MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————
LE DAO HAI AN
STABILITY OF NONLINEAR TIME-DELAY SYSTEMS
AND THEIR APPLICATIONS
Speciality: Differential and Integral Equations
Speciality code: 9 46 01 03
SUMMARY OF DOCTORAL THESIS IN MATHEMATICS
HANOI-2019
This dissertation has been written on the basis of my research work carried at:
Hanoi National University of Education
Supervisor:
Assoc. Prof. Le Van Hien
Dr. Tran Thi Loan
Referee 1: Professor Nguyen Minh Tri, Institute of Mathematics, Vietnam Academy
of Science and Technology
Referee 2: Professor Cung The Anh, Hanoi National University of Education
Referee 3: Associate Professor Nguyen Xuan Thao, Hanoi University of Science and
Technology
The thesis will be presented to the examining committee at Hanoi National University
of Education, 136 Xuan Thuy Road, Hanoi, Vietnam
At the time of ...., 2020
This dissertation is publicly available at:
- HNUE Library Information Centre
- The National Library of Vietnam
INTRODUCTION
1. Motivation
Time delays are widely used in modeling practical modelsin control engineering, bi-
ology and biological models, physical and chemical processes or artificial neural networks.
The presence of time-delay is often a source of poor performance, oscillation or instability.
Therefore, the stability of time-delay systems has been extensively studied during the past
decades. It is still one of the most burning problems in recent years due to the lack or the
absence of its complete solution.
A popular approach in stability analysis for time-delay systems is the use of the
Lyapunov-Krasovskii functional (LKF) method to derive sufficient conditions in terms of
linear matrix inequalities (LMIs). However, it should be noted that finding effective LKF
candidates for time-delay systems is often connected with serious mathematical difficulties
especially when dealing with nonlinear non-autonomous systems with bounded or unbounded
time-varying delay. In addition, extending the developed methodologies and existing results
in the literature to nonlinear time-delay systems proves to be a significant issue. This re-
search topic, however, has not been fully investigated, which gives much room for further
development in particular for nonautonomous nonlinear systems with delays in the area of
population dynamics and network control. This motivates us for the present study in this
thesis.
2. Research aims
This thesis is concerned with the stability of some classes of nonlinear time-delay
systems in neural networks. Specifically, we consider the following problems
1. Investigating the problem of stability of non-autonomous neural networks with hetero-
geneous time-varying delays in the effect of destablizing impulses.
2. Stabilizing Hopfiled neural networks with proposition delays subject to stabilizing and
destablizing impulsive effects simultaneously.
3. Investigating the problem of exponential stability of positive equilibrium of inertial
neural networks with multiple time-varying delays.
4. Deriving conditions for the problem of exponential stability of a unique equilibrium of
positive BAM neural networks with multiple time-varying delays and nonlinear self-
excitation rates.
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3. Objectives
3.1. Global exponential stability analysis of a class of non-autonomous neural
networks with heterogeneous delays and time-varying impulses
The states of various dynamical networks in the fields of artificial systems such as
mechanics, electronic and telecommunications networks, often suffer from instantaneous dis-
turbances and undergo abrupt changes at certain instants. These may arise from switching
phenomena or frequency changes, and thus, they exhibit impulsive effects. With the effect of
impulses, stability of the networks may be destroyed. Therefore, delays and impulses heavily
affect the dynamical behaviors of the networks, and thus it is necessary to study both effects
of time-delay and impulses on the stability of neural networks. Up to now, considerable
effort of researchers has been devoted to investigating stability and asymptotic behavior of
neural networks with impulses.
However, the aforementioned works have been devoted to neural networks with constant
coefficients. As discussed in the many exsting literatures, non-autonomous phenomena often
occur in realistic systems, for instance, when considering a long-term dynamical behavior
of the system, the parameters of the system usually change along with time. Also, the
problem of stability analysis for non-autonomous systems usually requires specific and quite
different tools from the autonomous ones (systems with constant coefficients). There are
only few papers concerning stability of non-autonomous neural networks with heterogeneous
time-varying delays and impulsive effects.
In Chapter 2 we investigate the exponential stability of a class of non-autonomous
neural networks with heterogeneous delays and time-varying impulses
x
i(t) = di(t)xi(t) +
n
X
j=1
aij (t)fj(xj(t))
+
n
X
j=1
bij (t)gj(xj(tτij (t))) + Ii(t), t > 0, t 6=tk,
xi(tk),xi(t+
k)xi(t
k) = σikxi(t
k), k N.
(1)
Based on the comparison principle, an explicit criterion is derived in terms of inequal-
ities for M-matrix ensuring the global exponential stability of the model under destabilizing
impulsive effects. The obtained results are shown improve some recent existing results. Fi-
nally, numerical examples are given to demonstrate the effectiveness the proposed conditions.
3.2. Exponential stability of impulsive neural networks with proportional delay
in the presence of periodic distribution impulses
Typically, a model of neural networks is composed of layers with a large number of cells
and connections. This fact reveals that NNs usually have a spatial nature due to the number
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of parallel pathways, axon sizes and lengths. Thus, time delays encountered in the practical
implementation of NNs are usually time-varying. Proportional delays form a particular type
of unbounded time-varying delays, which are widely used in modeling various models in the
field of networking. It is realized that proportional delay provides most well-known quality of
service (QoS) models because of its controllable and predictable characteristics. Specifically,
when a network with proportional delays is utilized to represent an applied model, dynamics
of the system at time tis determined by its states x(t) and x(qt), where 0 < q < 1 is a
constant representing the ratio of time between current states and historical states. Thus,
the network’s running time can be controlled by the proportional factor q. Recently, the
problem of stability of various neural network models with proportional delays has attracted
considerably increasing research attention and, consequently, a large number of interesting
results have been reported in the literature.
On the other hand, impulsive dynamical systems (IDSs), in general, and impulsive
neural networks with delays (IDNNs) have received considerable research attention in recent
years. According to their strength, impulsive effects can be classified into two types named
as stabilizing impulses (SI) and destabilizing impulses (DI). An impulsive sequence is said
to be destabilizing if its effect can suppress the stability of dynamical systems while SI
can enhance the stability of dynamical systems. In most of the existing works concerning
stability of impulsive systems, SI and DI are considered separately.
In the second part of Chapter 2 we study the problem of exponential stability of the
following neural networks model
x
i(t) = dixi(t) +
n
X
j=1
aij fj(xj(t)) +
n
X
j=1
bij gj(xj(qt)) + ui(t), t 6=tk,
xi(tk) = xi(t+
k)xi(t
k) = σikxi(t
k),
(2)
Both stabilizing and destabilizing impulsive effects are introduced in the model simulta-
neously. Based on the comparison principle, a unified stability criterion is first derived.
Then, on the basis of the derived stability conditions, the problem of designing a local state
feedback control law with bounded controller gains is addressed.
3.3. Positive solutions and global exponential stability of positive equilibrium
of inertial neural networks with multiple time-varying delays.
Conventional neural networks are typically described by first-order differential equa-
tions with or without delays. Recently, many authors focused on dynamics behaviors of
networks models called inertial neural networks (INNs). In state-space models, INNs are
described by systems of second-order differential equations, where the first-order derivative
terms are referred to as inertial terms. On one hand, there exist strong biological and en-
gineering backgrounds for the introduction of inertial terms in neural systems, in particular
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