Luận án TS: Tính giải, ổn định PTVSPĐS nhiễu ngẫu nhiên - Nguyễn Hồng Sơn

VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE Nguyen Hong Son SOLVABILITY AND STABILITY OF DIFFERENTIAL-DIFFERENCE ALGEBRAIC THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN APPLIED MATHEMATICS Hanoi — 2022 VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE Nguyen Hong Son SOLVABILITY AND STABILITY OF DIFFERENTIAL-DIFFERENCE ALGEBRAIC EQUATION WITH RESPECT TO STOCHASTIC PERTURBATIONS Speciality: Probability theory and mathematical statistics Speciality Code: 9460112.02 THESIS FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY IN APPLIED MATHEMATICS Supervisors: ASSOC. DO DUC THUAN and ASSOC. PHAN VIET THU Hanoi — 2022 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 toán học Mã số: 9460112.02 LUẬN ÁN TIẾN SĨ TOÁN ỨNG DỤNG Người hướng dẫn khoa học: PGS. ĐỒ ĐỨC THUẬN PGS.

PHAN VIÊT THƯ Hà Nội - 2022 Declaration This work has been completed at the Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam National University, Ha noi, under the supervisions of Assoc. Do Duc Thuan and Assoc. Phan Viet Thu. I declare hereby that the results presented in the thesis are new and have never been published elsewhere.

Author: Nguyen Hong Son Acknowledgments First and foremost, I want to express my deep gratitude to Assoc. Do Duc Thuan and Assoc. Phan Viet Thu for accepting me as a PhD student and for their help and advice while I was working on this thesis. They have always encouraged me in my work and provided me with the freedom to 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 his have given to me during my PhD study. I am so lucky to get his support.

I would like to express my special appreciation to Prof. Dang Hung Thang, other members of seminar at Department of Probability theory and mathematical statistics and all friends in Professor Nguyen Huu Du’s group seminar for their valuable comments and suggestions to my thesis. I wish to thank the other professors and lecturers at Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science for their teaching, 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 care and kindness. Thank you for all the help and making the class like a family. Furthermore, I would like to thank Tran Quoc Tuan University and Unit 871 for support throughout my PhD study. This work was also partially supported by NAFOSTED.

Last, but not least, I would like to express my deepest gratitude to my family. Without their unconditional love and support, I would not be able to do what I have accomplished. Thanks all for your love and support! ii Abstract In this thesis we will study stability and robust stability of stochastic differential- algebraic equations as well as stability of stochastic implicit difference equations. 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 equa- tions. After that stability is studied by using the method of Lyapunov functions. Finally, the robust stability of differential-algebraic equations with respect to stochastic perturbations is considered and formulas of the stability radii are derived.

In the second part, we study stochastic implicit difference equations (SIDEs for short). 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 continu- ous dependence of solution on the initial condition is also considered for these equations.

After that, the mean square stability of stochastic implicit difference equations is studied by using the method of Lyapunov functions. Finally, we investigate the index-⁄ concept, solvability and stability of stochastic implicit difference equations with constant coefficients. Some examples are given to il- lustrate the obtained results. 11 Tóm tắt Trong luận án này, chúng tôi nghiên cứu về tính ổn định và ổn định vững cho phương trình vi phân đại số ngẫu nhiên cũng như tính ổn định cho phương trình sai phân an ngẫu nhiên.

Luận án này được chia thành hai phần chính. Phan đầu tiên, chúng tôi nghiên cứu phương trình vi phân đại số chịu nhiễu ngẫu nhiên. Chúng tôi giới thiệu khái niệm chỉ số và thiết lập công thức nghiệm cho phương trình này. Tiếp theo, tính ổn định được nghiên cứu bằng cách sử dụng phương 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 công thức bán kính ổn định được đưa ra. Phần thứ hai, chúng tôi 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 toán ổn định bình phương trung bình của phương trình sai phân an ngẫu nhiên được nghiên cứu bằng phương pháp hàm Lyapunov. Cuối cùng, chúng tôi 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.

iv Contents Page Abstract iii Tom tat iv List of Notations vii List of Figures viii Introduction 1 Chapter 1 Preliminary 7 1. Basic notations of probability theory .2 Stochastic differential equations. Unique existence and stability .3 Stochastic difference equations.4 Index con€epfS. Implicit difference equations of index-l.

