Luận án tiến sĩ toán ứng dụng: Tính giải được và tính ổn định của phương trình vi sai phân địa số với nhiễu ngẫu nhiên.

Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (17.51 MB, 103 trang )

<span class="text_page_counter">Trang 1</span><div class="page_container" data-page="1">

VIETNAM NATIONAL UNIVERSITY, HANOIVNU UNIVERSITY OF SCIENCE

Nguyen Hong Son

SOLVABILITY AND STABILITY OF

DIFFERENTIAL-DIFFERENCE ALGEBRAIC

<small>THESIS FOR THE DEGREE OF</small>

DOCTOR OF PHILOSOPHY IN APPLIED MATHEMATICS

<small>Hanoi — 2022</small>

</div><span class="text_page_counter">Trang 2</span><div class="page_container" data-page="2">

VIETNAM NATIONAL UNIVERSITY, HANOIVNU UNIVERSITY OF SCIENCE

Nguyen Hong Son

SOLVABILITY AND STABILITY OF

Supervisors: ASSOC. PROF. DR. DO DUC THUANand ASSOC. PROF. DR. PHAN VIET THU

<small>Hanoi — 2022</small>

</div><span class="text_page_counter">Trang 3</span><div class="page_container" data-page="3">

ĐẠI HỌC QUỐC GIA HÀ NỘI

TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

Nguyễn Hồng Sơn

TÍNH GIẢI ĐƯỢC VÀ TÍNH ON ĐỊNH CUA

PHƯƠNG TRÌNH VI-SAI PHÂN ĐẠI SỐ

VỚI NHIÊU NGAU NHIÊN

Chuyên ngành: Lí thuyết xác suất và thống kê tốn họcMã số: 9460112.02

LUẬN ÁN TIẾN SĨ TOÁN ỨNG DỤNG

Người hướng dẫn khoa học:

PGS. TS. ĐỒ ĐỨC THUẬN

PGS. TS. PHAN VIÊT THƯ

</div><span class="text_page_counter">Trang 4</span><div class="page_container" data-page="4">

This work has been completed at the Faculty of Mathematics, Mechanics andInformatics, VNU University of Science, Vietnam National University, Ha noi,

under the supervisions of Assoc. Prof. Dr. Do Duc Thuan and Assoc. Prof. Dr.

<small>Phan Viet Thu. I declare hereby that the results presented in the thesis are newand have never been published elsewhere.</small>

Author: Nguyen Hong Son

</div><span class="text_page_counter">Trang 5</span><div class="page_container" data-page="5">

First and foremost, I want to express my deep gratitude to Assoc. Prof. Dr. DoDuc Thuan and Assoc. Prof. Dr. Phan Viet Thu for accepting me as a PhDstudent and for their help and advice while I was working on this thesis. Theyhave always encouraged me in my work and provided me with the freedomto elaborate my own ideas. Without their help I could not have overcome the

difficulties in research and study.

I also want to express sincere thanks to Prof. Nguyen Huu Du for all the help

<small>his have given to me during my PhD study. I am so lucky to get his support.</small>

I would like to express my special appreciation to Prof. Dang Hung Thang, othermembers of seminar at Department of Probability theory and mathematicalstatistics and all friends in Professor Nguyen Huu Du’s group seminar for their

<small>valuable comments and suggestions to my thesis.</small>

I wish to thank the other professors and lecturers at Faculty of Mathematics,

<small>Mechanics and Informatics, Hanoi University of Science for their teaching, </small>

con-tinuous support, tremendous research and study environment they have created.I also thank to my classmates for their friendship. I will never forget their careand kindness. Thank you for all the help and making the class like a family.

<small>Furthermore, I would like to thank Tran Quoc Tuan University and Unit 871</small>

for support throughout my PhD study. This work was also partially supportedby NAFOSTED.

Last, but not least, I would like to express my deepest gratitude to my family.

<small>Without their unconditional love and support, I would not be able to do what Ihave accomplished.</small>

Thanks all for your love and support!

</div><span class="text_page_counter">Trang 6</span><div class="page_container" data-page="6">

In this thesis we will study stability and robust stability of stochastic

<small>differential-algebraic equations as well as stability of stochastic implicit difference equations.</small>

The thesis is divided into two parts. In the first part, we investigate

differential-algebraic equations (DAEs for short) subject to stochastic perturbations. We

introduce the index-y concept and establish a formula of solution for these tions. After that stability is studied by using the method of Lyapunov functions.

<small>equa-Finally, the robust stability of differential-algebraic equations with respect to</small>

stochastic perturbations is considered and formulas of the stability radii arederived.

In the second part, we study stochastic implicit difference equations (SIDEs forshort). We give a definition of solution of such kind of equations. An index-1

concept is introduced and a formula of solution is established. The ous dependence of solution on the initial condition is also considered for theseequations. After that, the mean square stability of stochastic implicit difference

<small>continu-equations is studied by using the method of Lyapunov functions. Finally, we</small>

investigate the index-⁄ concept, solvability and stability of stochastic implicitdifference equations with constant coefficients. Some examples are given to il-

<small>lustrate the obtained results.</small>

</div><span class="text_page_counter">Trang 7</span><div class="page_container" data-page="7">

Tóm tắt

Trong luận án này, chúng tơi nghiên cứu về tính ổn định và ổn định vững chophương trình vi phân đại số ngẫu nhiên cũng như tính ổn định cho phương trìnhsai phân an ngẫu nhiên. Luận án này được chia thành hai phần chính. Phan

đầu tiên, chúng tơi nghiên cứu phương trình vi phân đại số chịu nhiễu ngẫu

nhiên. Chúng tôi giới thiệu khái niệm chỉ số và thiết lập cơng thức nghiệm cho

phương trình này. Tiếp theo, tính ổn định được nghiên cứu bằng cách sử dụngphương pháp hàm Lyapunov. Cuối cùng, tính ổn định vững của phương trình

vi phân đại số chịu nhiễu ngẫu nhiên được xem xét và các cơng thức bán kính

ổn định được đưa ra.

Phần thứ hai, chúng tơi nghiên cứu phương trình sai phân ẩn ngẫu nhiên. Chúng

tôi định nghĩa nghiệm của phương trình này. Khái niệm chỉ số 1 được giới thiệu

và công thức nghiệm được thiết lập. Sự phụ thuộc nghiệm vào điều kiện ban

đầu cũng được xem xét đối với phương trình đã cho. Tiếp theo, bài tốn ổn địnhbình phương trung bình của phương trình sai phân an ngẫu nhiên được nghiên

cứu bằng phương pháp hàm Lyapunov. Cuối cùng, chúng tơi cũng nghiên cứu

khái niệm chỉ số v, tính giải được và tính ổn định cho phương trình sai phân ẩn

ngẫu nhiên hệ số hằng. Ví dụ được đưa ra minh họa cho kết quả đạt được.

