Approximation problems for dynamic equations on time scales = Bài toán xấp xỉ cho phương trình động lực trên thang thời gian. Luận án TS. Toán học: 624601

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.. 47 2.3 On the convergence of solutions for

VIETNAM NATIONAL UNIVERSITY, HANOI

HANOI UNIVERSITY OF SCIENCE

Nguyen Thu Ha

APPROXIMATION PROBLEMS FOR DYNAMIC

EQUATIONS ON TIME SCALES

THESIS FOR THE DEGREE OFDOCTOR 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 TOÁN 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

Page

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

1.5.1 Concept of the exponential stability 26

1.5.2 Exponential stability of linear dynamic equations with con-stant 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 58 3.1 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

3.3 Data-dependence of stability radii 793.3.1 Stability radius of linear implicit dynamic equations 803.3.2 Data-dependence of stability radii 82

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 wasworking on this thesis He has always encouraged me in my work and provided mewith 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 theyhave 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, chanics and Informatics, Hanoi University of Science for their teaching, continuoussupport, tremendous research and study environment they have created I also thank

Me-to my classmates for their friendship and suggestion I will never forget their careand 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 Ihave accomplished

Hanoi, December 27, 2017

PhD student

Nguyen Thu Ha

The characterization of analysis on time scales is the unification and expansion ofresults 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 andhas many applications in diverse fields One of the most important problems inanalysis on time scales is to study the quality and quantity of dynamic equationssuch as long term behaviour of solutions; controllability; methods for solving numer-ical solutions In this thesis we want to study the analysis theory on time scalesunder a new approach That is, the analysis on time scales is also an approxima-tion problem Precisely, we consider the distance between the solutions of differentdynamical systems or study the continuous data-dependence of some characters ofdynamic 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 ofsolutions 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 ofthese time scales tends to the time scale T in Hausdorff topology Moreover, we cancompare the convergent rate of solutions with the Hausdorff distance between Tnand T when the function f satisfies the Lipschitz condition in both variables.Next, we study the continuous dependence of some characters for the linear implicitdynamic equation on the coefficients as well on the variation of time scales Forthe first step, we establish relations between the stability regions corresponding asequence of time scales when this sequence of time scales converges in Hausdorfftopology; after, we give some conditions ensuring the continuity of the spectrum

of matrix pairs; finally, we study the convergence of the stability radii for implicitdynamic equations with general structured perturbations on the both sides underthe variation of the coefficients and time scales

Tóm tắt

Đặc trưng của giải tích trên thang thời gian là thống nhất và mở rộng các nghiêncứu đã đạt được đối với giải tích trên thời gian liên tục hoặc thời gian rời rạc Trongcác 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 Mộttrong những bài toán quan trọng của giải tích trên thang thời gian là nghiên cứutính chất định tính và định lượng của phương trình động lực Trong luận án này,chúng tôi muốn nghiên cứu lý thuyết giải tích trên thang thời gian theo cách tiếpcận mới Đó là giải tích trên thang thời gian còn là bài toán xấp xỉ Cụ thể hơn,chúng tôi xét khoảng cách giữa các nghiệm của các hệ động lực khác nhau và sẽnghiên cứu sự phụ thuộc liên tục của một số đặc trưng của phương trình động lựctheo dữ liệu của phương trình

Luận án bao gồm hai phần chính Trước hết, chúng tôi xét bài toán xấp xỉ nghiệmcủa 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 sẽhội tụ đến nghiệm x(t) của phương trình này trên thang thời gian T nếu như dãythang thời gian này hội tụ về thang thời gian T theo khoảng cách Hausdorff Hơnnữa, chúng tôi cũng đánh giá được tốc độ hội tụ của các nghiệm theo tốc độ hội tụcủa 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 thangthời gian của một số đặc trưng của phương trình động lực ẩn tuyến tính Bước đầutiên, ta thiết lập mối liên hệ giữa các miền ổn định tương ứng của dãy các thangthời 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 tuyếntính chịu nhiễu cấu trúc ở cả hai vế của phương trình khi cả hệ số và thang thờigian đều biến thiên

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

Author:

Nguyen Thu Ha

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

List of Notations

otherwise Tk = T

Crd1 (T, X) Space of rd-continuously differentiable functions

f : Tk → X

CrdR(T, X) Space of rd-continuous and regressive functions

f : T → X

b

valued-domain is [−iπ, iπ)

N, Q, R, C The set of natural, rational, real, complex numbers

R(Tk, X) The set of regressive functions, defined on T

and take the value on X

R+

(Tk, R) The set of positive regressive function defined on T

and taking values in R

scale T

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

sys-Besides, the theory of difference equations has a long term of history Right fromthe dawn of mathematics, it was used to describe the evolution of a lapin populationwith the name "Fibonacci sequence" Difference equations might define the simplestdynamical systems, but nevertheless, they play an important role in the investigation

of dynamical systems The difference equations arise naturally when we want tostudy 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 beillustrated as discretization of continuous time systems in computing process

On the other hand, in some last decades, the theory of time scales, under the nameAnalysis on time scales, was introduced by Stefan Hilger in his PhD thesis(supervised by Bernd Aulbach) in order to unify continuous and discrete analy-ses 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 dealtwith the theory of analysis on the time scale Many familiar results concerning to

qualitative theory such as stability theory, oscillation, boundary value problem incontinuous 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 investigatedynamic equations Many results concerning differential equations carry over quiteeasily 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 crepancies between the differential equations and difference equations Moreover, ithelps us avoid proving results twice, one for differential equations and one for dif-ference equations The general idea is to prove results for dynamic equations wherethe domain definition of the unknown function is a so-called time scale T, which is

dis-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 anordinary 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 resultyields a result for difference equations

However, studying the theory of dynamic equations on time scales leads to muchmore general results since there are many more other complex time scales than theabove two sets That is why so far the analysis on time scales is still an attractedtopic in mathematical analysis Especially, there are still many open problems inthe 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 cisely, we want to consider the distance between the solutions of different dynamicalsystems or to study the continuous dependence of some characters of dynamics equa-tions on data such as spectrum, stability radii when both the coefficients and timescales vary The two following topics will be dealt with in this thesis:

Pre-1 Approximation of solutions

We begin firstly by analyzing the Euler method for solving the stiff initial valueproblem It is known that there are not many classes of ordinary differential equa-tions for which we can represent explicitly their solutions via analysis formulas,especially for nonlinear differential equations Therefore, finding numerical solu-

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 differentialequation Among these algorithms, we have to take into account the Euler explicitand implicit methods Let us consider the initial value problem (IVP)







˙x(t) = f (t, x(t)),x(t0) = x0

x(n)0 = x0, x(n)i+1 = x(n)i + (t(n)i+1− t(n)i )f (t(n)i , x(n)i ), i = 0, 1, , kn − 1 (0.3)Then, the sequence of points (t(n)k , x(n)k ), k = 1, , knevaluates the points (t(n)k , x(t(n)k )),

k = 1, 2, , kn on the solution curve starting from x0 at t0 We call this tive way by Euler explicit method or Euler forward method

approxima-Our problem is to give conditions of the function f and the partition (0.2) ensuringthe convergence of explicit Euler method:

supk

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 thismethod However, it has accumulation error in the processes of calculation andEuler scheme can also be numerically unstable, especially for stiff equations sinceexplicit method requires impractically small time steps to keep the error in theresult bounded and the convergent rate is not good Therefore, one deals with thesecond Euler method, called Euler implicit method or Euler backward method Inthis method, instead of the equation (0.3), we consider the difference equation

x(n)0 = x0, x(n)i+1 = x(n)i + (t(n)i+1− t(n)i )f (t(n)i+1, x(n)i+1), i = 0, 1, , kn− 1 (0.5)This differs from the Euler explicit method in that the latter uses f (t(n)k , x(n)k ) inplace of f (t(n)i+1, x(n)i+1) In Euler implicit method, the new approximation x(n)i+1 appears

on both sides of the equation (0.5) and thus the method needs to solve an algebraic

to use the Euler implicit method because many problems arising in practice arestiff 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 generalunconditionally stable For a stiff problem, explicit methods need to take very smalltime steps Therefore, meanwhile the calculation is complicated, the Euler implicitmethod could be preferred in this case (see [19, 22, 44])

We now consider these two methods in different way That is, when one discretisesthe dynamic equation (0.1) on the real line [0, T ] by Euler explicit method, weobtain 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 Whenthe 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

of the sequence of solutions x(n)(·), the solution of above equations on the time scales

(0.7)

where either t ∈ T or t ∈ Tn Assume that on every time scale Tn (resp T), theCauchy 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

Further, how can we estimate the rate of this convergence?

