Content uploaded by Pratap Anbalagan
Author content
All content in this area was uploaded by Pratap Anbalagan on Nov 04, 2021
Content may be subject to copyright.
Received: 23 May 2021 Revised: 1 August 2021 Accepted: 12 September 2021
DOI: 10.1002/asjc.2726
REGULAR PAPER
Global dissipativity and adaptive synchronization for
fractional-order time-delayed genetic regulatory networks
Zizhen Zhang1Weishi Zhang1Pratap Anbalagan2Mani Mallika Arjunan3
1School of Management Science and
Engineering, Anhui University of Finance
and Economics, Bengbu, China
2Research Centre for Wind Energy
Systems, Kunsan National University,
Gunsan-si, South Korea
3Department of Mathematics, School of
Arts, Science and Humanities, SASTRA
Deemed to be University, Thanjavur, India
Correspondence
Zizhen Zhang, School of Management
Science and Engineering, Anhui
University of Finance and Economics,
Bengbu 233030, China.
Email: zzzhaida@163.com
Abstract
This manuscript examines global dissipativity and synchronization criteria for
time-delayed fractional-order gene regulatory networks (FGRNs). The Lya-
punov stability theory is used to examine a class of FGRNs with delay arguments
that are inspired by the integer-order delayed gene regulatory network model.
Some sufficient criteria are established using the fractional-order comparison
principle and stability theorem for fractional-order time-delayed systems to
ensure the global dissipativity solution of FGRNs with both feedback regula-
tion and translation time delays. Furthermore, for FGRNs, the global asymptotic
synchronization condition is accomplished by constructing an adaptive delayed
feedback control method for the master–slave system. Finally, two numerical
examples are provided to demonstrate the applicability and superiority of the
conclusions obtained.
KEYWORDS
fractional-order, gene regulatory networks, global adaptive Synchronization, global dissipativity
1INTRODUCTION
Fractional calculus and its growth have attracted numer-
ous scientists and engineers over the past few decades,
owing to its increasing applications in several disciplines,
such as biology, biophysics, chemistry, and signal pro-
cessing. Compared with traditional integer-order systems,
fractional-order systems have more degrees of freedom
and infinite memory [1,2]. Because of these advantages,
the integration of fractional-order calculus into nonlin-
ear dynamical system has gained attention and resulted in
several new developments in this domain [3–5].
Rudolf Emil Kalman drafted the development of genetic
regulatory networks back thirty years ago. Gene expression
in an organism is regulated by mRNA and protein. The
interconnections between those two were defined by the
gene regulatory networks (GRNs). GRNs are considered as
complex networks. Each gene is considered as a node, and
the regulatory connection between these genes is known
as the node relationship. Recently, GRNs complex dynam-
ical behaviors have drawn a lot of attention and numerous
applications have been identified in several fields [6,7].
Usually, there are two kinds of network models, such as
the boolean model and the differential equation model,
which are developing from GRNs. Among these, the dif-
ferential equation model is commonly used to describe the
gene regulation mechanism [8–10].
The concept of dissipativity, which was first discussed
in the early 1970s, is one of the most important char-
acteristics of nonlinear dynamical systems. As an exten-
sion of the Lyapunov theory, the dissipativity of frac-
tional dynamical system has already emerged as a hot
research trend and a lot of scientific results have been
well-documented in recent literature [11–13]. As a result,
© 2021 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
Asian J Control. 2021;1–10. wileyonlinelibrary.com/journal/asjc 1
2ZHANG ET AL.
several researchers have investigated at the dissipativity of
GRNs, although their findings are based on integer-order
cases only [14–16].
Nowadays, the synchronization of a dynamical system
with fractional-order cases is advancing research dom-
inant and has attracted much interest from engineers,
mathematicians, and researchers and their application in
many fields such as cryptography, secure communication,
and signaling. Several researchers have studied different
form of synchronization problems in the existing litera-
ture [17,18]. GRN synchronization is essential for knowing
the synergies behavior between the networks of more than
one gene through the connections between gene signals
and their products. The advantages of researching GRNs
synchronization are obtaining information about a gene's
internal mechanism even at cellular levels. In recent years,
synchronization analysis has drawn a remarkable amount
of attention among researchers (see Ali et al., Jiang et al.,
and Yue et al. [19–21]).
As we all know, the main trait of fractional-order
derivative is an infinite memory. Because of this fea-
ture, several researchers incorporated fractional calcu-
lus into genetic regulatory networks and developed the
fractional-order GRNs (FGRNs), yielding some interesting
scientific results, such as synchronization [22], stability
[23], and bifurcation [24]. Qiao et al. [22] designed two
different types of controllers, like state feedback control,
which is better and easier to execute than the other con-
trols, and adaptive feedback control, which is designed
to prevent high feedback gains and can adjust the cou-
pling weights by itself. They investigated the finite time
delay independent synchronization criteria for FGRNs
with feedback regulation delays by the one-norm method,
Banach contraction mapping, and Razumikhin approach.