</div><span class="text_page_counter">Trang 8</span><div class="page_container" data-page="8">

Abstract iii

Tom tat iv

<small>List of Notations vii</small>

List of Figures viiiIntroduction 1

<small>1.4 Index con€epfS... . . . 0.00000 2 eee 20</small>

1.4.1. Implicit difference equations of index-l ... 201.4.2. Stochastic differential algebraic equations of index-1.... 22

<small>1.4.3. The Drazin inverse and index-y ... 23</small>

</div><span class="text_page_counter">Trang 9</span><div class="page_container" data-page="9">

Chapter 2 Differential-algebraic equations with respect to

<small>stochas-tic perturbations 26</small>

2.1 Stochastic differential-algebraic equations of index-y... 26

<small>2.1.1. Solvability of stochastic differential-algebraic equations .. 27</small>

2.1.2. Stability of stochastic differential-algebraic equations ... 332.2 Stability radii of stochastic differential-algebraic equations with

<small>respect to stochastic perturbations ... 37</small>

2.3 Conclusion of Chapter2 ... 000.000 000. 44Chapter 3 Stochastic implicit difference equations 463.1 Stochastic implicit difference equations of index-l ... 463.1.1. Solution of stochastic implicit difference equations .... 473.1.2. The variation of constants formula for stochastic implicit

</div><span class="text_page_counter">Trang 10</span><div class="page_container" data-page="10">

List of Notations

The adjoint matrix of A

The transpose matrix of A

Almost surely, or P—almost surely, or with probability 1

The identity matrix in KX”

= Trace(vu) for all u,v € K"TM

Open left half complex plane.

The Borel -ø-algebra on R#

<small>The Borel -ø-algebra on |</small>

The determinant of matrix A

Real part of complex number À

A field, to be replaced by an element from {R, C}

Linear space of n x m— matrices on KThe image space of A

The kernel space of AThe rank of matrix A

The expectation of the random variable X

The family of R¢—valued random elements €

such that E||é||? < œ

The set of matrix valued random elements X

such that E||X ||? < œ

The family of Borel measurable functions h : [a,b] + R4

such that ƒ° ||h()||Pdt < s

The family of R¢-valued Z;-adapted processes

{f(t)}act<p such that [” || f(t)||Pdt < so as.

The family of processes { ƒ(f)}a<¿<» in L£?({a, b]; RR“) such

</div><span class="text_page_counter">Trang 11</span><div class="page_container" data-page="11">

o(A, B) The set of solutions of det(AA — B) = 0

aVb The maximum of a and bsup, inf Supremum, infimum

</div><span class="text_page_counter">Trang 12</span><div class="page_container" data-page="12">

The unstable solution X(t) = z0) KT aaaaa 44

The stable solution X(f) = ("M). .. 0... ee 45

The stable solution X(n) = (2(n),y(n))P. 2. ee 64The unstable solution X(n) = (x(n),y(n))P. 2. ee 64

Simulation of the stable solution X(z,,z)...- 81

</div><span class="text_page_counter">Trang 13</span><div class="page_container" data-page="13">

Stochastic modelling has come to play an important role in many branches of

<small>science and industry where more and more people have encountered stochastic</small>

differential equations as well as stochastic difference equations. Stochastic modelcan be used to solve problem which evinces by accident, noise, etc.

This thesis is concerned with differential-algebraic equations (DAEs) subject to

stochastic perturbations of the form

Eda(t) = (Aa(t) + g(t))dt + f(t, 2(t))dw(t), (0.1.1)

x(to) = #0,

where £,A € K”*”, the leading coefficient # is allowed to be a singular

ma-trix and w(t) is an m-dimensional Wiener process. While standard

<small>differential-algebraic equations without random noise are today standard mathematicalmodels for dynamical systems in many application areas, such as multibody sys-</small>

tems, electrical circuit simulation, control theory, fluid dynamics, and chemical

engineering (see, e.g., [11, 35, 36, 51]), the stochastic version is typically needed

to model effects that do not arise deterministically (see, e.g., [5, 10]). In fact, an

accurate mathematical model of a dynamic system in electrical, mechanical, orcontrol engineering often requires the consideration of stochastic elements. Elec-tronic circuit systems or multibody mechanism systems with random noise are

often modeled by stochastic differential algebraic equations (SDAEs for short),

or sometime called stochastic implicit dynamic systems. These models have been

studied recently in [5, 10, 14, 52, 53, 59]. It is well known that, due to the factthat the dynamics of (0.1.1) are constrained, some extra difficulties appear in

the analysis of stability as well as numerical treatments of stochastic differentialalgebraic equations. These difficulties are typically characterized by index con-

cepts, see [11, 35, 36]. Note that in [5, 10, 14, 52, 53, 59], the authors considered

</div><span class="text_page_counter">Trang 14</span><div class="page_container" data-page="14">

<small>stochastic differential algebraic equations only in the case of index-1.</small>

As mentioned above, electronic circuit systems or multibody mechanism systemswith random noise are often modeled by stochastic differential algebraic equa-tions, or sometime called stochastic implicit dynamic systems. However, the ad-

<small>vent of many modern-day sampled data control systems has necessitated a study</small>

of stochastic discrete systems because they invariably include some stochasticelements that can only change at discrete instants of time. Examples of sampled

<small>data systems are digital computers, pulsed radar units, and coding units in most</small>

communication systems. These lead to stochastic implicit difference equations

(SIDEs). They can also be obtained from SDAEs by some discretization

meth-ods. Moreover, in recent years, a class of stochastic singular systems called the

Markov jumping singular systems have also been investigated [7, 20, 60, 61, 63],

and the references therein. However, there are few report on the study of implicit

<small>difference equations with state-dependent random noise, which is a more </small>

realis-tic mathemarealis-tical model due to that in many branches of science and industry. Infact, these systems are often perturbed by various types of random environment

noises which is state-dependent (see, e.g. [62, 65)).

In the case of deterministic, an implicit difference equation (IDE for short) can

be described in the form

Euz{(n + 1) = Anx(n) + Qn, n EN, (0.1.2)

where „, An € R&%4, X(n), qn € R¢ and E, may be a singular matrix. IDEs

<small>are generalization of regular explicit difference equations, which have been well</small>

investigated in the literature; see [1, 21]. They arise as mathematical models

in various fields such as population dynamics, economics, systems and control

theory, and numerical analysis (see, e.g. [3, 4, 12, 37, 38, 39]). If gn = ƒ(n)t0„+1

is a random noise then we obtain a SIDE

E,x(n+ 1) = Aaz(n) + ƒ(n)ua+1,n € Ñ, (0.1.3)where Wn+1 is a stochastic variable which is independent of the state a(n). In

this case, if E,, is the identity matrix then (0.1.3) becomes a stochastic difference

equation which has attracted a good deal of attention from researchers in recent

years (see, e.g. [13, 40, 55, 56, 64]). Unlike stochastic difference equations, the

analysis of SIDEs is more complicated. Even the solvability analysis is not trivial.

<small>This can be illustrated by the folllowing example.</small>

</div><span class="text_page_counter">Trang 15</span><div class="page_container" data-page="15">

Example 0.1.1. We give a simple example for SIDEs:

where f(n),g(n) € R for cach n € Ñ, the initial conditions are defined by

x(0) = xo, y(0) = yo, and {w,} are a sequence of independent random variables

such that Ew, = 0,E(w2) = 1. We can rewrite this two-dimensional equations

in a system of two scalar equations

Then, it is easy to see that solutions of the initial value problem of (0.1.4)

either has infinitely many solutions if g(n) = 0 or otherwise no solution. Thus,

in comparison with stochastic difference equations, SIDEs present at least two

<small>major difficulties: the first lies in the fact that it is not possible to establish</small>

general existence and uniqueness results, due to their more complicate structure;the second one is that SIDEs need to the consistence of initial conditions and

<small>random noise.</small>

On the other hand, in a lot of applications there is a frequently arising

ques-tion, namely, how robust is a characteristic qualitative property of a system (e.g.

stability) when the system comes under the effect of uncertain perturbations.