We can also deal with a similar problem for the nabla dynamic equations on timescales That is instead of considering the equation (0.7), we study the equation







x∇(t) = f (t, x(t)),x(t0) = x0,

(0.9)

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 thespectrum and stability radius for linear implicit dynamic equations on time scales

In the recent years, several technical problems in electronic circuit theory and roboticdesigns, chemical engineering, etc., see [15, 16, 28, 55] lead to the problem of inves-tigating the dynamic implicit equation (IDEs for short)

where the leading term X∆ can not be explicitly solved from X(t) The linear form

of this equation (LIDEs for short) is

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 necessarybut 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 bymultiplying both sides of (0.11) by A−1 In this case, it is known that if the originalequation (0.11) is exponentially stable then it is still stable under sufficiently smallperturbations In general case where A may be degenerate, this property is nolonger valid for the equation (0.11) since the structure of the solutions of a LIDEsdepends 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 Underperturbations, 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 hasclose 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 indexcauses, without additional assumptions, the sharp change of the spectrum σ(A, B)and the continuity of spectrum on the data is no longer valid

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

n→∞(An, Bn, Tn) = (A, B, T)?

In parallel, we have a similar problem for stability radii of implicit dynamic tions It is well-known that if the trivial solution x ≡ 0 of the linear differentialsystem x0 = Bx (resp difference system xn+1 = Bxn) is exponentially stable, thenfor a small perturbation Σ, the system

It is defined as the smallest (in norm) complex or real perturbations destabilizing theequation If complex perturbations are allowed, this measure is called the complexstability 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 shownthat the complex stability radius of linear differential equation (0.13) is given by

maxt∈iR kE(tI − B)−1Dk

−1

Since then, this problems have been getting a lot of attentions from many researchgroups of mathematics in the world D Hinrichsen and N.K Son [52] (1989) haveinvestigated 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

maxω∈C:|ω|=1

kE(ωI − B)−1Dk

−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

−1

whereUT is the uniform stability region of the time scale T and UTc

is the ment of UT

comple-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 ofthe linear implicit dynamic equation

subject to the perturbations

[ ˜A, ˜B] = [A, B] + DΣE = [A + DΣE1, B + DΣE2], (0.20)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

−1

where G(λ) = (λE1− E2)(λA − B)−1D

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 stabilityradius of the system (0.19) depends strongly in variation of the coefficients A, B

Du-Lien-Linh for the first time in [36]; Du-Linh [32] and Du-Linh [34] consider thiscontinuous dependence under the statement of small parameters They claim that

if r(E + εF, A; B, C) is the stability radius of

(E + εF )x0 = (A + BΣC)x,Then, under some heavy assumptions, it is shown that

In order to generalize this result, in this thesis we are concerned with the followingproblem

Problem: Let us consider a sequence of systems

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 stabilityregions and their relation, give conditions ensuring the continuous dependence on thedata 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 knowledgeabout 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 dy-namic 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 dynamicequation 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 timescales {Tn}∞

n=1 converges to the time scale T in the Hausdorff topology, Theorem2.2.7 shows that lim

n→∞xn(t) = x(t) Moreover, in case f satisfies the Lipschitz dition in both variables t and x, the convergent rate of solutions is estimated as thesame degree as the Hausdorff distance between Tn and T, i.e.,

con-kxn(t) − x(t)k 6 CdH(T, Tn), for all t ∈ T ∩ Tn : t0 6 t 6 T, (0.24)(see Theorem 2.2.9) Using these results, we obtain the convergence of the explicitEuler method as a consequence and we give some illustrative examples It can be

considered as a novel approach to the convergence problems of the approximativesolutions.