Ren et al. [23] have developed several stability criteria for
FGRNs by using the one-norm method, Banach contrac-
tion mapping theorem, and Mittag–Leffler function. Tao
et al. [24] have established the local stability and instability
problems of FGRNs with time delays by stability and diffu-
sion analysis. To the best of the author's knowledge, there
is no work done on the global dissipativity and adaptive
synchronization criteria for FGRNs with both feedback
regulation and translation time delays, which is the main
objective of the present study.
Motivated by the above discussion, in this manuscript,
we developed a class of FRNs with time delays.
The highlights of our contribution are listed in brief
as follows:
1) On the basis of Ren et al. and T. Stamov and I. Stamova
[23,25], we add translation and feedback time delay
to analyze its dynamic characteristics. This gives our
research results more practical significance.
2) Based on the fractional-order comparison principle
and absolute value Lyapunov function of norm 1,
some sufficient conditions are obtained to ensure the
global dissipativity and synchronization analysis of
the considered FGRNs.
3) Themaintoolsusedinthismanuscriptarethealge-
braic matrix element method, fractional-order cal-
culus theory, the fractional-order comparison prin-
ciple, and some inequalities related to the Caputo
fractional-order derivative.
4) Compared with the quadratic Lyapunov functional or
Lyapunov–Krasovskii functional, the Lyapunov func-
tion of norm 1 is constructed to reduce the computa-
tional burden.
5) At the end, we have provided some numerical exam-
ples to exemplify the essence of our main concepts.
2BASIC TOOLS FOR
FRACTIONAL-ORDER
DERIVATIVES
The required notations are displayed as follows: Rrefers
to the set of real numbers and Crefers to the set of com-
plex numbers. N+denotes the set of positive integers and
Rmrefers to the space of m-dimensional space. A set
of all m×mreal matrix is described by Rm×m. If vector
=(1,…,
m)T∈Rm,wehave =m
x=1x.
M,and Mare the two parameter and one parameter
Mittag–Leffler functions, respectively. sign(·) indicates the
signum function. C([−, 0],Rm)indicates the set of all
continuous functions from [−,0]to
Rm,where>0. Γ(·)
refers to the Gamma function.
Definition 1 ([2]).The -th Caputo-type fractional
order derivative for a function (t) is denoted as fol-
lows:
C
0D
t(t)= 1
Γ(m−)∫t
0
(m)()
(t−)−m+1d,
where m−1<<m∈N+.
Definition 2 ([2]).The Mittag–Leffler function with
two parameters is denoted as follows:
M,()=
+∞
l=0
l
Γ(l+),
where , ∈R+,∈C.
Definition 3 ([2]).For m−1< <m, the Laplace
transform of the Mittag–Leffler function with two
ZHANG ET AL.3
parameter is denoted as follows:
t−1M,(t)=s−
s−,Re(s)>
,
where sand tare both variables in the Laplace domain
and time domain, respectively.
Before discussing the global dissipativity of the FGRNs,
we first present some basic results.
Now, consider the following Caputo-fractional-order lin-
ear delay differential system:
DG(t)=ΛG(t)+G(t1)+G(t2),0<<1,(1)
where Λ=xm×m∈Rm×m,G(t1)=
m
=1p1g(t−1
1),. …,m
=1pmg(t−1
m)T,G(t2)=
m
=1q1g(t−2
1),. …,m
=1qmg(t−2
m)T,
G(t)=(g1(t),…,gm(t))T∈Rm. Especially, if 1
x=1,
2
x=2,x=1,2,…,m,P=pxm×m∈Rm×m,
Q=qxm×m∈Rm×m, the system (1) can be written as
DG(t)=ΛG(t)+PG(t−1)+QG(t−2),0<<1,(2)
where G(t−1)=(g1(t−11),g2(t−12),…,gm(t−1m))T
and G(t−2)=(g1(t−21),g2(t−22 ),…,gm(t−2m))T.
Then, by taking the Laplace transform of (2), we get
s+Δ
11 Δ12 …Δ
1m
Δ21 s+Δ
22 …Δ
2m
⋮⋮⋱⋮
······························
Δm1Δm2⋮s+Δ
mm
Ξ1(s)
Ξ2(s)
.
.
.
Ξm(s)
=
1(s)
2(s)
.
.