The aspect of developing measures of stability robustness for linear uncertainsystems with state space description has received significant attention in systemand control theory. These measures can be characterized by stability radius. Theproblem of evaluating and calculating this stability radius is of great importance,from both theoretical and practical point of view and has attracted a lot of at-

tention from researchers (see, e.g., [18, 22, 26, 48, 49] and the references giventherein). For a systematic introduction to the topic, the interested readers arereferred to the earlier work due to Hinrichsen and Pritchard [25] and their morerecent monograph [28], which contains, along with rigorous theoretical develop-

<small>ments, also an extensive literature review on the subject. It is remarkable that</small>

</div><span class="text_page_counter">Trang 16</span><div class="page_container" data-page="16">

<small>the similar problems have been considered for many other types of linear </small>

dynam-ical systems, including time-varying and time-delay systems, implicit systems,positive systems, linear systems in infinite-dimensional spaces as well as linear

systems respect to stochastic perturbations (see, e.g., |2, 8, 30, 37, 44, 48|).

On the basis of the above discussion, we have chosen the doctoral thesis search topic as "Solvability and stability of differential-difference alge-braic equations with respect to stochastic perturbations". There arisesa natural question whether one can define measures of stability robustness forDAEs respect to stochastic perturbations and, moreover, how to calculate thesemeasures. To the best of our knowledge, such kind of questions has not beenaddressed so far in the literature, although different aspects of robust analysisfor stability of DAEs respect to deterministic perturbations has been studied al-

re-ready (see, e.g., [16, 18, 19, 57]). The first purpose of the present thesis is to fill

<small>this gap. In the second chapter, we will study the consistency condition of </small>

ran-dom noise and define the index-v concept for SDAEs. By using this index notion,

we can establish the explicit expression of solution and the variation of constantsformula. After that we shall derive the necessary and sufficient conditions for

the exponential L?-stability of SIDEs by using the method of Lyapunov

func-tions which is well known for the stability theory of dynamic systems. As a main

result in this chapter, we will establish formula of the stability radius of DAEs

respect to stochastic perturbations. A problem, however, occurs in the case thatthe equation may not be solvable under stochastic perturbations, because thenconsistency conditions arise. To deal with this problem either a reformulation ofthe system has to be performed which characterizes the consistency conditions

or the perturbations have to be further restricted.

In the third chapter, we are to perform the first investigation of SIDEs. Themost important qualitative properties of SIDEs are solvability and stability. Tostudy that, the index notion, which plays a key role in the qualitative theoryof SIDEs, should be taken into consideration in the unique solvability and the

stability analysis, (see, [35, 36, 37]). Motivated by the index-1 concept for SDAEsin [14, 52, 59] and the index- concept in the second chapter, in this chapter we

will derive the index-1 concept for time-varying SIDEs and the index- conceptfor SIDEs with constant coefficient matrices. By using this index notion, we canestablish the explicit expression of solution, the variation of constants formula

</div><span class="text_page_counter">Trang 17</span><div class="page_container" data-page="17">

and the continuous dependence on initial condition of solution. On the otherhand, the method of Lyapunov functions is well known for the stability theoryof dynamic systems. By using this method, we shall establish the necessaryconditions for the mean square stability of SIDEs. After that, characterizationsof the mean square stability in the form of the quadratic Lyapunov equations

<small>are discovered.</small>

The thesis is organized as follows.

<small>e In the first chapter, we recall concepts of stochastic processes, the Drazin </small>

in-verse, index of a matrix pair, stochastic differential equations and stochasticdifference equations. We also mention some results on stability of stochasticdifferential equations and stochastic difference equations.

<small>e In the second chapter, the solvability of SDAEs is presented and a formula</small>

of solution is derived. The mean square stability of SDAEs is studied and a

formula of the stability radius is established.

<small>e In the third chapter, the solvability of SIDEs is investigated and a formula</small>

of solution is provided. The mean square stability of SIDEs is derived byusing the method of Lyapunov functions and the comparison theorem.

Parts of the thesis have been published in

1. Do Duc Thuan, Nguyen Hong Son and Cao Thanh Tinh (2021), "Stability

radii of differential-algebraic equations with respect to stochastic

perturba-tions, Systems & Control Letters, 147, pp. 1-9, article 104834. (SCI, Q1).

2. Do Duc Thuan, Nguyen Hong Son (2020), "Stochastic implicit difference

equations of index-1", Journal of Difference Equations and Applications.,

26(11-12), pp. 1428-1449. (SCIE, Q2).

3. Nguyen Hong Son, Ninh Thi Thu (2020), "A comparison theorem for

sta-bility of linear stochastic implicit difference equations of index-1", VNU

Journal of Science Mathematics-Physics, 36(3), pp. 24-31.

4. Do Duc Thuan, Nguyen Hong Son (2020), "Solvability and stability of

stochastic singular difference equations with constant coefficient matricesof index-v", submitted for publication.

</div><span class="text_page_counter">Trang 18</span><div class="page_container" data-page="18">

<small>The results are also presented at the following conferences and seminars:</small>

<small>1.</small> The 2nd PPICTA, Dynamical Systems Session, November 13-17, 2017,

<small>Bu-san, Korea.</small>

. The 9th Vietnam Mathematical Congress, Vietnamese Mathematical

Soci-ety, Nhatrang, Vietnam, August 14-18, 2018.

. The 18th Workshop on Optimization and Scientific Computing, August

<small>20-22, 2020, HoaLac, Vietnam.</small>

. Seminar about "Stochastic Differential Equations and Dynamical Systems"at the 7th floor, VIASM, 2016.

<small>. Seminar on Probability and Statistics, Faculty of Mathematics, Mechanics</small>

and Informatics, VNU University of Science, Hanoi, 2020.

</div><span class="text_page_counter">Trang 19</span><div class="page_container" data-page="19">

Chapter 1

In this chapter, we survey some basic notions related to the theory of

<small>stochas-tic process. We introduce the so-called the Drazin inverse, concepts of index-1,</small>

stochastic differential equations, stochastic difference equations. The notions ofstability are also defined for these equations. The detail for these concepts and

definitions can be referred to the books of X. Mao [41], L. Arnold [6], P. E. den and E. Platen [34], P. Kunkel and V. Mehrmann [35], the papers |3, 44, 59].

Kloe-1.1 Stochastic processes

1.1.1. Basic notations of probability theory

Probability theory deals with mathematical models of trials whose outcomesdepend on chance. All the possible outcomes-elementary events-are grouped to-gether to form a set, 2, with typical element, w € 2. Not every subset of (2 is

<small>in general an observable or interesting event. So we only group these observableor interesting events together as a family, F, of subsets of Q. For the purpose of</small>

probability theory, such a family, F, should have the following properties:

a, Ú € F, where Ú denotes the empty set;

b, AC F => AC € F, where AT = 2 \ A is the complement of A in Q;

Cc, {Ai}isi CcCF=> Ue A; cư.

A probability measure on a measurable space (O,.Z) is a function P : F >(0, 1] such that

</div><span class="text_page_counter">Trang 20</span><div class="page_container" data-page="20">

F={AC0:4B,C €F suth that BC ACC, P(B) = P(C)}.

Then F is a ơ—algebra and is called the completion of Z. If F = F, the

prob-ability space (Q,F, P) is said to be complete. If not, one can easily extend P

to F by defining P(A) = P(B) = P(C) for A € 7, where B,C € F with the

properties that B C AC C and P(B) = P(C). Now (Q,Ff, P) is a completeprobability space, called the completion of (Q, F, P).