Stochastic differential algebraic equations of index-1. The Drazin inverse and index-y. 23 Chapter 2 Differential-algebraic equations with respect to stochas- tic perturbations 26 2.1 Stochastic differential-algebraic equations of index-y. Solvability of stochastic differential-algebraic equations.

Stability of stochastic differential-algebraic equations .2 Stability radii of stochastic differential-algebraic equations with respect to stochastic perturbations .3 Conclusion of Chapter2. 44 Chapter 3 Stochastic implicit difference equations 46 3.1 Stochastic implicit difference equations of index-l. Solution of stochastic implicit difference equations. The variation of constants formula for stochastic implicit difference equations.

Dependence on the consistent initial condition of solution .2 Stability of stochastic implicit difference equations of index-1. Stability of stochastic implicit difference equations. A comparison theorem for stability of linear stochastic im- plicit difference equations of index-l1.3 Stochastic implicit difference equations of index-y. Solvability of stochastic implicit difference equations of index-vy 2.

Stability of stochastic implicit difference equations of index- Vow 74 3.4 Conclusions of Chapter3. 80 Conclusion 82 The author’s publications related to the thesis 85 Bibliography 86 vi 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(v*u) for all u,v € K"*TM Open left half complex plane. The Borel -ø-algebra on R# Jạdxm The Borel -ø-algebra on | 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 K The image space of A The kernel space of A The 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 that E [” || f(t)||Pdt < 00 The set of natural, rational, real, complex numbers The set of positive real numbers The Euclidean norm of a vector x ={zc R2: ||z||< h} The set of the eigenvalues of the matrix A vii o(A, B) The set of solutions of det(AA — B) = 0 aVb The maximum of a and b sup, inf Supremum, infimum viii List of Figures 2.1 The unstable solution X(t) = z0) KT aaaaa 44 2.2 The stable solution X(f) = ("M).1 The stable solution X(n) = (2(n),y(n))P.2 The unstable solution X(n) = (x(n),y(n))P.3 Simulation of the stable solution X(z,,z).- 81 ix Introduction Stochastic modelling has come to play an important role in many branches of science and industry where more and more people have encountered stochastic differential equations as well as stochastic difference equations.

Stochastic model can 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) 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 differential- algebraic equations without random noise are today standard mathematical models for dynamical systems in many application areas, such as multibody sys- tems, electrical circuit simulation, control theory, fluid dynamics, and chemical engineering (see, e., [11, 35, 36, 51]), the stochastic version is typically needed to model effects that do not arise deterministically (see, e. In fact, an accurate mathematical model of a dynamic system in electrical, mechanical, or control 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 fact that the dynamics of (0.1) are constrained, some extra difficulties appear in the analysis of stability as well as numerical treatments of stochastic differential algebraic 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 stochastic differential algebraic equations only in the case of index-1. As mentioned above, electronic circuit systems or multibody mechanism systems with random noise are often modeled by stochastic differential algebraic equa- tions, or sometime called stochastic implicit dynamic systems. However, the ad- vent of many modern-day sampled data control systems has necessitated a study of stochastic discrete systems because they invariably include some stochastic elements that can only change at discrete instants of time. Examples of sampled data systems are digital computers, pulsed radar units, and coding units in most 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 difference equations with state-dependent random noise, which is a more realis- tic mathematical model due to that in many branches of science and industry.

In fact, these systems are often perturbed by various types of random environment noises which is state-dependent (see, e. 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, (0.2) where „, An € R&%4, X(n), qn € R¢ and E, may be a singular matrix. IDEs are generalization of regular explicit difference equations, which have been well investigated in the literature; see [1, 21].