INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 16, Issue: 4, Month: August, Year: 2021
Article Number: 4215, https://doi.org/10.15837/ijccc.2021.4.4215
CCC Publications
Finite Time Synchronization of Inertial Memristive Neural Networks
with Time Varying Delay
Y.J. Pang, S.M. Dong
Yajing Pang*
Hebei University of Science and Technology
Shijiazhuang 050000, China
*Corresponding author: pangyj2020@163.com
Shengmei Dong
Shijiazhuang Nanyangzhuang Primary School
Shijiazhuang 050000, China
dsm3012431882@163.com
Abstract
Finite time synchronization control of inertial memristor-based neural networks with varying
delay is considered. In view of drive and response concept, the sufficient conditions to ensure finite
time synchronization issue of inertial memristive neural networks is given. Based on Lyapunov
finite time asymptotic theory, a kind of feedback controllers is designed for inertial memristor-
based neural networks to realize the finite time synchronization. Based on Lyapunov stability
theory, close loop error system can be proved finite time and fixed time stable. Finally, illustrative
example is given to illustrate the effectiveness of theoretical results.
Keywords: finite time, neural networks, synchronization.
1 Introduction
At present, the dynamic systems appearing in the fields of engineering applications and natural
sciences are becoming more and more complex, and control purposes are also diversified. In this case,
it is inevitable that control theory needs to provide more effective control methods and strategies. In
the actual industrial production and manufacturing, the control precision of the equipment is contin-
uously improved. At the same time, industrial production often requires multiple devices to achieve
synchronization effects in terms of time, position, and speed, which requires reasonable coordination
between events that occur in the system. Synchronization in the system can also be called timely or
synchronized.
The phenomenon in which two phases are in harmony is called chaotic synchronization, which
is actually a common phenomenon in nature. In the 1990s, Peroca and Carroll proposed a chaos
control method (P-C), and first observed chaotic synchronization in electronic circuits. In fact, the
phenomenon that several coupling units achieve synchronization through coordination not only occurs
https://doi.org/10.15837/ijccc.2021.4.4215 2
widely in nature, but also common in actual production applications, and is very important and has
to be researched, so synchronization control has always been a hot topic in the academic engineering
field.
As we all know, neuron is the basic unit of neural network function realization, with complex
nonlinear properties, such as chaos, periodicity and bifurcation. The composition of a neural network
includes a large number of neuron structures. A single neuron cannot realize the processing and
processing of complex information, and it requires the coupling of many neuron clusters to complete
it. Synchronization is a manifestation of neuron coupling, so the study of neuron synchronization is
particularly important.
Nowadays, with the continuous emergence of control problems and practical needs in the actual
engineering field, the research on synchronization stability in the academic engineering field is becoming
more and more popular, and a large number of research results and research directions have also
been produced, such as brain network synchronization, complex networks Synchronization, transient
synchronization, global synchronization, etc., and scholars have studied synchronization problems
under complex conditions. Even though there have been a lot of results in the field of synchronization,
there are still a lot of problems to be studied and solved in the world of nonlinear systems.
Chaos, as a very interesting nonlinear phenomenon, has been intensively investigated in many
fields of science and technology over the last four decades. Recently, chaos synchronization has at-
tracted increasing attention from various communities due to its powerfully potential applications in
laser physics, chemical reactor, secure communication, biomedical and so on [10, 21]. Many methods
have been proposed to achieve chaos control and synchronization, such as the passive control method,
backstepping design method, impulsive control method, adaptive control method, sliding mode con-
trol, control Lyapunov function (CLF) method and nonlinear feedback method, etc. The controllers
derived from the above methods are nonlinear. In a real industry process, because the linear feedback
controllers are economic and easy to implement, they possess a high value in applications [1, 4].
Since the first publication of chaotic synchronization by Pecora in naval Laboratory in 1990 [12], the
synchronization problem has attracted wide attention from scholars due to the widespread existence
of complex chaotic phenomena in nature and engineering systems [7]. The synchronization theory has
been reflected and applied in both nature and engineering circles, such as laser network synchronization
[22], resonance synchronization [18] and epilepsy [15]. In order to solve this complex nonlinear chaotic
behavior and completely reconstruct the chaotic states of the two systems, scholars have proposed
many synchronization strategies, such as generalized synchronization [6], complete synchronization
[23], coupled synchronization [9], etc.