Let (Q, F, P) be a probability space. A filtration is a family {7;};>o of increasingsub-o-algebras of F (ie. Fy C Fs C F for all 0 < t < s < ow). The filtration is

said to be right continuous if F; = N.s:*; for all ý > 0. When the probabilityspace is complete, the filtration is said to satisfy the usual conditions if it is right

<small>continuous and Fp contains all P—null sets.</small>

From now on, unless otherwise specified, we shall always work on a given

com-plete probability space (Q,F,P) with a filtration {F;}:50 satisfying the usualconditions. We also define F,, = ø(U¿>o.#;), ie. the o-algebra generated by

A family {X;};cr of IR“— valued random variables is called a stochastic processwith parameter set (or index set) J and state space IR“. The parameter set Ï

is usually the halfline Ry = [0,0o), but it may also be an interval [a, b|, the

nonnegative integers or even subsets of IR“. Note that for each fixed t € I we

<small>have a random variable</small>

Q30 — X,(w) € Rẻ.

On the other hand, for each fixed w € 2 we have a function

I3t— X,() € RY

which is called a sample path of the process, and we shall write X (w) for the

path. Sometimes it is convenient to write X(t,w) instead of X;(w), and the

</div><span class="text_page_counter">Trang 21</span><div class="page_container" data-page="21">

stochastic process may be regarded as a function of two variables (t,w) from

Ix Q to Rẻ. Similarly, one can define matrix-valued stochastic processes etc. We

often write a stochastic process {X;}is0 as {X¿}, X, or X(t).

We introduce some further notions of the stochastic process.

Definition 1.1.1 ([6, 41]). Let {X;};>o be an R?7— valued stochastic process.

(i) It is said to be continuous (resp. right continuous, left continuous) if foralmost all w € Q function X;(w) is continuous (resp. right continuous, leftcontinuous) on t > 0.

(ii) It is said to be cadlag (right continuous and left limit) if it is right continuous

and for almost all w € © the left limit lim, X,(w) exists and is finite for

all t > 0.

(iii) It is said to be integrable if for every £ > 0, X; is an integrable random

variable. It is said to be {F;,}-adapted (or simply, adapted) if for every t, X;is {F,}-measurable.

(iv) It is said to be measurable if the stochastic process regarded as a function

of two variables (t,w) from R, x Q to R¢ is B(R,) x F,—measurable, where

B(R,) is the family of all Borel sub-sets of R,.

(v) The stochastic process is said to be progressively measurable or progressiveif for every T > 0, {Xt }o<rer regarded as a function of (t, w) from [0,7] x Q

to R¢ is B(R,) x F—measurable, where B({0,7]) is the family of all Borel

sub-sets of [0, 7].

(vi) A real-valued stochastic process { 4;};>o is called an increasing process if

for almost all w € 2, A;(w) is nonnegative nondecreasing right continuous

on t > 0. It is called a process of finite variation if Ay = A; — A with {A;}and {A;} both increasing processes.

1.1.2. Martingales

Definition 1.1.2 ((41]). A random variable r : Q — [0,00] is call an stopping time (or simply, stopping time) if {w: 7(0) < t} € F; for any £ > 0.

</div><span class="text_page_counter">Trang 22</span><div class="page_container" data-page="22">

<small>{F;}-Let 7 and p be two stopping times with 7 < ø, a.s.. We define</small>

(Ir ll= {(t,w) ER, x O: rw) <£< plw)}

and call it a stochastic interval. Similarly, we can define stochastic intervals

(7, el], |]7, p]] and l]r, ø[[. If 7 is a stopping time, define

7; ={Ac 7: AN {w:7(w) < t} € Ff; for all t > 0}

which is a sub—o—algebra of Z#. If 7 and p are two stopping times with 7 < p

Definition 1.1.3 ([41]). An R?—valued {F,}—adapted integrable process { M;};>o

is called a martingale with respect to {F;} (or simply, martingale) ifE(Mi|F;)= M, as. for all0<s<£< œ

A real-valued {F,}—adapted integrable process { Mƒ;};>o is called a gale (with respect to {F;}) if

supermartin-1(M,|Z:) > M, as. for allO<s<t<o

It is called a submartingale (with respect to {F;}) if we replace the sign > in

the last formula with <.

Clearly, {A⁄,} is is submartingale if and only if {—M;,} is supermartingale.

For a real-valued martingale {M;}, {M,* := max(M,,0)} and {Mz}, {Mp :=

max(0, —M;)} are submartingales. For a supermartingale (resp. submartingale),

EM; is monotonically increasing (resp. decreasing). Moreover, if p > 1 and {1;}

is an R4—valued martingale such that M; € L?(Q;R®%), then {||M;||?} is a

non-negative submartingale.

A stochastic process X = {X;}is0 is called square-integrable if E||X;|| < co

for every t > 0. If M = {AM;};>ọ is a real-valued square-integrable continuous

<small>martingale, then there exists a unique continuous integrable adapted </small>

increas-ing process denoted by {(M,M),} such that {Mỹ — (M, M),} is a continuous

martingale vanishing at t = 0. The process {(M, M),} is called the quadratic

<small>variation of M. In particular, for any finite stopping time 7,</small>

</div><span class="text_page_counter">Trang 23</span><div class="page_container" data-page="23">

and call {(M, N);} the joint quadratic variation of M and N. It is useful to knowthat {(M, N),} is the unique continuous integrable adapted process of finitevariation such that {M,N,; — (M, N)¿} is a continuous martingale vanishing at

t = 0. In particular, for any finite stopping time 7,

EM,N, =E(M,N),.

1.1.3. Stochastic integral

Definition 1.1.4 (Í6, 41]). Let (Q,F, P) be a probability space with a filtration

{Fi}iso. A (standard) one-dimensional Wiener process (also called Brownianmotion) is a real-valued continuous {F;,}— adapted process {w;}:50 with the

following properties:

(i) to =0 as.;

(ii) for 0 < s < t < œ, the increment w; — ws is normally distributed with

<small>mean zero and variance t — s;</small>

(iii) for 0 < s <t < ow, the increment w; — ws, is independent of Fs.

Definition 1.1.5 ({6, 41]). A d-dimensional process {w; = (0‡,...,0#)};¿>o is

called a d-dimensional Wiener process if every {wi} is a one-dimensional Wiener

process, and {w}},...,{w#} are independent.

It is easy to see that a d-dimensional Wiener process is a d-dimensional uous martingale with the joint quadratic variations

<small>Now, we recall some notations as well as some known results on stochastic </small>

anal-ysis; see, e.g. [6, 41].

Let 0 <a <b < œ, denote by M?({a, R) the space of all real-valued

measur-able {F;}—adapted processes f = {f(t)}asesp such that

2, =F [i I F(t) [Pat < co. (1.44)

</div><span class="text_page_counter">Trang 24</span><div class="page_container" data-page="24">

We identify ƒ and ƒ¡ in M?((a,6];R) if ||f — fill, = 0, we say that f and f;

are equivalent and write f = ƒl.

Definition 1.1.6 ([41]). A real-valued stochastic process g = {ø(f)}a<¿<b iscalled a simple (or step) process if there exists a partition a = to < ty <... <tụ, = b of [a,b], and bounded random variables €;,0 <i < k — 1 such that &; is

<small>F;, measurable and</small>

g(t) = E01 [to ,t1](t) + So mm, (1.1.2)

Denote by Mo({a, b|; R) the family of all such processes.

Definition 1.1.7 ({41]). For a simple process g with the form of (1.1.2) in

Mo([a, b|; R), define

/ 9()đụy = ` Ei(wiy, — we) (1.1.3)

and call it the stochastic integral of g with respect to the Wiener process {w;}

<small>or the ltô integral</small>

Clearly, the stochastic integral f g(t)dw;, is F,—measurable.