In the last chapter, we study some problems concerning the stability regions, thespectrum of matrix pairs, the exponential stability and their robustness measurefor linear implicit dynamic equations of arbitrary index (0.19) subjected to generalstructured perturbations of the form

[ eAn, eBn] [An, Bn] + DnΣnEn, (0.25)where Σn, n ∈ N, are unknown disturbance matrices; Dn, En are known scalingmatrices defining the “structure" of the perturbations On each time scale Tn, thestable 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 arederived In more details, Theorem 3.1.7 tells us that the stability regions dependcontinuously on the time scales Concretely, if Tn tends to T in Hausdorff topologythenUT ⊂ S∞

n=1

Tm>nUTm and T∞

n=1

Sm>nUTm\ R ⊂ UT \ R Next, Proposition3.2.11 shows that if Ind(A, B) = 1 and lim

n→∞(An, Bn) = (A, B) and (An−A)Q = 0 forall n ∈ N, then we have lim

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

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

n→∞σ(An, Bn) = σ(A, B).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 and3.3.8 we see that with some further conditions, if lim

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

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

n→∞ r(An, Bn, Dn, En; Tn)

In conclusion, we think that the theory of dynamic systems on an arbitrary time scalewas found promising because it demonstrates the interplay between the theories ofcontinuous-time and discrete-time systems, see [7, 27, 43, 47, 65] It enables toanalyze the stability of dynamical systems on non-uniform time domains whichare subsets of real numbers [66] Based on this theory, stability analysis on timescales has been studied for linear time-invariant systems [61], linear time-varyingdynamic equations [26], implicit dynamic equations [39, 68], switched systems [66,67] and finite-dimensional control systems [10, 29] Therefore, it is meaningful toinvestigate the dependence of stability characteristics of these systems on time scalesand coefficients as well

Parts of the thesis have been published in

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 dependence of stability regions, exponential stability and stability radii for im-plicit linear dynamic equations", Math Control Signals Systems, 28(2), pp.13-28

data-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 uniformstability region of a time scale and the uniformly exponential stability for the dy-namic equations on time scales The details for these concepts and definitions can

be referred to the famous papers of S Hilger [43], C P¨otzsche, 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 Weassume throughout of this thesis that the time scale T has the topology inheritedfrom the standard topology of real numbers R

On the time scale T, we define the forward jump operator and backward jump ator as follow

oper-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 ifσ(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; pointsthat are right-scattered and simultaneously left-scattered are called scattered or

Definition 1.1.1 On time scale T the forward graininess function µ : T → R+ isdefined 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 κT = T \ {Tmin},otherwise κT = T.

For any t ∈ T we write fσ(t) for f (σ(t))

We give some examples of typical time scales

• 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]

1

k.Consider the time scale

1 A function f : T −→ R is called regulated, provided its right-sided limits exist(finite) at all right-dense points in T and its left-sided limits exist (finite) at allleft-dense points in T

2 A function f : T −→ R is said to be rd-continuous (right-dense continuous) if

it is continuous at right-dense points in T and its left-sided limits exist (finite)

at all left-dense points in T

3 A function f : T −→ R is ld-continuous (left-dense continuous) if it is tinuous at left-dense points in T and its right-sided limits exist (finite) at allright-dense points in T

con-4 Let A be an m × n−matrix valued function on time scale T We say A(·) isrd-continuous (resp ld-continuous) if each entry of A is rd-continuous (resp.ld-continuous) on T

For two variables functions we have the following definition Let X be a Banachspace

5 A mapping

(t, x) 7−→ f (t, x)

is said to be rd-continuous if it satisfies the following conditions

(a) f is continuous in (t, x) at each (t, x) where t is right-dense or t = max T,(b) The limits

f (t−, x) := lim

(s,y)→(t,x),s<t

f (s, y) and lim

y→xf (t, y)exist at each point (t, x) where t is left-dense

Theorem 1.2.2 Consider the function f : T −→ R, we have:

i) If f is continuous, then f is rd-continuous

ii) If f is rd-continuous, then f is regulated

iii) The jump operator σ is rd-continuous

iv) If f is regulated (rd-continuous), then so is fσ

v) Assume that f rd-continuous If g : T −→ R is regulated (rd-continuous) then

f ◦ g has that property too

vi) f is a continuous function iff it is ld-continuous and rd-continuous at the sametime