.
m(s)
with
x(s)=s−1𝓁x(0)+px1e−s1
x1∫0
−1
x1
e−st𝓁1(t)dt
+qx1e−s2
x1∫0
−2
x1
e−st𝓁1(t)dt
+px2e−s1
x2∫0
−1
x2
e−st𝓁2(t)dt
+qx2e−s2
x2∫0
−2
x2
e−st𝓁2(t)dt
+..... +pxme−s1
xm ∫0
−1
xm
e−st𝓁m(t)dt
+qxme−s2
xm ∫0
−2
xm
e−st𝓁m(t)dt,
where Δ11 =−11 −p11e−s1
11 −q11e−s2
11 ,Δ12 =
−12 −p12e−s1
12 −q12e−s2
12 ,Δ1m=−1m−p1me−s1
1m−
q1me−s2
1m,Δ21 =−21 −p21e−s1
21 −q21e−s2
21 ,Δ22 =
−22 −p22e−s1
22 −q22e−s2
22 ,Δ2m=−2m−p2me−s1
2m−
q2me−s2
2m,Δm1=−m1−pm1e−s1
m1−qm1e−s2
m1,Δm2=
−m2−pm2e−s1
m2−qm2e−s2
m2,Δmm =−mm −pmme−s1
mm −
qmme−s2
mm ,Υ(s) represent the characteristic matrix of sys-
tem (2) and det (Υ(s))stands for the characteristic polyno-
mial of Υ(s). It is obvious that the stability of system (2) is
completely determined by the distribution of eigenvalues
of Υ(s).
Lemma 1 ([26]).Consider the following fractional-
order delay differential inequality
Du(t)≤−u(t)+u(t−1)+u(t−2),t>0,0<≤1
u(t)=(t),t∈[−,0],
(3)
and fractional-order linear system with time delays
Dv(t)=−v(t)+v(t−1)+v(t−2),t>0,0< ≤1
v(t)=(t),t∈[−,0],
(4)
where the non-negative continuous functions u(t)and
v(t)aredefinedin[0,+∞), and v(t)≥0, t∈[−,0]. If ,
,>0, then u(t)≤v(t), ∀t∈[0, +∞).
Lemma 2 ([26]).If 0 <<1, all the eigenvalues esof
Ωholds arg (e)>
2and the characteristic equation
det (Λ(s))=0has no pure imaginary roots for any 1,
2>0, then the zero solution of system (1) is Lyapunov
globally asymptotically stable.
Lemma 3 ([27]).For 0 <<1, a non-decreasing deriv-
able function (t) is defined on positive, then there exists
a constant >0 such that
C
0D
t((t)−)2≤2((t)−)C
0D
t(t).
3RESEARCH PROBLEM
In this study, we consider a class of FGRNs represented by
the following fractional delay differential equations:
C
0D
tux(t)=−dxux(t)+m
=1hxv(t−1)+Kx,
C
0D
tvx(t)=−wxvx(t)+gxux(t−2),
(5)
where x=1,2, .., m,0< <1 is the fractional order,
ux(t),vx(t)∈Rmare the concentrations of mRNA and
protein of pth node at time t, respectively. Decay rates of
mRNA and protein molecules, respectively, are described
4ZHANG ET AL.
by dxand wx. The translation rate is described by gx.1>0
is the feedback regulation delay, and 2>0isthetransla-
tion delay. hxy is the element of coupling matrix. Besides,
the functions fy(·) is the nonlinear protein feedback regula-
tion. The coupling matrix of the network H=(hx)m×m∈
Rm×mis described as follows:
hx=
−x,is a repressor of gene x
0,does not regulate gene x
x,is a initiator of gene x.
Now, we define Kxas Kx=∈
Ghx,where
Gis
the set of all repressor of gene x. The initial val-
ues of FGRNs (5) can be described as:ux(t)=
x(t),vx(t)=x(t),t∈[−=−max{1,
2},0],
where x(t),
x(t)∈C([−,0],Rm)and its norm is
denoted by =m
x=1sup−1≤𝓁≤0{x(𝓁)}, =
m
x=1sup−2≤𝓁≤0{x(𝓁)}.
In the development of the main results, the following
definition and assumptions are important.
Definition 4. FGRNs (5) is said to be a dissipative
system, if there exists a compact set R2m,
such that, for all uT
0,vT
0T∈R2m,∃T,when
t≥t0+T,uT(t,t0,u0),vT(t,t0,v0)T∈where
uT(t,t0,u0),vT(t,t0,v0)T∈R2mis the solution of
FGRNs (5) from initial state uT
0,vT
0T∈R2mand
initial time t0.Inthiscase,is known as a globally
attractive set.
Assumption 1. The feedback function fy(·)ismono-
tonically increasing; it is fulfilled that
0≤(u)−(v)
u−v≤,=1,2,…,m,
for all u,v∈Rwith u≠v.
Assumption 2. For all x, =1,2, .., m,
min
1≤x≤m{dx,wx}sin
2>max
1≤x≤m{gx}+max
1≤x≤mm
=1
xhx.