Lemma 1.1.8 ({41]). 1ƒ g € Mo([a, 6]; R), then

: [aden =0 (1.1.4)al fo el = là (1.1.5)

Lemma 1.1.9 ([41]). For any f € M?( , there exists a sequence {g,} of

<small>simple processes such that</small>

ame [iso (t)|Pdt = 0. (1.1.6)

<small>Thus, by Lemmas 1.1.8 and 1.1.9,</small>

(an(t) ~ g(t) deve

{f. gn(t)dw;} is a Cauchy sequence in L?(Q;R). So the limit exists and we define

the limit as the stochastic integral. This leads to the following definition.

</div><span class="text_page_counter">Trang 25</span><div class="page_container" data-page="25">

Definition 1.1.10 ({41]). (Definition of Itô's integral) Let ƒ € M?({a,b]; R).

The Ito integral of ƒ with respect to {w;} is defined by

/ f(t)du;, = lim | gn(t)dw, in L*(Q;R), (1.1.7)

where {g,(t)} is a sequence of simple processes such that

jim E Ƒ lLƒ () (#)|| dt = 0. (1.1.8)

The stochastic integral has the following many nice properties

Proposition 1.1.11 (|6, 41]). Let f,g € M?({a,b];R), and let a, 8 be two real

numbers. Then

(i) J° f(t)dw; is Fy—measurable;(ii) Ef? f(t)dw, = 0;

(iti) Ell J2 f()dwi|? = E [2 | FO) Pat;

(iv) J (aƒ() + Bg(t))dw: = a f? f(t)dw:t 8 f? g(t)dwi;

(uv) If € is a real-valued bounded F,—measurable random variable, then Ef €

M?({a, b|:IR) and

fe (1)dw, = lầm (du.

Definition 1.1.12 ( . Let ƒ € M?((0, . Define

ve fo )du,, 1(0)=0 for0<t<T.

We call I(t) the indefinite Ito integral of ƒ, and {J(t)} is {Z;}—adapted.

Theorem 1.1.13 ([41]). Let ƒ € M2?((0,7];R). Then the indefinite integral

I = {I (t)}ocrer is a square-integrable continuous martingale and its quadratic

<small>variation 1s given by</small>

(1), = [ L/(s)|2ds, forO<t<T. (1.1.9)

Theorem 1.1.14 ({41]). Let ƒ € M?({0,7];R), and let a,8 be two stopping

times such thatO <a<6B<T. Then

z / ˆ f(s)dw, = 0:

</div><span class="text_page_counter">Trang 26</span><div class="page_container" data-page="26">

<small>We shall now extend the It6 stochastic integral to the multi-dimensional case.</small>

Let {uy = (w,..., wi”) }iso be an m— dimensional Wiener process defined on

the complete probability space (Q,F,P) adapted to the filtration {F;}. Let

M?((0, T};R%TM) denote the family of all d x m—matrix-valued measurable

{F,}—adapted processes ƒ = {(fij(t))axm}ocecr such that

5 l/0)|4t<œ,

where || f(t) |] = y/Trace(f(#)*f(t)).

Definition 1.1.15 (6, 41]). Let f € M?([0,7];R*TM). Using matrix notation,

we define the multi-dimensional indefinite It6 integral

<small>{Fif-We shall extend the stochastic integral to a larger class of stochastic processes.</small>

Let L?(R,;R‘%TM) denote the family of all d x m—matrix-valued measurable

{F,}— adapted processes f = {ƒ(f)};>o such that

</div><span class="text_page_counter">Trang 27</span><div class="page_container" data-page="27">

1.1.4. It6’s formula

Let w(t) = (wi(t),...,wm(t))’, t > 0 be an m—dimensional Wiener process

defined on the complete probability space (0,7, P) adapted to the filtration

Definition 1.1.16 ([41]). A d—dimensional Itô's process is an JR“— valued tinuous adapted process x(t) = (x1(t),.. ))? on t >0 of the form

con-v(t) = (0) + fa " \dw(s

where g = (91, +5 ga)” € £!{R,;R®) and ƒ = (fij)axm = Z2(R.; Rdxm), We

shall say that x(t) has stochastic differential đz(f) on t > 0 given by

dx(t) = g(t)dt + f(t)dw(t).

Let C?! (IR; R, xR“) denote the family of all real-valued functions V(t, x) definedon R, x R“ such that they are continuously twice differentiable in z and oncein t. If V € C?"(R; Ry x R®), we set

OV OV OV

Vi= 3p V„ = (Gn on)

9*V OPV

22V O0x,0r, ~ ÔzyÔzaOxjOx;/ dxd 22V ev

OxgOr, —— Ơzza

Clearly, when V € C^®!{(R;lR, x R4), we have W„ = ov and W+„ = 0 ove<sub>Ox Ox?</sub>

Theorem 1.1.17 ([41]). Let x(t) be a d—dimensional It6’s process ơn t > 0

with the stochastic differential

dx(t) = g(t)dt + f(t)dw(t),

where g € L'(Ry;R*) and ƒ € £2(R.;RfS”), Let V € C?(R; Ry x R°). Then

V(t, x(t)) is again an It6’s process with the stochastic differential given by

dV(t,2(t)) = ee z(1)) + Volt, (6))9(@)

+5 * Trace (fT (t)Vee(t, w(t)) F(0)) |dt + V,(t,0(t) f(t)dw(t) a.s..

</div><span class="text_page_counter">Trang 28</span><div class="page_container" data-page="28">

<small>Now, we introduce formally a multiplication table:dtdt = 0, dw dt = 0,</small>

dw;,dw; = dt, dwjdw; = 0 if 7 z 1.

1.2 Stochastic differential equations

1.2.1. Definitions

Throughout this subsection, we let w(t) = (wi(t),...,Wm(t))",t > 0 be an m—<sub>2</sub>

dimensional Wiener process defined on the space (Q, F, P). Let 0 < <small>to <T < œ.</small>

Let xp be an F;,—measurable R4—valued random variable such that E||xo||? <

oo. Let g : [to,T] x R4 > R“ and ƒ : [fo,T] x R4 — R4“ be both Borel

measurable. Consider the d-dimensional stochastic differential equation of ltô

dx(t) = g(t, x(t))dt + f(t, x(t))\dw(t) onto <t<T, (1.2.1)with initial value x(to) = xo. By the definition of stochastic differential, this

equation is equivalent to the following stochastic integral equation:

x(t) = xo +/ g(s, x(s))ds + f(s,x(s))dw(s) onto <t<T. (12.2)

Definition 1.2.1 ((6, 41]). An R¢—valued stochastic process {x(t)}ipcter is

called a solution of equation (1.2.1) if it has the following properties:

(i) {zŒ)} is continuous and {F;,}-adapted;

(ii) {ø(,z(9)} € L([to, T];R) and {f(t,«(t))} € L((to, T|;RTM);

(iii) equation (1.2.2) holds for every £ € [to, 7] with probability 1.

A solution {x(t)} is said to be unique if any other solution {Z(t)} is guishable from {x(t)}, that is

</div><span class="text_page_counter">Trang 29</span><div class="page_container" data-page="29">

But, this is a stochastic differential equation on [s,7] with initial valuex(s) = #(s,fo,#o), whose solution is denoted by z(t, s,#(s,fo,#o)). There-fore, we see that the solution of equation (1.2.1) satisfies the following flow

<small>or semigroup property</small>

x(t, to, to) = x(t, s, #(s, to, #o)}) to<s<t<T.

(b) The coefficients g and f can depend on w in a general manner as long as

they are adapted.