1.2.2 Delta derivative

The derivative and difference operators are basic notions in analysis theory fore, in order to unify them, we give a definition of the so-called delta (or Hilger)derivative for functions defined on time scales

There-Definition 1.2.3 Assume f : T → R is a function and let t ∈ Tκ Then we definethe ∆-derivative of the function f at t to be a number f∆(t) (provided it exists)with the property that given any ε > 0, there is a neighbourhood U of t such that

|f (σ(t)) − f (s) − f∆(t)(σ(t) − s)| 6 ε|σ(t) − s|,for all s ∈ U

We say that f is ∆-differentiable (or in short differentiable) on Tκ if f∆(t) existsfor all t ∈ Tκ

The ∆-derivative f∆ of a vector function f = (f1, f2, , fd) is understood as f∆ =(f1∆, f2∆, , fd∆)

The set of the ∆-differentiable functions on T is denoted by

Crd1 (T, Rd) = {f : T −→ Rd : f is differentiable and f∆ is rd-continuous}.Similar to the derivative concept of the real functions, we have the following prop-erties

Theorem 1.2.4 Assume f: T −→ R and let t ∈ Tκ Then we have the followingstatements:

1 If f is differentiable at t then f is continuous at t

2 If f is continuous at t and t is right-scattered then f is differentiable at t with

f∆(t) = f (σ(t)) − f (t)

3 If t is right-dense f is differentiable at t if and only if the limit lim

s→t

f (t) − f (s)

t − sexists as a finite number In this case

Remark 1.2.5 We consider two cases T = R and T = Z

1 If T = R then f : R −→ R is ∆− differentiable at t ∈ Tκ = T if and only if

lims→t

f (t) − f (s)

t − s exists,that mean f is differentiable (in ordinary sense) at t In this case we have

Theorem 1.2.6 Assume f , g: T −→ R are differentiable at t ∈ Tκ Then:

1 The sum f + g: T −→ R is differentiable at t with

∆(t) = f

∆(t)g(t) − f (t)g∆(t)

The nabla derivative has the similar properties as the delta one For the details, wecan refer to the book of M Bohner and A Peterson (2001) [12].

1.3 Delta and nabla integration

In this section we present some basic notions of the integral on time scales Thereare some ways to define the integral For example, we can define the Riemanndefinitive integral of a function by considering the partitions of the interval [t0, T ]into subintervals, the Darboux sums and then establishing the limit if it exists

In the following we introduce the basic definitions and properties about the Lebesgueintegral on time scales by Carathéodory extension For the details we can refer to[42]

1.3.1 ∆ and ∇ measures on time scales

Denote by A0 the family (collection) of all left closed and right open intervals of T

of the form [a, b) = {t ∈ T : a 6 t < b} with a, b ∈ T and a 6 b The interval [a, a)

is understood as the empty set A0 is a semi-ring of subsets of T

Let m1 : A0 → [0, ∞] be the set function on A0(whose values belong to the extendedreal half-line [0, ∞]) that assigns each interval [a, b) to its length b − a Then, m1 isaccountably additive measure on A0 We introduce the Carathéodory extension m∆

of m1 First, using the pair (A0, m1), we generate an outer measure m∗1on the family

of all subsets of T as follows Let E be any subset of T If Vj ∈ A0 (j = 1, 2, ) isone finite or countable system of intervals in T such that E ⊂ ∪jVj, we say that

Vj ∈ A0 (j = 1, 2, ) is a covering of E In supposing that there is at least onecovering of E, we put

m∗1(E) = infn X

j

m1(Vj)o,where the infimum is taken over all coverings of E Second, we define the family

A= A(m∗1) of all m∗1-measurable subsets of T, consisting of all subsets A of T suchthat m∗1(E) = m∗1(E ∩ A) + m∗1(E ∩ Ac) holds for all E ⊂ T, where Ac denotes thecomplement of A : Ac = T \ A The family A(m∗1) of all m∗1 -measurable subsets of