4MAIN RESULTS
In this portion, we will present the dissipative and synchro-
nization criteria for FGRNs with both feedback regulation
and translation time delays.
4.1 Dissipative criteria for FGRNs
Theorem 1. Under the Assumption 1, the FGRNs (5) is
a globally dissipative system and the set
=uT,vTT∈R2m∕(u(t)1+v(t)1)≤
−−
is globally attractive set, where =min1≤x≤m
{dx,wx}, =max1≤x≤mm
=1xhx, =
max1≤x≤m{gx},=max1≤x≤m{Kx}.
Proof. Consider the following Lyapunov functional
H(u(t),v(t))=
m
x=1ux(t)+
m
x=1vx(t).(6)
ccording to Assumption 1, we have
C
0D
tH(u(t),v(t))=
m
x=1
C
0D
tux(t)+
m
x=1
C
0D
tvx(t)
≤m
x=1−dxux(t)+
m
=1hxv(t−1)+Kx
+
m
x=1
{−wxvx(t)+gxux(t−2)}
≤−H(u(t),v(t))+H(u(t−1),v(t−1))
+H(u(t−2),v(t−2))+.
(7)
Consider the fractional-order linear system with
time delays as follows:
C
0D
tG(u(t),v(t))=−G(u(t),v(t))
+G(u(t−1),v(t−1))
+G(u(t−2),v(t−2))+,
(8)
where G(u(t),v(t))has same initial condition with
H(u(t),v(t))and assume G(u(t),v(t))≥0. Then,
by using Lemma 1, one has 0 ≤H(u(t),v(t))≤
G(u(t),v(t)),∀t∈[0,+∞). Based on Caputo
fractional-order derivative properties, then
Equation (8) can be written as
C
0D
tG(u(t),v(t))=−G(u(t),v(t))
+G(u(t−1),v(t−1))+G(u(t−2),v(t−2)).
(9)
where G(u(t),v(t))=G((u(t),v(t))−)and =
−−. Based on the similar procedure for Υ(s)in
ZHANG ET AL.5
Section 2, one can obtain that the characteristic
equation of det(Υ(s)) = 0 fulfills
det (Υ(s))=s+−e−s1−e−s2=0.(10)
Next, we demonstrate that, there is no purely imag-
inary roots for characteristic equation of det(Υ(s)) = 0
for any 1,2≥0 by contradiction. Suppose that s=
i=(cos(
2)+isin(
2)) has purely imaginary roots
of (10), where is real number. If
<0,s=i=cos
2−isin
2,and if
>0,s=i=cos
2+isin
2.
Substituting s=i=cos(
2)+isin(±
2)into s+
−e−s1−e−s2gives
cos(
2)+isin(±
2)+−[cos(1)
−isin(1)]−[cos(2)−isin(2)]=0.
(11)
Splitting the real and imaginary parts of (11) gives
cos(p
2)+−cos(1)−cos(2)=0
sin(±
2)+sin(1)+sin(2)=0.
From the above equations, we have
2cos2(
2)+2+2cos(
2)+2sin2(±
2)
=[cos(1)+cos(2)]2+[sin(1)+sin(2)]2.
(12)
Consider
[cos(1)+cos(2)]2
+[sin(1)+sin(2)]2
=2+2+2 cos (1−2).
By using Assumption 2, we get +<sin(
2),which
means the characteristic equations det (Υ(s))has no
purely imaginary roots for any 1,2>0. Then, we
show that all the eigenvalues of B=−++holds
arg (e(B)>
2.As,
+<sin
2<,
the matrix B=−++have negative eigenval-
ues, that is arg (e(B))>
2. Based on Lemma 2, the
zero solutions of system (9) is globally asymptotically
stable. Hence
G((u(t),v(t))−)→0ast→+∞.
Then, there exists a sufficiently small number >0
such that
G((u(t),v(t))) <+,
where G((u(t),v(t))) ≥0. In view of Lemma 1, one has
0≤H((u(t),v(t))) ≤G((u(t),v(t))) ,t∈[0,+∞).
Then, there exists T>0suchthat
H((u(t),v(t))) <+, t>T.
That is,
H((u(t),v(t))) ≤=
−−,t>T,
which leads to
u(t)1+v(t)1≤
−−,t>T.
Therefore, for uT
0,vT
0T∈R2m,thereexistst>T>0
such that
uT(t,t0,u0),vT(t,t0,v0)T∈R2m,
the FGRNs (5) is a globally dissipative system.
4.2 Synchronization criteria for FGRNs
FGRNs (5) acts as the master system and the correspond-
ing slave system is
C
0D
thx(t)=−dxhx(t)+
m
=1
hxs(t−1)+Kx+ux(t),
C
0D
tsx(t)=−wxsx(t)+gxhx(t−2)+vx (t).
