1 2 3) of DFGRN (66) with q 04 are shown inFigure 6

Remark 8 If τ1(t) τ2(t) 2 and q 04 in DFGRN (66)then system (66) converts to system (41) in [16] When

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 2 Transient states of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

m1m2m3

p1p2p3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

(b)

Figure 1 Transient states of DFGRN (57) with (a) q 095 and (b) q 06

Complexity 15

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

t⟶ +infin it is proved that system (41) is unstable in thesense of infinite-time stability [16] which means that thefinite-time stability is different from the infinite-time sta-bility of DFGRNs

If we take K ΔK(t) H ΔH(t) 0 and c3 c4 0in DFGRNs (57) (62) (64) and (66) systems (57) (62) (64)and (66) convert to the corresponding fractional-order generegulatory networks without time delays (FGRNs)

In order to investigate the effects of structured un-certainties controllers and time delays on the stability of theDFGRNs we calculate the ldquoestimated timerdquo Te of finite-timestability for above four examples and the correspondingFGRNs with different fractional-order q the results areshown in Tables 1 and 2 respectively

From Table 1 or Table 2 we have the followingconclusions

(i) e effect of the controllers comparing column 2with 3 (or column 4 with 5) we can know thatthe controllers can shorten the ldquoestimated timerdquoof finite-time stability under the same condi-tions of fractional-order q and structureduncertainties

(ii) e effect of the structured uncertainties com-paring column 3 with 5 we can know that thestructured uncertainties can shorten the ldquoestimatedtimerdquo of finite-time stability under the same frac-tional-order it q

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 3 Numerical simulations of DFGRN (62) with (a) q 095 and (b) q 06

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 4 Transient states of DFGRN (64) with (a) q 095 and (b) q 06

16 Complexity

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

(iii) e difference between the structured uncertaintiesand the controllers comparing column 3 with 4 wecan know that the size of ldquoestimated timerdquo of finite-time stability for DFGRN (62) with structured

uncertainties is longer than DFGRN (64) withcontrollers under the same fractional-order q

(iv) e effect of the fractional-order q in the samecolumn we can know that decreasing the fractional-

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

(b)

Figure 5 Transient states of DFGRN (66) with (a) q 095 and (b) q 06

0 02 04 06 08 10

02

04

06

08

1

12

14

Time

mRN

A an

d pr

otei

n co

ncen

trat

ions

m1m2m3

p1p2p3

Figure 6 Transient states of DFGRN (66) with q 040 and τ1(t) τ2(t) 2

Table 1 e ldquoestimated timerdquo Te of finite-time stability with different fractional-order q

q DFGRN (57) Te DFGRN (62) Te DFGRN (64) Te DFGRN (66) Te

095 03339 03585 03697 03984085 02607 02824 02933 03192075 01910 02093 02192 02416060 00993 01116 01187 01344050 00519 00598 00646 00751045 00338 00396 00431 00511

Complexity 17

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

order q will be useful to decrease the ldquoestimatedtimerdquo of finite-time stability for DFGRNs or FGRNs

(v) e effect of time delays comparing Table 1 withTable 2 we can know that the ldquoestimated timerdquo offinite-time stability is reduced under the samefractional-order q when considering time delays

5 Concluding Remarks

is paper deals with the existence and uniqueness of thesolution and the finite-time stability for a class of DFGRNswith structured uncertainties and controllers In particularwe design the memory state-feedback controllers forDFGRNs with structured uncertainties and give the suffi-cient conditions for the system to achieve the finite-timestability

It should be pointed out that the conditions of finite-time stability in the present paper are dependent on thefractional-order q which is more different from theprevious stability results for the case of integer order iethe finite-time stability is independent of the integerorder

In addition from the numerical results we find that allof the controllers uncertain terms fractional-order q andtime delays can affect the ldquoestimated timerdquo of finite-timestability Particularly (i) the size of ldquoestimated timerdquo offinite-time stability with controllers is shorter than thecase without controller but only with structured un-certainties which means that the controllers are morebeneficial for controlling the ldquoestimated timerdquo than thestructured uncertainties (ii) the size of ldquoestimated timerdquoof finite-time stability with time delays is shorter than thecase without time delays which means that time delaysdegrade the GRN performance

