(Luận án tiến sĩ) approximation problems for dynamic equations on time scales , bài toán xấp xỉ cho phương trình động lực trên thang thời gian 624601

VIETNAM NATIONAL UNIVERSITY, HANOI
HANOI UNIVERSITY OF SCIENCE

Nguyen Thu Ha

APPROXIMATION PROBLEMS FOR DYNAMIC
EQUATIONS ON TIME SCALES

THESIS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI – 2017


VIETNAM NATIONAL UNIVERSITY, HANOI
HANOI UNIVERSITY OF SCIENCE

Nguyen Thu Ha

APPROXIMATION PROBLEMS FOR DYNAMIC
EQUATIONS ON TIME SCALES

Speciality: Differential and Integral Equations
Speciality Code: 62 46 01 03

THESIS FOR THE DEGREE OF

DOCTOR OF PHYLOSOPHY IN MATHEMATICS

Supervisor: PROF. DR. NGUYEN HUU DU

HANOI – 2017


ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

Nguyễn Thu Hà

BÀI TỐN XẤP XỈ CHO PHƯƠNG TRÌNH ĐỘNG
LỰC TRÊN THANG THỜI GIAN

Chuyên ngành: Phương trình Vi phân và Tích phân
Mã số: 62 46 01 03

LUẬN ÁN TIẾN SĨ TOÁN HỌC

Người hướng dẫn khoa học:
GS. TS. NGUYỄN HỮU DƯ

HÀ NỘI – 2017


Contents
Page
Abstract

v

Tóm tắt

vi

List of Figures

vii

List of Notations

ix

Introduction

1

Chapter 1

Preliminary

11

1.1

Definition and example

. . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2

Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1. Continuous function . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2. Delta derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3. Nabla derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3

Delta and nabla integration . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1. ∆ and ∇ measures on time scales . . . . . . . . . . . . . . . . 17
1.3.2. Integration

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.3. Extension of integral . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.4. Polynomial on time scales . . . . . . . . . . . . . . . . . . . . 21
1.4

Exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1. Regressive group . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2.

Exponential function

. . . . . . . . . . . . . . . . . . . . . . 23

1.4.3. Exponential matrix function . . . . . . . . . . . . . . . . . . . 25
1.5

Exponential stability of dynamic equations on time scales . . . . . . . 26
i

1.5.1. Concept of the exponential stability

. . . . . . . . . . . . . . 26

1.5.2. Exponential stability of linear dynamic equations with constant coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6

Hausdorff distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 2

On the convergence of solutions for dynamic equations

on time scales

34

2.1

Time scale theory in view of approximative problems . . . . . . . . . 34

2.2

Convergence of solutions for ∆-dynamic equations on time scales . . . 36
2.2.1. The existence and uniqueness of solutions . . . . . . . . . . . 36
2.2.2. Convergence of solutions . . . . . . . . . . . . . . . . . . . . . 38
2.2.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3

On the convergence of solutions for nabla dynamic equations on time
scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.1. Nabla exponential function . . . . . . . . . . . . . . . . . . . . 51
2.3.2. Nabla dynamic equation on time scales . . . . . . . . . . . . . 52
2.3.3. Convergence of solutions for nabla dynamic equations . . . . . 53
2.3.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4

Approximation of implicit dynamic equations . . . . . . . . . . . . . 55

Chapter 3

On data-dependence of implicit dynamic equations on

time scales
3.1

58

Region of the uniformly exponential stability for time scales . . . . . 58
3.1.1. Stability region of time scales . . . . . . . . . . . . . . . . . . 59
3.1.2. Dependence of stability regions on time scales . . . . . . . . . 64

3.2

Data-dependence of spectrum and exponential stability of implicit
dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.1. Index of pencil of matrices . . . . . . . . . . . . . . . . . . . . 70

3.2.2. Solution of linear implicit dynamic equations with constant
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.3. Spectrum of linear implicit dynamic equations with constant
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
ii


3.3

Data-dependence of stability radii . . . . . . . . . . . . . . . . . . . . 79
3.3.1. Stability radius of linear implicit dynamic equations . . . . . . 80
3.3.2. Data-dependence of stability radii . . . . . . . . . . . . . . . . 82