(13)
where x=1,2, .., m,hx(t)∈Rmand sx(t)∈Rm
indicate the concentrations of mRNA and protein,
respectively. ux(t)andvx(t) are adaptive feedback con-
troller and all others are same as one of system (5).
The initial values of FGRNs (13) can be described
as follows: hx(t)= x(t),sx(t)= x(t),t∈[−, 0],
where x(t),x(t)∈C([−,0],Rm)and its norm is
denoted by =m
x=1sup−1≤𝓁≤0{x(𝓁)}, =
6ZHANG ET AL.
m
x=1sup−2≤𝓁≤0{x(𝓁)}.Let zux(t)=hx(t)−ux(t)and
zvx(t)=sx(t)−vx(t), then based on (5) and (13), the
synchronization error system is described by
C
0D
tzux(t)=−dxzux(t)+m
=1hxzv(t−1)+ux(t),
C
0D
tzvx(t)=−wxzvx(t)+gxzux (t−2)+vx (t),
(14)
where x=1,2, .., m,zv(t−1)=s(t−1)−
v(t−1). Next, we introduce the following adaptive
feedback control strategy:
ux(t)= −bx(t)zux(t)−xsign (zux(t))zux (t−2),
C
0D
tbx(t)=lxzux(t),
vx(t)= −cx(t)zvx (t)−xsign (zvx(t))zvx (t−1),
C
0D
tcx(t)=nxzvx (t),
(15)
where x=1,2, .., m,x,x,lx,nxare positive constants;
bx(t)andcx(t) are adaptive coupling strengths.
Theorem 2. Under the Assumption 1, master sys-
tem (5) synchronizes with slave system (13) under the
adaptive feedback controller (15) if
1=min
1≤x≤m{dx+bx}>0,
2=min
1≤x≤m{wx+cx}>0,
3=max
1≤x≤mx−
m
=1hxx>0,
4=max
1≤x≤m{x−gx}>0,
where bxand cxare arbitrary positive constants.
Proof. Consider the following Lyapunov functional
H(u(t),v(t))=
m
x=1zux(t)+
m
x=1zvx(t)
H1(u(t),v(t))
+
m
x=1
1
2lx
(bx(t)−bx)2+
m
x=1
1
2nx
(cx(t)−cx)2
H2(u(t),v(t))
.
(16)
According to Assumption 1 and Lemma 3, we have
C
0D
tH(u(t),v(t))=
m
x=1
C
0D
tzux(t)+
m
x=1
C
0D
tzvx(t)
+
m
x=1
1
2lx
C
0D
t(bx(t)−bx)2+
m
x=1
1
2nx
C
0D
t(cx(t)−cx)2
≤m
x=1−dxzux(t)+
m
=1hxzv(t−1)
−bx(t)zux(t)−xzux (t−2)
+
m
x=1
{−wxzvx(t)+gxzux (t−2)
−cx(t)zvx(t)−xzvx (t−1)}
+
m
x=1
(bx(t)−bx)zux(t)+
m
x=1
(cx(t)−cx)zvx(t)
=∶−
m
x=1
(dx+bx)zux(t)
+
m
x=1−x+
m
=1hxxzvx(t−1)
−
m
x=1
(wx+cx)zvx(t)+
m
x=1
(−x+gx)zux(t−2).
(17)
Selecting the values of bx,cx,x,xyields dx+bx>
0,wx+cx>0,
x−m
=1hxx>0,
x−gx>0.
Further, we get
C
0D
tH(u(t),v(t))≤−1
m
x=1zux(t)+3
m
x=1zvx(t−1)
−2
m
x=1zvx(t)+4
m
x=1zux(t−2)
≤−1
m
x=1zux(t)−2
m
x=1zvx(t)
=∶ −H(u(t),v(t)),
(18)
where =min{1,
2}. From (18), there exists a
non-negative function Π(t) holds
C
0D
tH(u(t),v(t))+ Π(t)=−H1(u(t),v(t)).(19)
Taking the Laplace transform of (19), we have
sH(u(s),v(s))−s−1H(u(0),v(0))+Π(s)=−H1(u(s),v(s)),
(20)
where H(u(0),v(0))is the initial condition of
H(u(t),v(t)),H(u(s),v(s))={H(u(t),v(t))},
ZHANG ET AL.7
H1(u(s),v(s))={H1(u(t),v(t))} and Π(s)={Π(t)}.
From (16), we have H1(u(t),v(t))≤H(u(t),v(t)).
Hence, there exists a function Π1(t)≥0suchthat
H(u(t),v(t))=H1(u(t),v(t))+Π
1(t).(21)
Taking the Laplace transform of (21), we have
H(u(s),v(s))=H1(u(s),v(s))+Π
1(s),(22)
where Π1(s)={Π1(t)}. Substitute (22) into (20), we
have
H1(u(s),v(s))=s−1H(u(0),v(0))
s+−sΠ1(s)
s+−Π(s)
s+.