As is known to all, neural network has attracted much attention in the field of academic engineering
for its unique characteristics. With the in-depth study of neural network and the development of syn-
chronization theory, network system synchronization has attracted extensive attention from scholars
in recent years. In Chu’s study [2], two consistent fractal-order differential inequalities are estab-
lished to deal with the global exponential quasi-synchronization of a consistent FCDN, and the global
exponential quasi-synchronization of a conformal fractal-order complex dynamic network (FCDN) is
realized by periodic and intermittent pinning control. Gan et al. [5] believed that a controller based on
Markov scheduling protocol was designed to solve the mean-square inverse synchronization problem
of periodic BAM neural network (NN) with time-varying delay. The theory of differential inclusion is
combined with the definition of Filippov solution, and the interval parameter system is established to
study the exponential synchronization of CVMDNN based on complex valued memristor [11]. And
in related study, sufficient conditions for finite time synchronization bounds are derived according to
Markov observations, thus verifying the finite time synchronization of coupled neural networks with
hopping internal coupling and non-fragile controllers [8]. Cui et al. [3] considered that the fixed-
time synchronization of Markov jumping fuzzy neural networks with random disturbance and leakage
time-varying delays was studied by designing time-dependent controllers with or without fuzzy terms.
According to the existing results, several control strategies can be given to realize the synchroniza-
tion strategy mentioned above. For example, sliding mode control [17], Backstepping control [19],
adaptive control ![13] and active control [20], etc. The above control strategies obviously have their
own advantages in different situations. Among them, there is a kind of linear feedback control [14]
https://doi.org/10.15837/ijccc.2021.4.4215 3
with extremely superior effect for chaotic system synchronization and differential synchronization, and
chaotic system synchronization has extremely high value in some practical applications [16].
The rest of this brief is organized as follows. The considered model and some necessary assumptions
are given in Section 2, Section 3 presents finite time and fixed time synchronization criteria and rigorous
mathematical proof. Section 4 given the numerical simulation and conclusion is given in Section 5.
2 Preliminaries
The inertial memristor-based neural networks model:
ẍi (t) = −aiẋi (t) − bixi (t) + cij (xi (t)) fj (xj (t)) + dij (xi (t)) fj (xj (t− τ (t))) + Ii (t)
i = 1, 2, . . . ,n (1)
where xi(t) represents the state of the i-th neuron; ai > 0,bi > 0 are constants, fi() is activation
function, τ(t) is varying delay, which satisfies 0 ≤ τ(t) ≤ τ, Ii(t) is external input, cij(xi(t)),dij(xi(t))
are memristive connection weights, which are given by
cij (x) =
{
c′ij, |x| ≤ Ti
c′′ij, |x| > Ti
dij (x) =
{
d′ij, |x| ≤ Ti
d′′ij, |x| > Ti
i,j = 1, 2, . . .n
(2)
where c′ij,c”ij,d′ij,d”ij and Ti are known constants.
By introducing the following variable substitution x1i,x2i as
x1i = xix2i = ẋi (3)
Then, the system can be rewritten as
ẋ1i = x2i
ẋ2i = −aix2i (t) − bix1i (t) + cij (x1i (t)) fj (x1j (t))
+dij (x1i (t)) fj (x1j (t− τ (t))) + Ii (t)
(4)
This is referred to as drive system, and the response system can be described by
ÿi (t) = −aiẏi (t) − biyi (t) + cij (yi (t)) fj (yj (t)) + dij (yi (t)) fj (yj (t− τ (t))) + Ii (t) (5)
Similarly, by introducing variable substitution
ẏ1i (t) = y2i (t) + u1i (t)
ẏ2i (t) = −aiy2i (t) − biy1i (t) + cij (y1i (t)) fj (y1j (t)) + dij (y1i (t)) fj (y1j (t− τ (t)))
+Ii (t) + u2i (t)
(6)
The synchronization errors as
e1i = y1i −x1ie2i = y2i −x2i (7)
Then we have {
ė1i (t) = e2i (t) + u1i (t)
ė2i (t) = −aie2i (t) − bie1i (t) + Fi (t) + u2i (t)
(8)
where,
Fi (t) = cij (y1i (t)) fj (y1j (t)) − cij (x1i (t)) fj (x1j (t))
+dij (y1i (t)) fj (y1j (t− τ (t))) −dij (x1i (t)) fj (x1j (t− τ (t)))
(9)
To design the finite time and fixed-time synchronization theory, some lemmas are proposed based
on a general nonlinear system
ẋ = f (x) (10)
where x is error system state.