Conclusion

91

The author’s publications related to the thesis

93

Bibliography

94


Acknowledgments
First and foremost, I want to express my deep gratitude to Prof. Dr. Nguyen Huu
Du for accepting me as a PhD student and for his help and advice while I was
working on this thesis. He has always encouraged me in my work and provided me

with the freedom to elaborate my own ideas.
I also want to thank Dr. Do Duc Thuan and Dr. Le Cong Loi for all the help they
have given to me during my graduate study. I am so lucky to get their support.
I wish to thank the other professors and lecturers at Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science for their teaching, continuous
support, tremendous research and study environment they have created. I also thank
to my classmates for their friendship and suggestion. I will never forget their care
and kindness. Thank you for all the help and making the class like a family.
Last, but not least, I would like to express my deepest gratitude to my family.
Without their unconditional love and support, I would not be able to do what I
have accomplished.

Hanoi, December 27, 2017
PhD student

Nguyen Thu Ha.

iv


Abstract
The characterization of analysis on time scales is the unification and expansion of
results obtained on the discrete and continuous time analysis. In some last decades,
the study of analysis theory on time scales leads to much more general results and
has many applications in diverse fields. One of the most important problems in
analysis on time scales is to study the quality and quantity of dynamic equations
such as long term behaviour of solutions; controllability; methods for solving numerical solutions... In this thesis we want to study the analysis theory on time scales
under a new approach. That is, the analysis on time scales is also an approximation problem. Precisely, we consider the distance between the solutions of different
dynamical systems or study the continuous data-dependence of some characters of
dynamic equations.
The thesis is divided into two parts. Firstly, we consider the approximation problem

to solutions of a dynamic equation on time scales. We prove that the sequence of
solutions xn (t) of dynamic equation x∆ = f (t, x) on time scales {Tn }∞
n=1 converges
to the solution x(t) of this dynamic equation on the time scale T if the sequence of
these time scales tends to the time scale T in Hausdorff topology. Moreover, we can
compare the convergent rate of solutions with the Hausdorff distance between Tn
and T when the function f satisfies the Lipschitz condition in both variables.
Next, we study the continuous dependence of some characters for the linear implicit
dynamic equation on the coefficients as well on the variation of time scales. For
the first step, we establish relations between the stability regions corresponding a
sequence of time scales when this sequence of time scales converges in Hausdorff
topology; after, we give some conditions ensuring the continuity of the spectrum
of matrix pairs; finally, we study the convergence of the stability radii for implicit
dynamic equations with general structured perturbations on the both sides under
the variation of the coefficients and time scales.

v


Tóm tắt
Đặc trưng của giải tích trên thang thời gian là thống nhất và mở rộng các nghiên
cứu đã đạt được đối với giải tích trên thời gian liên tục hoặc thời gian rời rạc. Trong
các thập niên vừa qua, việc nghiên cứu lý thuyết giải tích trên thang thời gian cho
ta nhiều kết quả tổng quát và có nhiều ứng dụng vào các lĩnh vực khác nhau. Một
trong những bài tốn quan trọng của giải tích trên thang thời gian là nghiên cứu
tính chất định tính và định lượng của phương trình động lực. Trong luận án này,
chúng tơi muốn nghiên cứu lý thuyết giải tích trên thang thời gian theo cách tiếp
cận mới. Đó là giải tích trên thang thời gian cịn là bài tốn xấp xỉ. Cụ thể hơn,
chúng tôi xét khoảng cách giữa các nghiệm của các hệ động lực khác nhau và sẽ
nghiên cứu sự phụ thuộc liên tục của một số đặc trưng của phương trình động lực

theo dữ liệu của phương trình.
Luận án bao gồm hai phần chính. Trước hết, chúng tơi xét bài tốn xấp xỉ nghiệm
của phương trình động lực trên thang thời gian và chứng minh được dãy các nghiệm
xn (t) của phương trình x∆ = f (t, x) trên dãy thang thời gian tương ứng {Tn }∞
n=1 sẽ
hội tụ đến nghiệm x(t) của phương trình này trên thang thời gian T nếu như dãy
thang thời gian này hội tụ về thang thời gian T theo khoảng cách Hausdorff. Hơn
nữa, chúng tôi cũng đánh giá được tốc độ hội tụ của các nghiệm theo tốc độ hội tụ
của dãy thang thời gian khi hàm f thỏa mãn điều kiện Lipschitz theo cả hai biến.
Tiếp theo, ta nghiên cứu sự phụ thuộc theo tham số và theo sự biến thiên của thang
thời gian của một số đặc trưng của phương trình động lực ẩn tuyến tính. Bước đầu
tiên, ta thiết lập mối liên hệ giữa các miền ổn định tương ứng của dãy các thang
thời gian khi dãy thang thời gian này hội tụ theo tô pô Hausdorff. Cuối cùng, chúng
ta nghiên cứu sự hội tụ của bán kính ổn định của phương trình động lực ẩn tuyến
tính chịu nhiễu cấu trúc ở cả hai vế của phương trình khi cả hệ số và thang thời
gian đều biến thiên.