(23)
Taking the inverse Laplace transform of (23), we
have
H1(u(t),v(t))=H(u(0),v(0))M(−t)
−Π
1(t)t−1∗M,0(−t)
− Π(t)t−1∗M, (−t),
where ∗refers to the convolution operator. It should be
pointed out that M, (−t),M,0(−t),t−1,andt−1
are non-negative functions. Therefore
H1(u(t),v(t))=H(u(0),v(0))M(−t).(24)
From (24), one can simply get
limt→+∞H1(u(t),v(t))=0. For H1(u(t),v(t))=
m
x=1zux(t)+m
x=1zvx(t),wehavezux(t)→0and
zvx(t)→0. Therefore, master system (5) synchronizes
with slave system (13) under the adaptive feedback
controller (15).
Remark 1. There are many results about dissipa-
tive and synchronization criteria for fractional-order
dynamical models [11–13,18], and a few results dis-
cuss synchronization of FGRNs [22]. However, no one
has looked into the dissipative and synchronization
criterion for FGRNs that have both feedback and trans-
lation time delays. Hence, our proposed research work
is purely different from the existing literature in the
sense of novelty.
Remark 2. If lx=nx=0in(15),basedonC
0D
tbx(t)=
0,C
0D
tcx(t)=0, we conclude that bx(t)=bx,cx(t)=cx,
where bx,cxare some scalars. Then adaptive feed-
back controllers (15) reduces into the following state
feedback controller:
ux(t)=−bxzux(t)−xsign (zux (t))zux(t−2),
vx(t)=−cxzvx(t)−xsign (zvx (t))zvx (t−1),
bx>0andcx>0 are constants.
Remark 3. This study constitutes the first attempt to
determine the dissipative and synchronization criteria
for FGRNs with both feedback regulation and transla-
tion time delays. This analysis takes into consideration
of the fractional-order comparison principle, feedback
regulation time delays 1, translation time delays 2,
adaptive feedback control, and algebraic inequalities.
The key benefit of this manuscript is that it simply
explains how to deal with the effects of feedback reg-
ulation time delays and translation time delays of the
involved system using the fractional-order compari-
son principle and stability theorem for fractional-order
time delay system.
5NUMERICAL SIMULATIONS
This part provides two examples to verify the superiority
and advantages of obtained dissipative and synchroniza-
tion criteria.
Example 1. Consider the 3-D FGRNs model:
C
0D0.99
tux(t)=−dxux(t)+3
=1hxv(t−1)+Kx,
C
0D0.99
tvx(t)=−wxvx(t)+gxux(t−2),t≥0,x=1,2,3,
(25)
where d1=d2=d3=3, w1=w2=w3=3.5,
g1=g2=g3=2, K1=3.2,K2=−4.1,K3=2,
1=2=0.9, (v(t)) = v2
(t)
1+v2
(t),andhx3×3=
0.80.45 0.75
0.2−0.25 0.4
−0.22 0.30.7.The initial conditions of (25)
are u0=[−0.2,0.5,−0.3]Tand v0=[0.2,−0.3,0.1]T.
From Assumption 1, we have 1=2=3=0.3.
According to Theorem 1, the FGRNs (25) is a globally dissi-
pative with (u(t)1+v(t)1)≤6.508, which is displayed
in Figure 1.
Example 2. Consider the 2-D FGRNs model:
C
0D0.985
tux(t)=−dxux(t)+2
=1hxv(t−1)+Kx,
C
0D0.985
tvx(t)=−wxvx(t)+gxux(t−2),t≥0,x=1,2,
(26)
where d1=d2=1.5, w1=w2=1.3, g1=g2=2.7,
K1=K2=0.1, 1=2=0.5, (v(t)) = v2
(t)
1+v2
(t),and
8ZHANG ET AL.
FIGURE 1 Time Response state variable of FGRNs (25) [Color figure can be viewed at wileyonlinelibrary.com]
FIGURE 2 Time response of master–slave synchronization error states with controller [Color figure can be viewed at
wileyonlinelibrary.com]
FIGURE 3 Time response curves of b(t)andc(t) [Color figure can be viewed at wileyonlinelibrary.com]
hx2×2=−2.12.8
2.4−2.6.The initial conditions of (26)
are u0=[2,1.2]Tand v0=[1,2]T. The corresponding
slave system is described by
C
0D0.985
thx(t)=−dxhx(t)+
2
=1
hxs(t−1)
+Kx+ux(t),
C
0D0.985
tsx(t)=−wxsx(t)+gxhx(t−2)+vx (t).