T is a σ-algebra Third, we take the restriction of m∗1 to A(m∗1), which we denote by

m∆ This m∆ (the Lebesgue ∆-measure) is a countably additive measure on A(m∗1).All intervals of family A0 including the empty set are ∆-measurable By definition,

it is obvious that the T is also ∆-measurable Suppose that T has a finite maximum

Tmax Obviously, the set X = T \ {Tmax} can be represented as a finite or countableunion of intervals of the family A0 and, therefore is ∆-measurable Consequently,the single-point set {Tmax} = T \ X is ∆-measurable as the difference of two mea-surable sets T and X Evidently, the single-point set {Tmax} does not have a finite

or countable covering by intervals of A0 Therefore, the single-point set {Tmax} andalso any ∆-measurable subset of T containing Tmax have ∆-measure infinity

Theorem 1.3.1 For each t0 ∈ T\{Tmax}, the single-point set {t0} is ∆-measurableand its ∆-measure is given by

Since each single-point subset of T is ∆-measurable and since every kind of intervalcan be obtained from an interval of the form [a, b) by adding or subtracting theend points a and b, each interval of T is ∆-measurable The following theorem givesformulas for evaluating the ∆-measure of any interval of T

Theorem 1.3.2 If a, b ∈ T and a 6 b, then

If a, b ∈ T \ {Tmax} and a 6 b, then

Now we define the concept of the Lebesgue ∇-measure on T Let B denote the family

of all right closed and left open intervals of T of the form (a, b] = {t ∈ T : a < t 6 b}with a, b ∈ T and a 6 b The interval (a, a] is understood as the empty set B is

a semi-ring of subsets of T Further, let m2 : B → [0, ∞] be the set function on

B assigned to each interval (a, b] its length b − a Then m2 is accountably additivemeasure on B We denote by m∇ the Carathéodory extension of the set function

m2 associated with the family B as above and call m∇ the Lebesgue ∇-measure on

T The following two theorems can be proved analogously to Theorems 1.3.1 and1.3.2

Theorem 1.3.3 For each t0 in T \ {Tmin} the ∇-measure of the single-point set{t0} is given by m∇({t0}) = t0− ρ(t0)

Theorem 1.3.4 If a, b ∈ T and a 6 b, then

Theorem 1.3.5 If a, b ∈ T \ {Tmin} and a 6 b, then

m∇([a, b)) = ρ(b) − ρ(a), m∇([a, b]) = b − ρ(a) (1.5)Remark 1.3.6 In the case T = R both measures m∆ and m∇ coincide with theusual Lebesgue measure on R In the case T = Z we also have m∆(E) = m∇(E)coincide with the number of points of the set E for any subset E ⊂ Z

1.3.2 Integration

The Lebesgue integrals associated with the measures m∆ and m∇ on T are calledLebesgue ∆ -integral and the Lebesgue ∇-integral on T, respectively For a measur-able set E ⊂ T and a measurable function f : E → R, we denote the integrals of

f over E corresponding to ∆ and ∇ measures by REf (t)∆t and REf (t)∇t, tively In case E = [a, b) we write Rabf (t)∆t for R[a,b)f (t)∆t Similarly, if E = (a, b]

respec-we write Rabf (t)∇t for R(a,b]f (t)∇t

It is easy to see that any regulated function is integrable because it has at most

a countable number of discontinuous points In particular, all rd-continuous andld-continuous functions are both nabla and delta integrable

All theorems of the general Lebesgue integration theory, including the Lebesguedominated convergence theorem, will be held for the Lebesgue ∆ and ∇ integrals

on T

In general, there is no relation between the delta integral and nabla one However,

in case f is continuous we have

In the following theorem, we present some unusual, but useful properties of integral

f∆(t)g(σ(t)) ∆t

4 If |f (t)| 6 g(t), for all t ∈ [a, b), then

bRa

f (t) ∆t

6

bRag(t)∆t

5 If f (t) > 0, for all t ∈ [a, b), then

bRa

where mes is the Lebesgue measure on the interval [a, b]T

1.3.3 Extension of integral

Let T1 and T2 be two time scales Suppose that T1 ⊂ T2 Let f be an integrablefunction defined on the interval [a, b)T1 Consider an extension ef of f to [a, b)T2

as the following: for any t ∈ [a, b]T2, put ef (t) = f (s) if t ∈ [s, σ1(s)], where σ1 isforward operator on T1 Then we have