vi


Declaration
This work has been completed at Hanoi University of Science, Vietnam National
University under the supervision of Prof. Dr. Nguyen Huu Du. I declare hereby that
the results presented in it are new and have never been used in any other thesis.
Author:

Nguyen Thu Ha

vii

List of Figures
1.1

Points of the time scale T. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1

The graph of the solution xn (t) on the time scale Tn . . . . . . . . . . 49

2.2

The graph of the solution x4 (t) on the time scale T4 . . . . . . . . . . 50

2.3

The graph of the solution xn (t) on the time scale Tn . . . . . . . . . . 55

viii


List of Notations
Time scale

T
κ

T \ {Tmax } if T has a left-scattered maximum Tmax

κT

T \ {Tmin } if T has a right-scattered minimum Tmin

T

otherwise T k = T
Tτ

= {t ∈ T : t

τ}

σ(t)

Forward jump operator

ρ(t)

Backward jump operator

µ(t)

Graininess (forward) of a time scale

ν(t)

Graininess (backward) of a time scale

C(X, Y )

The set of continuous functions from X to Y

Crd (T, X)

Space of rd-continuous functions f : T → X

1
Crd
(T, X)

Space of rd-continuously differentiable functions
f : Tk → X .

Crd R(T, X)

Space of rd-continuous and regressive functions
f :T→X

C− , C+

open left and right half complex plane.

d(x, X)

Distance between point x and subset X

d(X, Y )

Hausdorff distance of two non-empty subsets X and Y

det A

The determinant of matrix A

eα (t, s)

∆- Exponential function with parameter α

eα (t, s)

∇- Exponential function with parameter α

f (t−)

limσ(s)↑t f (s)

GL(Rm )

The set of linear automorphism of Rm

λ

Imaginary part of complex number λ

λ

Real part of complex number λ
A field, to be replaced by an element from {R, C}

K

K

m×n

Linear space of m × n− matrices on K

Im A

Valued domain of operator A

Ker A

Kernel of operator A

rank A

Rank of matrix A

L(X)

Space of linear continuous operator f: X → X

Ln

The principal logarithm function with the
valued-domain is [−iπ, iπ)
ix


N, Q, R, C

The set of natural, rational, real, complex numbers

N0

= N ∩ {0}

R(Tk , X)

The set of regressive functions, defined on T
and take the value on X

+

k

R (T , R)

The set of positive regressive function defined on T
and taking values in R

R+

The set of positive real numbers

UT = UT(T)

The uniformly exponential stability domain of time
scale T.

σ(A)

The set of the eigenvalues of the matrix A

σ(A, B)

The set of solutions of det(λA − B) = 0

S(T)

Exponential stability domain of time scale T

sup, inf

supremum, infimum

x


Introduction
The theory of ordinary differential equations (ODEs for short) is a theoretical system, rather academic but going into the practical issues. In practical problems,
ordinary differential equations occur in many scientific disciplines, for instance in
physics, chemistry, biology, and economy. Under the name "modeling", the ordinary
differential equation is a useful tool to describe the evolution in times of population.
Therefore, studying the qualitative and quantitative properties of differential equations is important both in theory and practice. For the qualitative properties, the
long term behavior of the solutions has got many interests. The main tools in studying the stability are Lyapunov functions, Lyapunov exponents or spectral analysis of
matrices. For the quantitative analysis, we have to find numerical approximations to
the solutions since almost ODEs can not be solved analytically. The Euler methods
are well-known because it is simple and useful, see [9, 19, 22, 63].
Besides, the theory of difference equations has a long term of history. Right from