If we take ΔA(t) ΔW(t) ΔK(t) ΔC(t) ΔD

(t) Δ H(t) ΔQ1(t) ΔQ2(t) 0 and controllers termsu1(t) u2(t) 0 meanwhile in the special case constanttime delay system (3) convert to (22) in [16] and we findthat numerically as t⟶ +infin DFGRN (62) in this paper isunstable however DFGRN (41) in [16] is globally as-ymptotically stable which means that the structured un-certainty can change the stability of DFGRNs Furthermorefrom Remark 8 we know that DFGRN (66) is finite-timestable while the corresponding system (41) in [16] isinfinite-time unstable which means that an infinite-timeunstable system can change to a finite-time stable one underextra conditions e analytical study on above questions isdesirable in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Hunan Provincial NaturalScience Foundation (nos 2019JJ50222 and 13JJ4065) and theScientific Research Fund of Hunan Provincial EducationDepartment (no 19C0911)

References

[1] L Chen and K Aihara ldquoStability of genetic regulatory net-works with time delayrdquo IEEE Transactions on Circuits andSystems I Fundamental eory and Applications vol 49no 5 pp 602ndash608 2002

[2] H-D Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Bi-ology vol 9 no 1 pp 67ndash103 2002

[3] N Friedman M Linial I Nachman and D Persquoer ldquoUsingbayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[4] L Wu K Liu J Lu and H Gu ldquoFinite-time adaptive stabilityof gene regulatory networksrdquo Neurocomputing vol 338pp 222ndash232 2019

[5] M B Elowitz and S Leibler ldquoA synthetic oscillatory networkof transcriptional regulatorsrdquo Nature vol 403 no 6767pp 335ndash338 2000

[6] A Becskei and L Serrano ldquoEngineering stability in genenetworks by autoregulationrdquo Nature vol 405 no 6786pp 590ndash593 2000

[7] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403no 6767 pp 339ndash342 2000

[8] C Huang J Cao andM Xiao ldquoHybrid control on bifurcationfor a delayed fractional gene regulatory networkrdquo ChaosSolitons amp Fractals vol 87 pp 19ndash29 2016

[9] F Ren F Cao and J Cao ldquoMittag-Leffler stability andgeneralized Mittag-Leffler stability of fractional-order generegulatory networksrdquo Neurocomputing vol 160 pp 185ndash1902015

[10] B Tao M Xiao Q Sun and J Cao ldquoHopf bifurcation analysisof a delayed fractional-order genetic regulatory networkmodelrdquo Neurocomputing vol 275 pp 677ndash686 2018

[11] Y Zhang Y Pu H Zhang Y Cong and J Zhou ldquoAn ex-tended fractional Kalman filter for inferring gene regulatory

Table 2 e ldquoestimated timerdquo Te of finite-time stability without time delays

q FGRN (57) Te FGRN (62) Te FGRN (64) Te FGRN (66) Te

095 04204 04365 04477 04650085 03382 03529 03644 03804075 02575 02703 02813 02956060 01452 01545 01632 01740050 00824 00889 00952 01029045 00566 00616 00666 00726

18 Complexity

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19

networks using time-series datardquo Chemometrics and In-telligent Laboratory Systems vol 138 pp 57ndash63 2014

[12] T Yu X Zhang G Zhang and B Niu ldquoHopf bifurcationanalysis for genetic regulatory networks with two delaysrdquoNeurocomputing vol 164 pp 190ndash200 2015

[13] R Ji D Liu X Yan and X Ma ldquoModelling gene regulatorynetwork by fractional order differential equationsrdquo in Pro-ceedings 2010 IEEE 5th International Conference on Bio-In-spired Computing eories and Applications BIC-TA 2010pp 431ndash434 Changsha China September 2010

[14] X Min X Wei G Jiang and J Cao ldquoStability and bifurcationanalysis of arbitrarily high-dimensional genetic regulatorynetworks with hub structure and bidirectional couplingrdquoIEEE Transactions on Circuits amp Systems I Regular Papersvol 63 no 8 pp 1243ndash1254 2016

[15] X Fan Y Xue X Zhang and J Ma ldquoFinite-time state ob-server for delayed reaction-diffusion genetic regulatory net-worksrdquo Neurocomputing vol 227 pp 18ndash28 2017

[16] Z Wu Z Wang and T Zhou ldquoGlobal stability analysis offractional-order gene regulatory networks with time delayrdquoInternational Journal of Biomathematics vol 12 no 6 ArticleID 1950067 2019

[17] D Yue Z-H Guan J Li F Liu J-W Xiao and G LingldquoStability and bifurcation of delay-coupled genetic regulatorynetworks with hub structurerdquo Journal of the Franklin In-stitute vol 356 no 5 pp 2847ndash2869 2019

[18] H Zang T Zhang and Y Zhang ldquoBifurcation analysis ofa mathematical model for genetic regulatory network withtime delaysrdquoAppliedMathematics and Computation vol 260pp 204ndash226 2015

[19] X Zang and Q Han ldquoGlobal asymptotic stability analysis fordelayed neural networks using a matrix-based quadraticconvex approachrdquo Neural Networks vol 54 pp 57ndash69 2014