1.2.2. Unique existence and stability

Theorem 1.2.3 ([34, 41]). Assume that there exist two positive constants K

and K such that

(i) (Lipschitz condition) for all x,y € TRÍ and t € {to, T]

<small>We shall assume that the assumptions of the existence-and-uniqueness </small>

The-orem 1.2.3 are fulfilled. Hence, for any given initial value z(fo) = zo € R%,

equation (1.2.1) has a unique global solution that is denoted by x(t, to, 29). We

know that the solution has continuous sample paths and its every moment isfinite. Assume furthermore that

g(t,0) =0 and f(t,0)=0 forallf > ty.

So equation (1.2.1) has the solution #(t) = 0 corresponding to the initial value

x(to) = 0. This solution is called the trivial solution or equilibrium position.

Definition 1.2.4. Equation (1.2.1) is said to be L?-stable if f/* E(||z(f, to, zo)||?)dt< co for all x9 € R*. Equation (1.2.1) is said to be exponentially L?-stable if

there exist a, 8 > 0 such that

</div><span class="text_page_counter">Trang 30</span><div class="page_container" data-page="30">

for all £ > tp > 0 and zp € R4.

Theorem 1.2.5 ([44]). Equation (1.2.1) is exponentially L?-stable if and only

if there exists rị > 0 such that

[Bett to,20)IP)at < lool? (1.2.7)

for allt > tạ > 0 and xo € R?.

Besides, we shall need a few more notations. Let 0 < h < oo. Denote by

C?(R,;R, x S,) the family of all nonnegative functions V(t,x) defined on

R, x S, such that they are continuously twice differentiable in x and once in í.

Define the differential operator L associated with equation (1.2.1) by

If L acts on a function V € C®'(R,;R, x S),) then

LV (t,x) = Vi(,#) + Volt, x)g(t, 2) + 5 Trace [f*(t,2)Veo(t, 2) f(t,2)].

By Itô's formula, if x(t) € S;,, then

dV (t, x(t) = LV(t,x(t))dt + V(t, x(t)) f(t, 2(t))dw(t).

1.3 Stochastic difference equations

Now, let {O, Z, P} be a basic probability space, Z¿ € F,n € N, be a family of

o-algebraic, {wy}: tạ € R be a sequence of mutually independent F,,—adapted

random variables and independent to Zj,,k < ø satisfying Ew, = 0,Ew? = 1 for

all n € N. Consider the equation

z{ín +1) = Aaz(n) + R(n,z(n))u„+i, n EN, (1.3.1)

<small>with the initial condition</small>

x(0) = Xo. (1.3.2)

Here R: N x R? > R¢ is measurable. Then, for any given initial value z(0) =

zo € R4, equation (1.3.1) has unique global solution which is denoted by x(n; 0, 29).

Assume furthermore that R(n,0) = 0 for all n € N, so equation (1.3.1) has asolution x(n) = 0 corresponding to the initial value z(0) = 0. This is called the

<small>trivial solution or the equilibrium position.</small>

</div><span class="text_page_counter">Trang 31</span><div class="page_container" data-page="31">

Definition 1.3.1 ([55]). The trivial solution of equation (1.3.1) with the initial

condition (1.3.2) is called:

<small>e Mean square stable if for each e > 0 there exists a 6 > O such that</small>

Rllz(n)||Ÿ < ¢,Vn EN, if E|lzo||2 < 6.

<small>e Asymptotically mean square stable if it is mean square stable and with </small>

un-der the condition E|lzo|| < co the solution x(n) of equation (1.3.1) satisfies

limp ;+ El|a(n)||? = 0.

Consider a function V : Nx R? > R, with V(n, 0) = 0. Putting V, = V(n, 2(n)),

n EN, then the difference operator AV,, is defined by

AV, =V(n+1,2(n+1))—V(n,2(n)), neéN. (1.3.3)

Theorem 1.3.2 ({55]). Assume that there exists a nonnegative function V;, =V(n,x(n)) which satisfies the conditions

EV (0,2(0)) < eiEllzolẺ, (1.3.4)

EAV, < —œEllz(n)|”, 1 EN, (1.3.5)

where cị,ca are positive constants. Then the trivial solution of equation (1.3.1)

<small>is asymptotically mean square stable.</small>

Corollary 1.3.3 ([55]). Assume that there exists a nonnegative function V„ =Vín,z(n)), which satisfies condition (1.3.4) and

EAV, = -cRllz(n)|, n EN. (1.3.6)

<small>Then c > 0 ts a necessary and sufficient condition for asymptotic mean square</small>

stable of the trivial solution of (1.3.1).

<small>Example 1.3.4. Consider the following stochastic difference equation</small>

#{n + 1) = 1 (n) 5 sinZ(n)ti (1.3.7)

with the initial condition #(0) = zo.

We define the Lyapunov function V, = 2?(n), thenAV, = V(n+1,2(n + 1)) — V(n,2(n))

= ( z0) + sin v(n)tsr) — # (n)

</div><span class="text_page_counter">Trang 32</span><div class="page_container" data-page="32">

= Tạ? (9 + qin) sin(#(n))+0„+1 + 7 sin? x(n)w?,, — # (n).

Since =(qz(n) sin(0(n))twn41) = 0 and E(5 sin? z(n)#` 1) = iE sin? a(n), by

<small>simple calculation we get</small>

1.4.1. Implicit difference equations of index-1

Let (En, En-1, An) € RO? x R®TM4 x RTM? be a triple of matrices. Suppose thatrank E, = rank E,-; = r and let T, € GL(R%) such that T, |kerz, is an

<small>isomorphism between kerE, and kerE,_1, put E_,; = Ep. We can give such</small>

an operator 7 by the following way: let Q, (resp. Qn—1) be a projector ontokerE,, (resp. onto kerE,_1); find the non-singular matrices V, and V,_; such

that Qn = V2QE)V-! and Qạ_¡ = Vr-1Q), V4, where QỤ” = diag (0, Iu_y)

and finally we obtain 7¡, by putting 7„ = V„_+M„ 1,

Now, we introduce sub-spaces and matrices

<small>Gn := En — AnTnQn, Pạ:= I— Qn,</small>

Qn-1 = —TrQnG;,' An, „ni :=T— Ôn.

We have the following lemmas, see, e.g. |3, 4, 20].

Lemma 1.4.1 ((3]). The following assertions are equivalent

a) Sy, kerE,_1 = {0};

b) the matrices Gy, = E„ — AnTrQn is non-singular;

</div><span class="text_page_counter">Trang 33</span><div class="page_container" data-page="33">

c) R¢=S,, @ kerE„_.

Lemma 1.4.2 (|3|). Suppose that the mmatri Œạ is non-singular. Then, there

hold the following relations:

i) Pạ = GE, where Dạ = 1 — Qn;

1) On-1 is the projector onto kerE,_1 along Sy;

iu) PrGn' An = PrGh' AnPn-1,QnGn' An = QnGn' AnPa-1 — Th *Qn-13

0) TrQnG,,! does not depend on the choice of Ty and Qn.

We now consider a linear implicit difference equation

E„z(n + 1) = Anz(n)+ qn, n EN, (1.4.1)

and the homogeneous system associated with (1.4.1) is given by

Enz(n + 1) = Anz(n), n€Ñ, (1.4.2)

where E,, An € IR“**“# a, € R“ and the matrix E, may be singular. By virtue

of Lemma 1.4.1, the index-1 concept for linear implicit difference equations isgiven in the following definition.