the dawn of mathematics, it was used to describe the evolution of a lapin population
with the name "Fibonacci sequence". Difference equations might define the simplest
dynamical systems, but nevertheless, they play an important role in the investigation
of dynamical systems. The difference equations arise naturally when we want to
study mathematical models describing real life situations such as queuing problems,
stochastic time series, electrical networks, quanta in radiation, genetics in biology,
economy, psychology, sociology, etc., on a fixed period of time. They can also be
illustrated as discretization of continuous time systems in computing process.
On the other hand, in some last decades, the theory of time scales, under the name
Analysis on time scales, was introduced by Stefan Hilger in his PhD thesis
(supervised by Bernd Aulbach) in order to unify continuous and discrete analyses. As soon as this theory was born, it has been received a lot of attention, see
[6, 7, 13, 39, 41, 43]. Until now, there are thousands of books and articles dealt
with the theory of analysis on the time scale. Many familiar results concerning to
1


qualitative theory such as stability theory, oscillation, boundary value problem in
continuous and discrete times were "shifted" and "generalized" to the time scale.
Studying the theory of time scales leads also to some important applications such
as in the study of insect density model, nervous system, thermal conversion process,
the quantum mechanics and disease model.
One of the most important problems in the analysis on time scales is to investigate
dynamic equations. Many results concerning differential equations carry over quite
easily to corresponding results for the difference equations, while other results seem
to be completely different in nature from their continuous counterparts. The study
of dynamic equations on time scales gives a better perspective and reveals such discrepancies between the differential equations and difference equations. Moreover, it
helps us avoid proving results twice, one for differential equations and one for difference equations. The general idea is to prove results for dynamic equations where
the domain definition of the unknown function is a so-called time scale T, which is
an arbitrary nonempty closed subset of real numbers R. By choosing the time scale
to be the set of real numbers R, the general result yields a result concerning an

ordinary differential equation as studies in a first course in differential equations.
On the other hand, if the time scale is the set of integers Z, the same general result
yields a result for difference equations.
However, studying the theory of dynamic equations on time scales leads to much
more general results since there are many more other complex time scales than the
above two sets. That is why so far the analysis on time scales is still an attracted
topic in mathematical analysis. Especially, there are still many open problems in
the studying dynamic equations on time scales.
The aim of this thesis is to consider the analysis on time scales under a new point
of view. That is not only a unification, but also in the view of approximation. Precisely, we want to consider the distance between the solutions of different dynamical
systems or to study the continuous dependence of some characters of dynamics equations on data such as spectrum, stability radii when both the coefficients and time
scales vary. The two following topics will be dealt with in this thesis:
1. Approximation of solutions
We begin firstly by analyzing the Euler method for solving the stiff initial value
problem. It is known that there are not many classes of ordinary differential equations for which we can represent explicitly their solutions via analysis formulas,
especially for nonlinear differential equations. Therefore, finding numerical solu2


tions of differential equations plays an important role in both theory and practice.
So far one proposes many algorithms to solve numerically solutions of a differential
equation. Among these algorithms, we have to take into account the Euler explicit
and implicit methods. Let us consider the initial value problem (IVP)

x(t)
˙
= f (t, x(t)),
x(t ) = x .
0
0

(0.1)

The approximation of the solution x(t) of (0.1) will carry out at some different
values of times, say mesh points, on the interval [t0 , T ]. To do that, for every n ∈ N,
we consider a partition of [t0 , T ] consisting of points
(n)

(n)

(n)

(n)

t0 = t0 < t1 < · · · < tkn −1 < tkn := T, kn ∈ N.

(0.2)

Based on the points in (0.2), one constructs a difference equation
(n)

x0 = x0 ,

(n)

(n)

xi+1 = xi

(n)

(n)

(n)

(n)

+ (ti+1 − ti )f (ti , xi ),
(n)

i = 0, 1, . . . , kn − 1.

(n)

(0.3)

(n)

(n)

Then, the sequence of points (tk , xk ), k = 1, ..., kn evaluates the points (tk , x(tk )),
k = 1, 2, ..., kn on the solution curve starting from x0 at t0 . We call this approximative way by Euler explicit method or Euler forward method.
Our problem is to give conditions of the function f and the partition (0.2) ensuring
the convergence of explicit Euler method:
(n)

(n)

sup |xk − x(tk )| → 0 khi n → ∞.