(27)
where ux(t), vx (t) are adaptive feedback control, the
initial conditions of (26) are h0=[−1.5,2]Tand s0=
[−1,−2.3]T, respectively. From Assumption 1, we have
1=2=0.5. According to adaptive feedback con-
trol (15), we fix 1=2=3, 1=2=4, l1=l2=
0.45,n1=n2=0.5, b1(0)=b2(0)=0.1andc1(0)=
c2(0)=0.2. Let us take bx=cx=0.1. Thus, all the con-
ditions of Theorem 2 hold. It follows from Theorem 2
that FGRNs (26) synchronize with FGRNs (27) based
on the adaptive feedback controller (15). Under the
adaptive feedback controller (15), the time response of
synchronization error states between (26) and (27) is
presented in Figure 2. The adaptive coupling weights
are shown in Figure 3.
6CONCLUSIONS
In this research work, the global dissipativity and syn-
chronization criteria for FGRNs with both feedback regu-
lation and translation time delays have been investigated
in brief. Theorem 1 is derived to investigate the global
dissipativity solution of addressed FGRNs based on the
fractional-order Lyapunov functional and fractional-order
comparison principle. Theorem 2 is established to dis-
cuss the global asymptotic synchronization conditions
for addressing master–slave systems by means of the
Laplace transform, Mittag–Leffler function properties, and
designed adaptive delayed feedback control strategy. Fur-
thermore, the main results obtained in this manuscript
have been expressed in terms of matrix elements, which
greatly reduces the computational burden. Finally, the
theoretical results of Theorems 1 and 2 are verified numer-
ically through two numerical examples. In the future, the
ZHANG ET AL.9
above analysis method can be extended to some other
dynamical behavior analysis such as stability, stabilization,
passivity, synchronization, and so on for fractional-order
time-delayed GRNs with leakage, discrete, and distributed
time delays.
AUTHOR CONTRIBUTIONS
Zizhen Zhang: Weishi Zhang: Pratap Anbalagan:
Mani Mallika Arjunan: conceptualization, formal anal-
ysis, methodology, software, validation, visualization.
ORCID
Zizhen Zhang https://orcid.org/0000-0002-2879-4434
REFERENCES
1. K. S. Miller and B. Ross, An introduction to the fractional calculus
and fractional differential equations, Wiley, New York, 1993.
2. I. Podlubny, Fractional differential equations, Academic Press,
San Diego California, 1999.
3. I. Stamova, T. Stamov, and X. Li, Global exponential stability of
a class of impulsive cellular neural networks with supremums,In.
J. Adap. Control Sig. Process. 28 (2014), 1227–1239.
4. M. Wang, B. Jia, C. Chen, X. Du, and F. Du, Discrete fractional
Bihari inequality and uniqueness theorem of solutions of nabla
fractional difference equations with non-Lipschitz nonlinearities,
Appl. Math. Comput. 376 (2020), 125118.
5. H. Zhang, R. Ye, J. Cao, A. Ahmed, X. Li, and Y. Wan, Lyapunov
functional approach to stability analysis of Riemann-Liouville
fractional neural networks with time-varying delays, Asian J.Con-
trol 20 (2018), no. 5, 1938–1951.
6. A. Becskei and L. Serrano, Engineering stability in gene networks
by autoregulation, Nature 405 (2000), 590–593.
7. P. Smolen, D. A. Baxter, and J. H. Byrne, Mathematical modeling
of gene networks,Neuron.26 (2000), 567–580.
8. P. Balasubramaniam and L. J. Banu, Robust state estimation
for discrete-time genetic regulatory network with random delays,
Neurocomputing 122 (2013), 349–369.
9. J. H. Koo, D. H. Ji, S. C. Wona, and J. H. Park, An improved
robust delay-dependent stability criterion for genetic regulatory
networks with interval time delays, Commun. Nonlinear Sci.
Numer. Simul. 17 (2012), no. 8, 3399–3405.
10. S. Lakshmanan, J. H. Park, H. Y. Jung, P. Balasubramaniam, and
S. M. Lee, Design of state estimator for genetic regulatory networks
with time-varying delays and randomly occurring uncertainties,
Biosystems 111 (2013), no. 1, 51–70.
11. Z. Ding and Y. Shen, Global dissipativity of fractional-order
neural networks with time delays and discontinuous activations,
Neurocomputing 196 (2016), 159–166.
12. Y. Fan, X. Huang, Z. Wang, and Y. Li, Global dissipativity and
quasi-synchronization of asynchronous updating fractional-order
memristor-based neural networks via interval matrix method,J.
Franklin Inst. 355 (2018), no. 13, 5998–6025.
13. G. Velmurugan, R. Rakkiyappan, V. Vembarasan, J. Cao, and
A. Alsaedi, Dissipativity and stability analysis of fractional-order
complex-valued neural networks with time delay, Neural Netw. 86
(2017), 47–53.