1.3.4 Polynomial on time scales

In this subsection, we start with the definition of the generalized polynomial, which

for all t, τ ∈ Tκ and k = 1, 2,

The above definition obviously implies h1(t, τ ) = t − τ , for all t, τ ∈ T However,finding hk(t, τ ) is not easy in general But for a particular time scale it may be easy

to find these function For example, by induction, we will show that

a

f (t)∆t =

bZ

a

f (t) dt,where f is regulated function

2 If T = Z then

bZ

−Pa−1 t=b f (t) if a > b,where f : Z −→ R, is arbitrary function on Z

3 If T = hZ then

bZ

−Pah−1 k=hb f (kh)h if a > b,where f is arbitrary function: hZ −→ R

1.4 Exponential function

1.4.1 Regressive group

Let K be a real or complex field and T be a time scales

Definition 1.4.1 The function p: T −→ K is called delta regressive (for shortregressive) if 1 + µ(t)p(t) 6= 0 holds for every t ∈ Tκ

Theorem 1.4.2 The set R = R(T, K) of all regressive function on T and the

"circle" addition ⊕ defined by

(p ⊕ q)(t) := p(t) + q(t) + µ(t)p(t)q(t), p, q ∈ Rform an Abel group The inverse of element q of this group is defined by

1 + µ(t)q(t).This group is called the regressive group

Note that we accept that (p q)(t) as (p ⊕ q)(t) So, we have

1.4.2 Exponential function

First, we introduce the so-called cylinder transformation which allows us to defineexplicitly the exponential function on an arbitrary time scale After, we show thatthis exponential function solves the initial value problem of first order linear dynamicequation on an arbitrary time scale

Definition 1.4.5 For h > 0, we define the Hilger complex numbers set Ch and thetrip Zh as follows:

Ch :=nz ∈ C : z 6= −1

h

o,

where Ln is principal logarithm function with the valued-domain [−iπ, iπ)

It is known that ξh(z) is an rd-continuous function (see [12])

We note that, the inverse transformation of the cylinder transformation

where the cylinder transformation ξh(z) is introduced in above definition

Lemma 1.4.8 If p(·) is rd-continuous and regressive, then the exponential functionhas semigroup property

ep(t, r)ep(r, s) = ep(t, s), for all r, s, t ∈ T

Theorem 1.4.9 If p(·) is rd-continuous and regressive, then for each t0 ∈ Tκ fixed,

ep(·, t0) is a solution of the initial value problem

6 If p(·), q(·) ∈ R+ and p6 q, then ep(t, t0) 6 eq(t, t0) for all t > t0

7 If p(·) ∈ R+, then ep(t, t0) > 0 for any t ∈ T

8 If τ ∈ T is a point such that 1 + µ(τ )p(τ ) < 0, then ep(τ, t0)ep(σ(τ ), t0) < 0.The explicit formula of the exponential function is given by

ep(t, t0) = exp

Z

[t0,t)Tp(s)ds

Y

2 If T = N20 = {n2, n ∈ N}, we have σ(t) = (√t + 1)2, µ(t) = 1 + 2√

t Whenp(t) = 1, by using (1.11) we obtain

e1(t, 0) = 2

√

t(√t)!

3 Consider the time scale Pa,b as in Example 1.1.2 with a = b = 1

1.4.3 Exponential matrix function

Theorem 1.4.13 ([43]) Assume that A(·) is rd-continuous matrix valued function.Then, for each t0 ∈ Tκ, the initial value problem

t > t0, even if A(·) is not regressive If we assume further that A(·) is regressivethen Cauchy operator ΦA(t, t0) is defined for all t, t0 ∈ Tκ

Theorem 1.4.14 (Constant variation formula) Let A : Tκ → Rm×m and f :