(0.4)

k

These conditions can be found in [19, 22, 44].
Although the explicit Euler method is quite simple and easy to implement, even
we can carry out by portable calculators, and we can show the convergence of this
method. However, it has accumulation error in the processes of calculation and
Euler scheme can also be numerically unstable, especially for stiff equations since
explicit method requires impractically small time steps to keep the error in the
result bounded and the convergent rate is not good. Therefore, one deals with the
second Euler method, called Euler implicit method or Euler backward method. In
this method, instead of the equation (0.3), we consider the difference equation
(n)

x0 = x0 ,

(n)

(n)

xi+1 = xi

(n)

(n)

(n)

(n)

+ (ti+1 − ti )f (ti+1 , xi+1 ),

i = 0, 1, . . . , kn − 1.
(n)

(0.5)
(n)

This differs from the Euler explicit method in that the latter uses f (tk , xk ) in
(n)

(n)

(n)

place of f (ti+1 , xi+1 ). In Euler implicit method, the new approximation xi+1 appears
on both sides of the equation (0.5) and thus the method needs to solve an algebraic
3


equation
x = y + hf (t, x),

(0.6)

where t, h and y are known and x is unknown. We can solve numerically the solution
x of (0.6) by iterative method
xk+1 = y + hf (t, xk ), k = 0, 1, ...
By fixed point theorem, if h is sufficiently small and f satisfies Lipschitz condition
then xk → x, which solves (0.6). It is clear that implicit methods require an extra

computation and they can be much harder to implement. However, people prefer
to use the Euler implicit method because many problems arising in practice are
stiff and it can achieve a higher convergent rate. The error at a specific time t
is O(h) (using the big O notation) and the Euler implicit method is in general
unconditionally stable. For a stiff problem, explicit methods need to take very small
time steps. Therefore, meanwhile the calculation is complicated, the Euler implicit
method could be preferred in this case (see [19, 22, 44]).
We now consider these two methods in different way. That is, when one discretises
the dynamic equation (0.1) on the real line [0, T ] by Euler explicit method, we
obtain the difference equation (0.3). On the time scale languages, this equation is
in fact the dynamic equation x∆ (t) = f (t, x(t)) on time scale Tn described by (0.2).
Similarly, the equation (0.5) is the dynamic equation x∇ (t) = f (t, x(t)) on Tn . When
the mesh steps of Euler methods tend to 0, the sequence of time scales Tn converges
to T in some senses and the convergence of Euler methods means the convergence
(n)

of the sequence of solutions x(·) , the solution of above equations on the time scales
Tn .
Therefore, for the first part of this thesis, we follow this idea to set up an approximation problem in a more general context: Let T be a time scale and let {Tn }∞
n=1
be a sequence of time scales, which converges to T in some senses. Consider the
dynamic equation

x∆ (t) = f (t, x(t)),
x(t ) = x ,
0
0

(0.7)

where either t ∈ T or t ∈ Tn . Assume that on every time scale Tn (resp. T), the
Cauchy problem of the equation (0.7) has a unique solution xn (t) (resp x(t)). Then,
the question here is whether we can specify conditions to have
xn (·) → x(·) as n → ∞.
Further, how can we estimate the rate of this convergence?
4

(0.8)


We can also deal with a similar problem for the nabla dynamic equations on time
scales. That is instead of considering the equation (0.7), we study the equation

x∇ (t) = f (t, x(t)),
(0.9)
x(t ) = x ,
0

0

and put similar questions on the convergence of solutions.
2. Continuous dependence of the spectrum and stability radius on data
The second topic dealt with in this thesis is to consider the data dependence of the
spectrum and stability radius for linear implicit dynamic equations on time scales.
In the recent years, several technical problems in electronic circuit theory and robotic
designs, chemical engineering, etc., see [15, 16, 28, 55] lead to the problem of investigating the dynamic implicit equation (IDEs for short)
f (X ∆ (t), X(t)) = 0,

(0.10)

where the leading term X ∆ can not be explicitly solved from X(t). The linear form
of this equation (LIDEs for short) is
AX ∆ (t) − BX(t) = 0,