https://doi.org/10.15837/ijccc.2021.4.4215 4
Lemma 1. For xi ∈ Randxi ≤ 0, i = 1, 2, . . . ,n, 0 < p < 1,q > 1, then(
n∑
i=1
xi
)p
≤
n∑
i=1
xi
p (11)
n1−q
(
n∑
i=1
xi
)q
≤
n∑
i=1
xi
q (12)
Lemma 2. Suppose the positive definite and continuous function V (t) meets the following differential
inequality fV̇ (t) ≤ −ρV η(t), t ≥ 0,V (0) ≥ 0, where 0 < η < 1 and ρ > 0 are constants. Then, V (t)
meets the following inequality V 1−η(t) ≤ V 1−η(0) − ρ(1 − η), 0 ≤ t ≤ T. Then the origin of system
(10) can achieve finite time stability, and T ≤ V
1−η(0)
ρ(1−η)
Lemma 3. Suppose that V (ů) : Rn → R + ∪0 is a continuous radically unbounded function and the
following two conditions hold:
(1)V (x) = 0 ⇔ x = 0
(2)Any solution x(t) of system equation reference goes here satisfies,V (x(t)) ≤ −aV p(x(t)) −
bV q(x(t)), for some a,b > 0, 0 ≤ p < 1andq > 1. Then the origin of system (10) can achieve
fixed-time stability, and Tmax = 1a(1−p) +
1
b(q−1)
3 Synchronization control
To derive the finite time synchronization criteria, the finite time synchronized feedback controller.
Theorem 1. Choose the synchronization controller as following:{
u1i (t) = −e2i (t) −k1ieη1i
u2i (t) = aie2i (t) + bie1i (t) −Fi (t) −k2ieη2i
(13)
where 0 < η < 1, then, drive system (1) and response system (5) can achieve finite time synchroniza-
tion, furthermore, finite time is
T ≤
V
1−η
2 (0)
2k (1 −η)
(14)
Proof 1. Choose the Lyapunov candidate functional:
V =
n∑
i=1
e21i (t) +
n∑
i=1
e22i (t) (15)
Calculate the derivative of V along the trajectories of the error system (8) as
V̇ = 2
n∑
i=1
ė1i (t) e1i (t) + 2
n∑
i=1
ė2i (t) e2i (t) (16)
Based on the controller (10), the error system (8) can be get as{
ė1i (t) = −k1ieη1i
ė2i (t) = −k2ieη2i
(17)
Take (13) into (12) as
V̇ = −
n∑
i=1
2k1ieη+11i (t) −
n∑
i=1
2k2ieη+12i (t) ≤−kV
η+1
2 (18)
where k = min(k1i,k2i), based on Lemma 2, the error system is finite time asymptotic stable is
T ≤ V
1−η
2 (0)
2k(1−η) , the proof is completed.
https://doi.org/10.15837/ijccc.2021.4.4215 5
Theorem 2. Choose the synchronization controller as following:{
u1i (t) = −e2i (t) −k1iep1i − l1ie
q
1i
u2i (t) = aie2i (t) + bie1i (t) −Fi (t) −k2iep2i − l2ie
q
2i
(19)
Proof 2. Choose the Lyapunov candidate functional:
V =
n∑
i=1
e21i (t) +
n∑
i=1
e22i (t) (20)
Calculate the derivative of V along the trajectories of the error system (8) as
V̇ = 2
n∑
i=1
ė1i (t) e1i (t) + 2
n∑
i=1
ė2i (t) e2i (t) (21)
Based on the controller (10), the error system (8) can be get as{
ė1i (t) = −k1iep1i − l1ie
q
1i
ė2i (t) = −k2iep2i − l2ie
q
2i
(22)
Take (13) into (12) as
V̇ = −
n∑
i=1
2k1iep+11i (t) −
n∑
i=1
2k2iep+12i (t) −
n∑
i=1
2l1ieq+11i (t) −
n∑
i=1
2l2ieq+12i (t) (23)
Based on Lemma 1
n∑
i=1
2k1iep+11i +
n∑
i=1
2k2iep+12i ≥ a
(
n∑
i=1
(
e21i
)p+1
2 +
n∑
i=1
(
e22i
)p+1
2
)
≥ a
(
n∑
i=1
e21i +
n∑
i=1
e22i
)p+1
2
(24)
where a = min(k1i,k2i).
n∑
i=1
2l1ieq+11i (t) +
n∑
i=1
2l2ieq+12i (t) ≥ b1
(
n∑
i=1
(
e21i
)q+1
2 +
n∑
i=1
(
e22i
)q+1
2
)
≥ b
(
n∑
i=1
e21i +
n∑
i=1
e22i
)q+1
2
(25)
where , b = (2n)
1−q
2 b1,b1 = min(2l1i, 2l2i), therefore we have
V̇ ≤−aV
p+1
2 − bV
q+1
2 (26)
Based on Lemma 3, the error system is fixed-time asymptotic stable is Tmax = 2a(1−p) +
2
b(q−1) , the
proof is completed.