Definition 1.4.3 ([3]). The linear implicit difference equations (1.4.2) is said to

be of index-1 tractable (index-1 for short) if for all n € N the following conditions

(i) rank E,, = r = constant;

(ii) kerE, 1S, = {0},

It is known that if equation (1.4.2) has index-1 then with the consistent initial

condition z(0) = P_129 it has the unique solution and this solution can be given

by an explicit formula. In the case EF, = E, A, = A then the index-1 property of

(1.4.2) is equivalent to that the pair (, A) can be transformed to

Weierstraf-Kronecker canonical form, i.e., there exist nonsingular matrices W, U € R¢*¢

such that

U Ln-r :

</div><span class="text_page_counter">Trang 34</span><div class="page_container" data-page="34">

where Ï„, I, are identity matrices of indicated size, J € 7X” (see, e.g. [19, 35]).

Then, we have

0 I,

P,=P,=U U, Q,=Q, =U un.

1.4.2. Stochastic differential algebraic equations of index-1

In this subsection, we present the definition of index-1 in [59] to stochastic

algebraic differential equations (SDAEs) of the form

Eda(t) + g(t, x(t))dt + f(t, 2(t))dw(t) = 0. (1.4.3)

Here g : [to, 7] x R” — R" is a continuous vector-valued function of dimensionn, f : [to, T] x R” — R"X” is continuous n x m— dimensional matrix-function,

FE isa constant singular matrix in R"*" with rank E = r < n, and w denotes an

m-—dimensional Wiener process given on the probability space (Q, F, P) with afiltration {Z7;};>¡¿¿. For a mathematical treatment of (1.4.3), we understand it as

<small>a stochastic integral equation</small>

Ez(s)lj„ + / 9(s,#(s))ds + | f(s,2(s))dw(s) = 0, (1.4.4)

where the second integral is an It6 integral. We are interested in strong solutions

defined as follows. A solution x is a vector-valued stochastic process of dimension

n that depends both on the time £ and an element w of the probability space 2.

The argument w is omitted in the notations above. The unknown z(t) = z(t, -)

is a vector-valued random variable in L?(Q,F,P) and the identity in (1.4.4)

<small>means identity for all ¢ and almost surely in w.</small>

Definition 1.4.4 ([59]). A strong solution of (1.4.4) is a process #(-) = {x(t) }reftor

<small>with continuous sample paths that fulfills the following conditions:</small>

e x(-) is adapted to the filtration {Fi}iefto,7),

° f. lgi(s, #(s))|ds < co a.s., Vi=1,...,n, Vt € [to, T],

fi. fij(s,v(s))?ds < œ as., Vi =1,...,n, Vj =1,...,m, Vt € [to, TI,

<small>e (1.4.4) holds a.s.</small>

<small>Let R be a projector along Im E.</small>

</div><span class="text_page_counter">Trang 35</span><div class="page_container" data-page="35">

Definition 1.4.5 ([59]). Equation (1.4.3) is called tractable with index-1 (or,

for short, of index-1) if + Rg/,(t, x) is nonsingular and Rf = 0.

1.4.3. The Drazin inverse and index-v

In this subsection we introduce the basic definitions and properties about the

Drazin inverse. For the details we can refer to [35].

Definition 1.4.6 ([35]). A matrix pair (F, A), E, A € K"*” is called regular if

there exists s € C such that det(sE — A) is different from zero. Otherwise, if

det(sE — A) = 0 for all s € C, then we say that (EF, A) is singular.

If (E,A) is regular, then a complex number À is called a (generalized finite)eigenvalue of (E, A) if det(XE — A) = 0. The set of all (finite) eigenvalues of

(E, A) is called the (finite) spectrum of the pencil (E, A) and denoted by o(E, A).If # is singular and the pair is regular, then we say that (#, A) has the eigenvalue

Regular pairs (E, 4) can be transformed to Weierstrafs-Kronecker canonical

form, see [11, 35, 36], ¡.e., there exist nonsingular matrices W, T € K”X” such

where I,, J„ „ are identity matrices of indicated size, J € KẾ”, and N €

K'"—")*("—") are matrices in Jordan canonical form and N is nilpotent. If E

<small>is invertible, then r = n, i.e., the second diagonal block does not occur.</small>

Definition 1.4.7 ((59]). Consider a regular pair (E, 4) with E,A € K"*”" in

Weierstra-Kronecker form (1.4.5). If r <n and N has nilpotency index v €

{1,2,...}, le., NY = 0, N’ 40 for i = 1,2,...,y —1, then v is called the index

of the pair (E, A) and we write Ind(E, A) = v. If r = n then the pair has index

We note that v = Ind(, A) does not depend on the special transformation to

canonical form. If ⁄ € K"*” then the quantity v = Ind(£, I) is called index (ofnilpotency) of # and is denoted by v = Ind(£).

Definition 1.4.8 ((35]). Let E € K"*” have v = IndE. A matrix X € K"X”

</div><span class="text_page_counter">Trang 36</span><div class="page_container" data-page="36">

EX = XE, (1.4.6a)

XEX=X, (1.4.6b)

XE! = EY, (1.4.6c)

is called a Drazin inverse of EF.

Theorem 1.4.9 ([35]). Every E € K"*" has one and only one Drazin inverse

ED. Moreover, if E € K"*” is nonsingular then

Theorem 1.4.11 ((35]). Let ⁄, A € K"Š" satisfy AE = EA. Then we have

EA? = APE, EPA=AE?, EPA? = APE”. (1.4.12)

</div><span class="text_page_counter">Trang 37</span><div class="page_container" data-page="37">

Note that for the commuting matrices # and A, condition (1.4.13) is equivalent

to the regularity of (, A) and in formula (1.4.5) we can choose T = W.

</div><span class="text_page_counter">Trang 38</span><div class="page_container" data-page="38">

Chapter 2 Differential-algebraic equationswith respect to stochastic

In this chapter, we investigate differential-algebraic equations subject to

<small>stochas-tic perturbations. We introduce the index- concept and establish a formula of</small>

solution for these equations. After that stability is studied by using the method of

Lyapunov functions. Finally, the robust stability of differential-algebraic tions with respect to stochastic perturbations is considered. Formulas of thestability radii are derived. An example is given to illustrate the obtained results.

<small>equa-This chapter is written on the basis of paper 1. in the list of the publications</small>

used in this thesis.

2.1 Stochastic differential-algebraic equations of index-v

In this section, we consider linear stochastic differential-algebraic equations

(SDAEs) with constant coefficients of the form

Edz() = (Ax(t) + g(t))dt + f(t, 2(t))dw(t),

where #, A € K"*” are constant matrices, g : [to, oo) + K” is a (v—1)-times

con-tinuously differentiable vector-valued function, w(t) is an m-dimensional Wienerprocess, ƒ : [to,00) x K" —> K"*TM plays the role of a perturbation such that it

</div><span class="text_page_counter">Trang 39</span><div class="page_container" data-page="39">

is Lipschitz continuous in #, f(t,x(t)) is {F;,}-adapted and f(t,0) is square

in-tegrable on [to, 7].

Definition 2.1.1. A function z : [to,co) x 2 > K” is called a solution of the

initial value problem (2.1.1) if x is continuous and {F;}-adapted, f. \|x(t)||dt <

% a.s., fi. Il f(t, e(t))|/?dt < co as. for T > ty and

Ex(t) = Ex +f (Az(s) + g(s))ds + f(s, x(s))dw(s)

a.s. for all t € [to,00). The functions ƒ, ø and the initial condition x is called

consistent with equation (2.1.1) if the associated initial value problem has atleast one solution. Equation (2.1.1) is called solvable if for every consistent ƒ, g

and zọ, the associated initial value problem has a solution.

Remark 2.1.2. If = ï„ then equation (2.1.1) becomes a stochastic differentialequation (SDE) and with the above assumption it has a unique solution (see,e.g., [23]). Moreover, the well-posedness of the SDAEs (2.1.1) in the case ofindex-1 also has been studied in [14, 52, 53, 59].