Tκ× Rm → Rm be rd-continuous If x(t), t > t0 is a solution of dynamic equation

then we have

x(t) = ΦA(t, t0)x0+

tZ

in-Lemma 1.4.15 (Gronwall-Bellman lemma) Let x(t) be a continuous function and

k > 0, x0 ∈ R Assume that x(t) satisfies the inequality

x(t) 6 x0+ k

Z t

t0x(s)∆s, for all t ∈ T, t > t0 (1.16)Then, the relation

x(t) 6 x0ek(t, t0) for all t ∈ T, t > t0 (1.17)holds

1.5 Exponential stability of dynamic equations on time scales

1.5.1 Concept of the exponential stability

Let T be a time scale, t0 ∈ T and for any a ∈ T we put Ta = [a, ∞)T Consider thedynamic equation of the form

where f : T × Rm → Rm is rd-continuous Suppose that f (t, 0) = 0 With thisassumption we see that this dynamic equation has the trivial solution x ≡ 0.Denote by x(t; t0, x0) the solution of Cauchy problem (1.18) Suppose that for any

x0 ∈ Rm, there exists uniquely a solution satisfying x(t0; t0, x0) = x0 and thissolution is defined on Tt0

There are two concepts of exponential stability for a dynamic equations on timescales The first is to compare the solutions of (1.18) with exponential functions onthe time scale T meanwhile the other is based on classical exponential functions Infact two these definitions are equivalent

Definition 1.5.1 ([26, 43]) The trivial solution x ≡ 0 of the dynamic equation(1.18) is said to be exponentially stable if there exist a positive constant α with

−α ∈ R+ and a positive number δ > 0 such that for each t0 ∈ T there exists an

N = N (t0) > 1 for which, the solution of (1.18) with the initial condition x(t0) = x0satisfies

kx(t; t0, x0)k 6 N kx0ke−α(t, t0),for all t > t0, t ∈ T and kx0k < δ

Definition 1.5.2 ([41, 61]) The trivial solution x ≡ 0 of the dynamic equation(1.18) is said to be exponentially stable if there exist a positive constant α and a

positive number δ > 0 such that for each t0 ∈ T, there exists an N = N(t0) > 1 forwhich, the solution of (1.18) with the initial condition x(t0) = x0 satisfies

kx(t; t0, x0)k 6 N kx0ke−α(t−t0),for all t > t0, t ∈ T and kx0k < δ

If the constant N can be chosen independently of t0 ∈ T then the solution x ≡ 0 of(1.18) is called uniformly exponentially stable

Note that when applying Definition 1.5.1, the condition −α ∈ R+ is equivalent toµ(t) < α1 This means that we are working on time scales with bounded graininess.Theorem 1.5.3 On time scales with bounded graininess, two above definitions areequivalent

Proof If −α ∈ R+, t > t0 then

e−α(t, t0) = expn

tZ

t0

limu&µ(s)

ln(1 − αu)

o

Since

limu&µ(s)

ln(1 − αu)

u 6 −α, ∀s ∈ T =⇒ e−α(t, t0) 6 e−α(t−t0)for all α > 0, −α ∈ R+ and t > t0 Thus, the exponential stability in Definition1.5.1 implies the exponential stability in Definition 1.5.2

Conversely, with α > 0 put

Evidently, we have

−α(·) ∈ R+ and e−α(·)(t, t0) = e−α(t−t0).Let µ∗ := supt∈Tµ(t) If µ∗ = 0 (µ(t) ≡ 0) we see that α(t) ≡ α In case µ∗ > 0, it isclear that the function y = 1−eu−αu, defined on 0 < u6 µ∗, is a decreasing function.Therefore,

α(t) > 1 − e

−αµ∗

µ∗ := β, ∀ t ∈ Tt0.Note that β > 0 and −β ∈ R+ Hence,

e−α(·)(t, t0) = e−α(t−t0) 6 e−β(t, t0), ∀ t > t0.Thus, Definition 1.5.2 implies Definition 1.5.1 We have the proof

1.5 Exponential stability of dynamic equations on time scales< /h3>

1.5.1 Concept of the exponential stability

Let T be a time scale, t0 ∈ T and for any... the exponential function on an arbitrary time scale After, we show thatthis exponential function solves the initial value problem of first order linear dynamicequation on an arbitrary time scale