(0.11)

where A and B are two constant matrices (see [24, 46]). According to [24] and [58],
the investigation of the so-called index of the pencil of matrices {A, B} is necessary
but the situation becomes more complicated. Note that if A is a nonsingular matrix,
then equation (0.11) can easily be reduced to an ordinary differential equation by
multiplying both sides of (0.11) by A−1 . In this case, it is known that if the original
equation (0.11) is exponentially stable then it is still stable under sufficiently small
perturbations. In general case where A may be degenerate, this property is no
longer valid for the equation (0.11) since the structure of the solutions of a LIDEs
depends strongly on the index of {A, B} (see [31, 45, 46, 58]) and the solutions of
(0.11) contain several components, which are related by algebraic relations. Under
perturbations, the index of the system might be changed, which leads to the change
of the algebraic relations.
In view of spectral theory, it is known that the uniformly exponential stability has
close relations with the spectrum σ(A, B) of the matrices pencil {A, B}, i.e., the set
of roots λ of the equation det(λA − B) = 0. The changing in parameters of index
causes, without additional assumptions, the sharp change of the spectrum σ(A, B)
and the continuity of spectrum on the data is no longer valid.
5


Then, the question whenever the spectrum σ(A, B) depends continuously in {A, B}
is important in both theory and practice. Thus, we come to the following problem
on time scales.
Problem: consider a family of linear implicit dynamic equations on the time scales

Tn
An x∆n (t) = Bn x(t), n ∈ N,

(0.12)

with An , Bn ∈ Cm×m . Which conditions ensure that the system Ax∆ (t) = Bx(t) on
the time scale T is exponentially stable if the system (0.12) is exponentially stable
for every n and lim (An , Bn , Tn ) = (A, B, T)?
n→∞

In parallel, we have a similar problem for stability radii of implicit dynamic equations. It is well-known that if the trivial solution x ≡ 0 of the linear differential
system x = Bx (resp. difference system xn+1 = Bxn ) is exponentially stable, then
for a small perturbation Σ, the system
x = (B + DΣE)x

(0.13)

xn+1 = (B + DΣE)xn ,

(0.14)

and respectively,

is still exponentially stable. Where Σ is an unknown disturbance matrix and D, E are
known scaling matrices defining the “structure" of the perturbation. The question
rises here: how large Σ is possible in order to keep the stability of the system (0.13)
(resp. (0.14)). The threshold between the stability and instability, which measures
the stability robustness of system to such perturbation, is called the stability radius.
It is defined as the smallest (in norm) complex or real perturbations destabilizing the
equation. If complex perturbations are allowed, this measure is called the complex

stability radius, if only real perturbation are considered the real radius is obtained.
The concept of stability radii was introduced by Hinrichsen and Pritchard [48] in
1986 for linear time-invariant systems of ordinary differential equations with respect
to time-invariant input, i.e., static perturbations. In this work, authors have shown
that the complex stability radius of linear differential equation (0.13) is given by
−1
−1

max E(tI − B) D
t∈iR

.

(0.15)

Since then, this problems have been getting a lot of attentions from many research
groups of mathematics in the world. D. Hinrichsen and N.K Son [52] (1989) have
investigated the difference equation with the perturbation Σ of the form xn+1 =
(B + DΣE)xn and have shown that the complex stability radius is computed by the
6


formula

−1
−1

E(ωI − B) D

max

.

(0.16)

ω∈C:|ω|=1

To unify the equation (0.15) and (0.16), recently in [30], the authors give a formula
of the stability radius for the dynamic equation on time scales
x∆ = Bx,

(0.17)

subjected to structure perturbations of the form
x∆ = (B + DΣE)x.
The stability radius of (0.17) is given by
−1

maxc E(tI − B)−1 D

,

t∈UT

(0.18)

where UT is the uniform stability region of the time scale T and UTc is the complement of UT.
The natural extension of the formula (0.18) to the implicit linear dynamic equation
on time scales belongs to [39]. In this work, authors consider the stability radius of
the linear implicit dynamic equation

Ax∆ (t) = Bx(t),

(0.19)

˜ B]
˜ = [A, B] + DΣE = [A + DΣE1 , B + DΣE2 ],
[A,

(0.20)

subject to the perturbations

where A, B ∈ Cm×m , D ∈ Cm×l , E1 , E2 ∈ Kq×m , E = [E1 , E2 ], the perturbation
Σ ∈ Cl×q . With these perturbations, the system (0.19) leads to
Ax∆ (t) = Bx(t),

(0.21)

authors have shown that the complex stability radius of the equation (0.19) is given
by the formula
−1

r(A, B; D, E; T) =

sup

G(λ)

,

(0.22)

λ∈UTc

where G(λ) = (λE 1 − E 2 )(λA − B)−1 D.
We emphasise that the form (0.20) says that both sides of (0.19) are perturbed by Σ.
As far as we know, the perturbation on the left side of (0.19) is very sensitive since
it can make its index change roughly. Following the analysis in [68], the stability
radius of the system (0.19) depends strongly in variation of the coefficients A, B.
7