2.1.1. Solvability of stochastic differential-algebraic equations

We first treat the special case where # and A commute, i.e.

EA= AE. (2.1.2)

According to Theorem 1.4.10, we have the decomposition F = C +N with the

properties of C and N as given there. We get the following lemma.

Lemma 2.1.3. Equation (2.1.1) with property (2.1.2) is equivalent to the system

Cdr,(t) = Azi(t)dt + E? Eg(t)dt + EP Ef (t,(t))dw(t), (2.1.3a)

Ndro(t) = Ara(t)dt+(I — E? E)g(t)dt + (I-E?E) f(t, x(t))dw(t), (2.1.3b)

zi() = BE? Ex(t), za( = (I — EPE)z(). (2.1.4)

Moreover, equation (2.1.3a) is equivalent to the stochastic differential equation

dax,(t) = E? Aa, (t)dt + E?g(t)dt + EP f(t, a(t))dw(t). (2.1.5)

</div><span class="text_page_counter">Trang 40</span><div class="page_container" data-page="40">

Luận án tiến sĩ toán ứng dụng: Tính giải được và tính ổn định của phương trình vi sai phân địa số với nhiễu ngẫu nhiên.

VIETNAM NATIONAL UNIVERSITY, HANOIVNU UNIVERSITY OF SCIENCE

Nguyen Hong Son

SOLVABILITY AND STABILITY OF

DIFFERENTIAL-DIFFERENCE ALGEBRAIC

DOCTOR OF PHILOSOPHY IN APPLIED MATHEMATICS

VIETNAM NATIONAL UNIVERSITY, HANOIVNU UNIVERSITY OF SCIENCE

Nguyen Hong Son

SOLVABILITY AND STABILITY OF

Supervisors: ASSOC. PROF. DR. DO DUC THUANand ASSOC. PROF. DR. PHAN VIET THU

ĐẠI HỌC QUỐC GIA HÀ NỘI

TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

Nguyễn Hồng Sơn

TÍNH GIẢI ĐƯỢC VÀ TÍNH ON ĐỊNH CUA

PHƯƠNG TRÌNH VI-SAI PHÂN ĐẠI SỐ

VỚI NHIÊU NGAU NHIÊN

Chuyên ngành: Lí thuyết xác suất và thống kê tốn họcMã số: 9460112.02

LUẬN ÁN TIẾN SĨ TOÁN ỨNG DỤNG

Người hướng dẫn khoa học:

PGS. TS. ĐỒ ĐỨC THUẬN

PGS. TS. PHAN VIÊT THƯ

differential-algebraic equations (DAEs for short) subject to stochastic perturbations. We

In the second part, we study stochastic implicit difference equations (SIDEs forshort). We give a definition of solution of such kind of equations. An index-1

Tóm tắt

Trong luận án này, chúng tơi nghiên cứu về tính ổn định và ổn định vững chophương trình vi phân đại số ngẫu nhiên cũng như tính ổn định cho phương trìnhsai phân an ngẫu nhiên. Luận án này được chia thành hai phần chính. Phan

đầu tiên, chúng tơi nghiên cứu phương trình vi phân đại số chịu nhiễu ngẫu

nhiên. Chúng tôi giới thiệu khái niệm chỉ số và thiết lập cơng thức nghiệm cho

phương trình này. Tiếp theo, tính ổn định được nghiên cứu bằng cách sử dụngphương pháp hàm Lyapunov. Cuối cùng, tính ổn định vững của phương trình

vi phân đại số chịu nhiễu ngẫu nhiên được xem xét và các cơng thức bán kính

ổn định được đưa ra.

Phần thứ hai, chúng tơi nghiên cứu phương trình sai phân ẩn ngẫu nhiên. Chúng

tôi định nghĩa nghiệm của phương trình này. Khái niệm chỉ số 1 được giới thiệu

và công thức nghiệm được thiết lập. Sự phụ thuộc nghiệm vào điều kiện ban

đầu cũng được xem xét đối với phương trình đã cho. Tiếp theo, bài tốn ổn địnhbình phương trung bình của phương trình sai phân an ngẫu nhiên được nghiên

cứu bằng phương pháp hàm Lyapunov. Cuối cùng, chúng tơi cũng nghiên cứu

khái niệm chỉ số v, tính giải được và tính ổn định cho phương trình sai phân ẩn

ngẫu nhiên hệ số hằng. Ví dụ được đưa ra minh họa cho kết quả đạt được.

Tom tat iv

List of Notations

= Trace(v*u) for all u,v € K"*TM

The Borel -ø-algebra on R#

A field, to be replaced by an element from {R, C}

The family of R¢—valued random elements €

such that E||é||? < œ

such that E||X ||? < œ

The family of Borel measurable functions h : [a,b] + R4

such that ƒ° ||h()||Pdt < s

The family of R¢-valued Z;-adapted processes

{f(t)}act<p such that [” || f(t)||Pdt < so as.

The family of processes { ƒ(f)}a<¿<» in L£?({a, b]; RR“) such

o(A, B) The set of solutions of det(AA — B) = 0

The unstable solution X(t) = z0) KT aaaaa 44

The stable solution X(f) = ("M). .. 0... ee 45

The stable solution X(n) = (2(n),y(n))P. 2. ee 64The unstable solution X(n) = (x(n),y(n))P. 2. ee 64

Simulation of the stable solution X(z,,z)...- 81

x(to) = #0,

ma-trix and w(t) is an m-dimensional Wiener process. While standard

to model effects that do not arise deterministically (see, e.g., [5, 10]). In fact, an

often modeled by stochastic differential algebraic equations (SDAEs for short),

studied recently in [5, 10, 14, 52, 53, 59]. It is well known that, due to the factthat the dynamics of (0.1.1) are constrained, some extra difficulties appear in

cepts, see [11, 35, 36]. Note that in [5, 10, 14, 52, 53, 59], the authors considered

(SIDEs). They can also be obtained from SDAEs by some discretization

Markov jumping singular systems have also been investigated [7, 20, 60, 61, 63],

noises which is state-dependent (see, e.g. [62, 65)).

In the case of deterministic, an implicit difference equation (IDE for short) can

where „, An € R&%4, X(n), qn € R¢ and E, may be a singular matrix. IDEs

investigated in the literature; see [1, 21]. They arise as mathematical models

theory, and numerical analysis (see, e.g. [3, 4, 12, 37, 38, 39]). If gn = ƒ(n)t0„+1

this case, if E,, is the identity matrix then (0.1.3) becomes a stochastic difference

years (see, e.g. [13, 40, 55, 56, 64]). Unlike stochastic difference equations, the

x(0) = xo, y(0) = yo, and {w,} are a sequence of independent random variables

such that Ew, = 0,E(w2) = 1. We can rewrite this two-dimensional equations

Then, it is easy to see that solutions of the initial value problem of (0.1.4)

ques-tion, namely, how robust is a characteristic qualitative property of a system (e.g.

systems respect to stochastic perturbations (see, e.g., |2, 8, 30, 37, 44, 48|).

re-ready (see, e.g., [16, 18, 19, 57]). The first purpose of the present thesis is to fill

the exponential L?-stability of SIDEs by using the method of Lyapunov

stability analysis, (see, [35, 36, 37]). Motivated by the index-1 concept for SDAEsin [14, 52, 59] and the index- concept in the second chapter, in this chapter we

perturba-tions, Systems & Control Letters, 147, pp. 1-9, article 104834. (SCI, Q1).

26(11-12), pp. 1428-1449. (SCIE, Q2).

= Trace(vu) for all u,v € K"TM