[20] X Zang Q Han X Ge and D Ding ldquoAn overview of recentdevelopments in Lyapunov-Krasovskii functionals and sta-bility criteria for recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 313 pp 392ndash401 2018

[21] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neuro-computing vol 71 no 4ndash6 pp 834ndash842 2008

[22] M P Lazarevic ldquoNon-Lyapunov stability and stabilization offractional order systems including time-varying delaysrdquo Re-cent Researches in System Science in Proceedings of the 15thWSEAS International Conference on Systems pp 196ndash201Corfu Greece July 2011

[23] V N Phat and N T anh ldquoNew criteria for finite-timestability of nonlinear fractional-order delay systemsa Gronwall inequality approachrdquo Applied Mathematics Let-ters vol 83 pp 169ndash175 2018

[24] I Stamova ldquoGlobal Mittag-Leffler stability and synchroni-zation of impulsive fractional-order neural networks withtime-varying delaysrdquo Nonlinear Dynamics vol 77 no 4pp 1251ndash1260 2014

[25] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[26] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks using bayesian inverse reinforcement learningrdquoIEEEACM Transactions on Computational Biology and Bio-informatics vol 16 no 4 pp 1250ndash1261 2019

[27] M Imani and U M Braga-Neto ldquoControl of gene regulatorynetworks with noisy measurements and uncertain inputsrdquo

IEEE Transactions on Control of Network Systems vol 5 no 2pp 760ndash769 2018

[28] Y He J Zeng MWu and C-K Zhang ldquoRobust stabilizationand controllers design for stochastic genetic regulatory net-works with time-varying delays and structured uncertaintiesrdquoMathematical Biosciences vol 236 no 1 pp 53ndash63 2012

[29] H-L Li J Cao H Jiang and A Alsaedi ldquoFinite-time syn-chronization of fractional-order complex networks via hybridfeedback controlrdquo Neurocomputing vol 320 pp 69ndash75 2018

[30] H Li L Zhang C Hu H Jiang and J Cao ldquoGlobal Mittag-Leffler synchronization of fractional-order delayed quater-nion-valued neural networks direct quaternion approachrdquoApplied Mathematics and Computation vol 373 Article ID125020 2020

[31] C Chen L Li H Peng and Y Yang ldquoAdaptive synchro-nization of memristor-based BAM neural networks withmixed delaysrdquo Applied Mathematics and Computationvol 322 pp 100ndash110 2018

[32] H Zhu R Rakkiyappan and X Li ldquoDelayed state-feedbackcontrol for stabilization of neural networks with leakagedelayrdquo Neural Networks vol 105 pp 249ndash255 2018

[33] Y Ebihara D Peaucelle and D Arzelier ldquoPeriodically time-varying memory state-feedback controller synthesis for dis-crete-time linear systemsrdquo Automatica vol 47 no 1pp 14ndash25 2011

[34] Y Wang Z Wang and J Liang ldquoOn robust stability ofstochastic genetic regulatory networks with time delaysa delay fractioning approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B (Cybernetics) vol 40 no 3pp 729ndash740 2010

[35] G Chesi and Y S Hung ldquoStability analysis of uncertaingenetic sum regulatory networksrdquo Automatica vol 44 no 9pp 2298ndash2305 2008

[36] T-H Kim Y Hori and S Hara ldquoRobust stability analysis ofgene-protein regulatory networks with cyclic activation-re-pression interconnectionsrdquo Systems amp Control Letters vol 60no 6 pp 373ndash382 2011

[37] W Zhang J-a Fang and Y Tang ldquoRobust stability for ge-netic regulatory networks with linear fractional un-certaintiesrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 4 pp 1753ndash1765 2012

[38] W Wang Y Dong S Zhong and F Liu ldquoFinite-time robuststability of uncertain genetic regulatory networks with time-varying delays and reaction-diffusion termsrdquo Complexityvol 2019 Article ID 8565437 18 pages 2019

[39] F-F Wang D-Y Chen X-G Zhang and Y Wu ldquoeexistence and uniqueness theorem of the solution to a class ofnonlinear fractional order system with time delayrdquo AppliedMathematics Letters vol 53 pp 45ndash51 2016

[40] I Podlubny Fractional Differential Equations ElsevierAmsterdam Netherlands 1999

[41] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquoJournal of Mathematical Analysis and Applications vol 328no 2 pp 1075ndash1081 2007

[42] K Diethelm N J Ford and A D Freed ldquoA predictor-cor-rector approach for the numerical solution of fractionaldifferential equationsrdquo Nonlinear Dynamics vol 29 no 14pp 3ndash22 2002

Complexity 19