Du-Lien-Linh for the first time in [36]; Du-Linh [32] and Du-Linh [34] consider this
continuous dependence under the statement of small parameters. They claim that
if r(E + εF, A; B, C) is the stability radius of
(E + εF )x = (A + BΣC)x,
Then, under some heavy assumptions, it is shown that
lim r(E + εF, A; B, C) = min{r(E, A; B, C), r(F22 , A22 ; B2 , C2 )},
ε↓0

where A =

A11 A12
A21 A22

; F =

F11 F12

and B = (B1 , B2 ) ; C = (C1 , C2 ) .

F21 F22

In order to generalize this result, in this thesis we are concerned with the following
problem.
Problem: Let us consider a sequence of systems
An x∆n (t) = Bn x(t),

(0.23)

where An , Bn ∈ Cm×m , t ∈ Tn and n ∈ N. The leading coefficients An , n ∈ N,
are allowed to be singular matrices. We want to investigate the structure of stability
regions and their relation, give conditions ensuring the continuous dependence on the
data of the stability radii for implicit dynamic equations (0.23) when (An , Bn , Tn )
converges.
The content of this thesis are as follows. Chapter 1 presents some basic knowledge
about time scales such as the definition of derivative, integration on time scales.
Besides, we give concepts of the exponential function, exponential stability region
as well as some results of the stability for the dynamic equations on time scales.
The second chapter is devoted to the study of the convergence of solutions to dynamic equations. We endow the set of time scales with the Hausdorff distance. Let
xn (t) (resp. x(t)) be the solution of the equation x∆ = f (t, x) (or the nabla dynamic
equation x∇ = f (t, x)) on time scale Tn (resp. on T). Under the assumption that
f (t, x) satisfies the Lipschitz condition in the variable x and the sequence of time
scales {Tn }∞
n=1 converges to the time scale T in the Hausdorff topology, Theorem
2.2.7 shows that lim xn (t) = x(t). Moreover, in case f satisfies the Lipschitz conn→∞

dition in both variables t and x, the convergent rate of solutions is estimated as the
same degree as the Hausdorff distance between Tn and T, i.e.,
xn (t) − x(t)

CdH (T, Tn ), for all t ∈ T ∩ Tn : t0

t

T,

(0.24)

(see Theorem 2.2.9). Using these results, we obtain the convergence of the explicit
Euler method as a consequence and we give some illustrative examples. It can be
8


considered as a novel approach to the convergence problems of the approximative
solutions.
In the last chapter, we study some problems concerning the stability regions, the
spectrum of matrix pairs, the exponential stability and their robustness measure
for linear implicit dynamic equations of arbitrary index (0.19) subjected to general
structured perturbations of the form
[An , Bn ]

[An , Bn ] + Dn Σn En ,

(0.25)

where Σn , n ∈ N, are unknown disturbance matrices; Dn , En are known scaling
matrices defining the “structure" of the perturbations. On each time scale Tn , the
stable radius of this system were defined by the formula (0.22). Some characteristics
of the stability regions corresponding to a convergent sequence of time scales are

derived. In more details, Theorem 3.1.7 tells us that the stability regions depend
continuously on the time scales. Concretely, if Tn tends to T in Hausdorff topology
then UT ⊂

∞
n=1

m n UTm

and

∞
n=1

m n UTm \ R

⊂ UT \ R. Next, Proposition

3.2.11 shows that if Ind(A, B) = 1 and lim (An , Bn ) = (A, B) and (An −A)Q = 0 for
n→∞

all n ∈ N, then we have lim σ(An , Bn ) = σ(A, B) in the Hausdorff distance. Further,
n→∞

Proposition 3.2.13 tells us that in case Ind(A, B) > 1 and lim (An , Bn ) = (A, B)
n→∞

and (An − A)Q = (Bn − B)Q = 0 for all n ∈ N we have lim σ(An , Bn ) = σ(A, B).
n→∞

Based on these propositions we can claim that the stability radii r(An , Bn , Dn ,
En ; Tn ) are lower semi continuous in An , Bn , Dn , En ; Tn . By Theorems 3.3.6 and
3.3.8 we see that with some further conditions, if lim (An , Bn , Dn , En ; Tn ) = (A, B, D,
n→∞