4 Numerical simulations
In this section, we will use an example to illustrate how to apply the theory results proposed in
this brief in applications. The chaotic neural network proposed is adopted as the drive system:
ẋ (t) = −dixi (t) +
∑3
j=1
aij (xi (t)) fi (xj (t)) +
∑3
j=1
bij (xi (t)) gj (xj (t− τj (t))) (27)
The corresponding response system as follows
ẏi (t) = −diyi (t) +
∑n
j=1
aij (yi (t)) fi (yj (t)) +
∑n
j=1
bij (yi (t)) gj (yj (t− τj (t))) + ui (t) (28)
https://doi.org/10.15837/ijccc.2021.4.4215 6
where,
a11 (x1) =
{
1.2, |x1| ≤ 1
1.8, |x1| > 1
,a12 (x1) =
{
−2.1, |x1| ≤ 1
−1.6, |x1| > 1
,a13 (x1) =
{
2.1, |x1| ≤ 1
1.6, |x1| > 1
a21 (x2) =
{
−3.6, |x2| ≤ 1
−2.3, |x2| > 1
,a22 (x2) =
{
1.4, |x2| ≤ 1
2.3, |x2| > 1
,a23 (x2) =
{
−1.4, |x2| ≤ 1
−2.2, |x2| > 1
a31 (x3) =
{
1.6, |x3| ≤ 1
1.3, |x3| > 1
,a32 (x3) =
{
1.4, |x3| ≤ 1
2.1, |x3| > 1
,a33 (x3) =
{
1.5, |x3| ≤ 1
2.2, |x3| > 1
b11 (x1) =
{
1.1, |x1| ≤ 1
1.7, |x1| > 1
,b12 (x1) =
{
1.5, |x1| ≤ 1
1.2, |x1| > 1
,b13 (x1) =
{
1.5, |x1| ≤ 1
1.2, |x1| > 1
b21 (x2) =
{
1.3, |x2| ≤ 1
1.1, |x2| > 1
,b22 (x2) =
{
−1.2, |x2| ≤ 1
−2.2, |x2| > 1
,b23 (x2) =
{
−1.1, |x2| ≤ 1
−2.3, |x2| > 1
b31 (x3) =
{
2.6, |x3| ≤ 1
2.3, |x3| > 1
,b32 (x3) =
{
2.4, |x3| ≤ 1
2.1, |x3| > 1
,b33 (x3) =
{
3.5, |x3| ≤ 1
3.2, |x3| > 1
The activation functions and time-varying delays are taken as f(x) = g(x) = tanh(x).
Figure 1: Time evolution of variable of coupled neural networks x1,y1
The time delay τ = 1, the initial condition choose as x1(0) = 3,x2(0) = 5,y1(0) = 4,y2(0) = 7.
Based on Theorem 1, choose the control gain as η = 1/3,k = 0.3, then the error system is finite
time stable. The simulated results are shown in Figure 1 and Figure 2, indicate the response system
synchronizes the driver system in finite time.
Figure 2: Time evolution of variable of coupled neural networks x2,y2
https://doi.org/10.15837/ijccc.2021.4.4215 7
The time delay τ = 1, the initial condition choose as x1(0) = 3,x2(0) = 5,y1(0) = 4,y2(0) = 7.
Based on Theorem 2, choose the control gain as p = 1/3,q = 5/3,k = 1, then the error system is
fixed-time stable. The simulated results are shown in Figure 3 and Figure 4, indicate the response
system synchronizes the driver system in fixed time. Simulation result indicated effectiveness and
feasibility of the proposed control method.
Figure 3: Time evolution of variable of coupled neural networks x3,y3
Figure 4: Chaotic attractor of neural networks
5 Conclusion
In this paper, the problem of finite time and fixed time synchronization of a class of chaotic neural
networks has been considered. The method introduced for linear feedback finite time and fixed time
control is very effective, and it is simple to implement in practice. Based on Lyapunov stability theory,
finite time and fixed time stability is proved. To the end, numerical simulation is given to show the
effectiveness and feasibility of the developed method.
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Cite this paper as:
Pang, Y.J.; Dong, S.M. (2021). Finite time Synchronization of Inertial Memristive Neural Networks
with Time Varying Dela, International Journal of Computers Communications & Control, 16(4), 4215,
2021.
https://doi.org/10.15837/ijccc.2021.4.4215
Introduction
Preliminaries
Synchronization control
Numerical simulations
Conclusion