14. Y. Ma, Q. Zhang, and X. Li, Dissipative control of Markovian
jumping genetic regulatory networks with time-varying delays and
reaction-diffusion driven by fractional Brownian motion, 2017.
https://doi.org/10.1007/s12591-017-0349-7
15. R. Manivannan, J. Cao, and K. T. Chong, Generalized dissipativ-
ity state estimation for genetic regulatory networks with interval
time-delay signals and leakage delays, Commun. Nonlinear Sci.
Numer. Simul. 89 (2020), 105326.
16. H. Shen, Z. Huang, J. Xia, J. Cao, and J. H. Park, Dissipativity
analysis of switched gene regulatory networks actuated by persis-
tent dwell-time switching strategy, 2019. https://doi.org/10.1109/
TSMC.2019.2956281
17. F. Liu, Z. Zhang, and X. Wang, Stability and synchronization con-
trol of fractional-order gene regulatory network system with delay,
J. Adv. Comput. Intell. Intell. Inform. 21 (2017), no. 1, 148–152.
18. R.Ye,X.Liu,H.Zhang,andJ.Cao,GlobalMittag-Leffler synchro-
nization for fractional-order BAM neural networks with impulses
and multiple variable delays via delayed-feedback control strat-
egy, Neural. Process. Lett. 49 (2019), no. 1, 1–18.
19. M. S. Ali, R. Agalya, and K. S. Hong, Non-fragile synchronization
of genetic regulatory networks with randomly occurring controller
gain fluctuation, Chin. J. Phys. 62 (2019), 132–143.
20. N. Jiang, X. Liu, W. Yu, and J. Shen, Finite-time stochastic syn-
chronization of genetic regulatory networks, Neurocomputing 167
(2015), 314–321.
21. D. Yue, Z. H. Guan, T. Li, R. Q. Liao, F. Liu, and Q. Lai,
Event-based cluster synchronization of coupled genetic regulatory
networks, Physica A 482 (2017), 649–665.
22. Y. Qiao, H. Yan, L. Duan, and J. Miao, Finite-time synchroniza-
tion of fractional-order gene regulatory networks with time delay,
Neural Netw. 126 (2020), 1–10.
23. F. Ren, F. Cao, and J. Cao, Mittag-Leffler stability and gener-
alized Mittag-Leffler stability of fractional-order gene regulatory
networks, Neurocomputing 160 (2015), 185–190.
24. B. Tao, M. Xiao, Q. Sun, and J. Cao, Hopf bifurcation analysis
of a delayed fractional-order genetic regulatory network model,
Neurocomputing 275 (2018), 677–686.
25. T. Stamov and I. Stamova, Design of impulsive controllers and
impulsive control strategy for the Mittag-Leffler stability behav-
ior of fractional gene regulatory networks, Neurocomputing 424
(2021), 54–62.
26. T. Manchun and P. Qi, Global stability analysis of delayed
complex valued neural networks with nodes of different
dimensions, Int. J. Mach. Learn. Cybern. 10 (2019), no. 5,
897–912.
27. J. Yu, C. Hu, and H. Jiang, Projective synchronization for frac-
tional neural networks, Neural Netw. 49 (2014), 87–95.
AUTHOR BIOGRAPHIES
Dr. Zizhen Zhang working as Assis-
tant Professor in School of Man-
agement Science and Engineering,
Anhui University of Finance and Eco-
nomics, Bengbu, China. His current
research interests include stability
and bifurcation theory of dynamical
10 ZHANG ET AL.
systems. He has authored and co-authored for more
than 35 publications in these research areas.
Mr. Weishi Zhang working as a
present master in School of Man-
agement Science and Engineering,
Anhui University of Finance and Eco-
nomics, Bengbu, China. His current
research interests include stability
and bifurcation theory of dynamical
systems.
Dr. Pratap Anbalagan working as
a Post Doctorate in Kunsan National
University, South Korea. His current
research interests include stabil-
ity theory of neural networks and
fractional-order systems. He has
authored and co-authored for more
than 25 publications in these research areas and also
he served as a reviewer for more than 10 journals.
Dr. Mani Mallika Arjunan work-
ing as Senior Assistant Professor in
Department of Mathematics, School
of Arts, Science and Humanities, SAS-
TRA Deemed to be University, Than-
javur, Tamil Nadu, India. He is having
15 years of experience in both teach-
ing and research. He has completed PhD in 2008 and
published 113 papers which include 40 SCI/SCIE and
60 SCOPUS indexed journals.
How to cite this article: Z. Zhang, W.
Zhang, A. Pratap, and M.M. Arjunan, Global
dissipativity and adaptive synchronization for
fractional-order time-delayed genetic regula-
tory networks, Asian J. Control (2021), 1–10.
https://doi.org/10.1002/asjc.2726