E; T) then
r(A, B; D, E; T) = lim inf r(An , Bn , Dn , En ; Tn ).
n→∞

In conclusion, we think that the theory of dynamic systems on an arbitrary time scale
was found promising because it demonstrates the interplay between the theories of
continuous-time and discrete-time systems, see [7, 27, 43, 47, 65]. It enables to
analyze the stability of dynamical systems on non-uniform time domains which
are subsets of real numbers [66]. Based on this theory, stability analysis on time
scales has been studied for linear time-invariant systems [61], linear time-varying
dynamic equations [26], implicit dynamic equations [39, 68], switched systems [66,
67] and finite-dimensional control systems [10, 29]. Therefore, it is meaningful to
investigate the dependence of stability characteristics of these systems on time scales
and coefficients as well.
Parts of the thesis have been published in
9


1. Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2016),
"On the convergence of solutions to dynamic equations on time scales", Qual.
Theory Dyn. Syst., 15(2), pp. 453–469.
2. Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2015),
"On the convergence of solutions to nabla dynamic equations on time scales",
Dyn. Syst. Appl., 24(4), pp. 451-465.
3. Nguyen Thu Ha, Nguyen Huu Du and Do Duc Thuan (2016), "On datadependence of stability regions, exponential stability and stability radii for implicit linear dynamic equations", Math. Control Signals Systems, 28(2), pp.

13-28.

10


Chapter 1

Preliminary

In this chapter we survey some basic notions related to the analysis on time scales.
We also introduce the so-called cylinder transformation order to define the uniform
stability region of a time scale and the uniformly exponential stability for the dynamic equations on time scales. The details for these concepts and definitions can
be referred to the famous papers of S. Hilger [43], C. Păotzsche, S. Siegmund and F.
Wirth [61] and the book of M. Bohner and A. Peterson [12].

1.1

Definition and example

A time scale T is an arbitrary nonempty closed subset of the real numbers R. We
assume throughout of this thesis that the time scale T has the topology inherited
from the standard topology of real numbers R.
On the time scale T, we define the forward jump operator and backward jump operator as follow
1. Forward jump operator: given by σ(t) := inf{s ∈ T : s > t}, t ∈ T.
2. Backward jump operator: defined by ρ(t) := sup{s ∈ T : s < t}, t ∈ T.
In this definition we put inf ∅ = sup T; sup ∅ = inf T. (i.e., σ(t) = t if t = max T;
and ρ(t) = t if t = min T).
A point t ∈ T is said to be right-scattered if σ(t) > t; it is called right-dense if
σ(t) = t and left-scattered if ρ(t) < t, left-dense if ρ(t) = t.
Points that are right-dense and left-dense at the same time are called dense; points

that are right-scattered and simultaneously left-scattered are called scattered or
11


isolated.
Definition 1.1.1. On time scale T the forward graininess function µ : T → R+ is
defined by µ(t) := σ(t)−t, t ∈ T and the backward graininess ν(t) := t−ρ(t), t ∈ T.
The left dense or right dense is illustrated by the Figure 3.1.1..

Figure 1.1: Points of the time scale T.

Denote (a, b)T = {t ∈ T : a < t < b}. Similarly, we can denote (a, b]; [a, b) or
[a, b]. To simplify notations, from now on we write (a, b); (a, b]; [a, b); [a, b] instead of
(a, b)T ; (a, b]T ; [a, b)T ; [a, b]T if there is no confusion.
If T has a left-scattered maximum Tmax , then define Tκ = T \ {Tmax }, otherwise
Tκ = T. Similarly, If T has a right-scattered minimum Tmin , then κ T = T \ {Tmin },
otherwise κ T = T.
For any t ∈ T we write f σ (t) for f (σ(t)).
We give some examples of typical time scales.
Example 1.1.2.
• If T = R then σ(t) = t = ρ(t) and µ(t) ≡ 0; if T = Z then σ(t) = t + 1, ρ(t) =
t 1 and à(t) 1.
ã Let h > 0 and
T = hZ = {hn : n ∈ Z} = {· · · , −3h, −2h, −h, 0, h, 2h, 3h, · · · }.
We have σ(t) = t + h, ρ(t) = t − h and µ(t) ≡ h.
• Let a, b > 0 be fixed real numbers. We define the time scale Pa,b by
Pa,b = ∪∞
k=0 [k(a + b), k(a + b) + a].
12