Thông tin tài liệu
VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE Ngô Thi Thanh Nga STABILITY AND ROBUST STABILITY OF SINGULAR LINEAR DIFFERENCE EQUATIONS THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI 2018 z VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE Ngô Thi Thanh Nga STABILITY AND ROBUST STABILITY OF SINGULAR LINEAR DIFFERENCE EQUATIONS Speciality: Dierential and Integral Equations Speciality Code: 62 46 01 03 THESIS FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY IN MATHEMATICS Supervisors: ASSOC PROF DR HABIL VŨ HOÀNG LINH and PROF DR NGUYỄN HỮU DƯ HANOI 2018 z ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN Ngô Thị Thanh Nga TÍNH ỔN ĐỊNH VÀ ỔN ĐỊNH VỮNG CỦA PHƯƠNG TRÌNH SAI PHÂN TUYẾN TÍNH SUY BIẾN Chun ngành: Phương trình Vi phân Tích phân Mã số: 62 46 01 03 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng dẫn khoa học: PGS.TSKH VŨ HOÀNG LINH GS.TS NGUYỄN HỮU DƯ HÀ NỘI – 2018 z Declaration This work has been completed at the Faculty of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, under the supervision of Assoc.Prof.Dr.habil Vu Hoang Linh and Prof.Dr Nguyen Huu Du I hereby declare that the results presented in the thesis are new and have never been published fully or partially in any other thesis/work Author: Ngô Thị Thanh Nga z Acknowledgments Firstly, I would like to thank my two supervisors Prof.Dr Nguyễn Hữu Dư and especially Assoc.Prof.Dr.habil Vũ Hoàng Linh for the continuous support of my PhD study and related research; for their patience, motivation and immense knowledge Without their help I could not have overcome the difficulties in research and study I would like to express sincere thanks to Assoc.Prof.Dr Lê Văn Hiện and Dr Nguyễn Trung Hiếu for their useful comments and suggestions that led to the improvement of the thesis I would also like to thank Dr Đỗ Đức Thuận for his collaboration in research My deepest appreciation goes to Prof Phạm Kỳ Anh and other members of "Seminar on Computational and Applied Mathematics", and also to the members of "Seminar on Differential Equations and Dynamical Systems" at the Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Hanoi, for their valuable comments and discussions I am grateful to my parents, brother, my beloved daughters, my husband and other members in my big family, who have provided me moral and emotional support throughout my life A very special gratitude goes to all Thang Long University, National Foundation for Science and Technology Development, the MOET project 911 for providing the funding for me in the period of my study Last but not least, I would like to thank my colleagues in Thang Long University, the staffs of Vietnam Institute for Advanced Study in Mathematics, my friends, and many other people beside me for their love, motivation and constant guidance Thanks all for your love and support! z Abstract This work is concerned with linear singular dierence equations (LSDEs) of rst order and second order For LSDEs of rst order, by using the projectorbased approach we characterize the stability of the system under perturbations and establish the relation between the boundedness of solutions of nonhomogeneous systems and the exponential/ uniform stability of the corresponding homogeneous systems We also extend the concept of Bohl exponent from regular dierence equations to LSDEs and investigate its properties For LSDEs of second-order, we use the strangeness-index approach Un- der the strangeness-free assumption we investigate the solvability of IVPs, the consistency of initial conditions, and the relation between the solution sets of the systems and those of the associated reduced regular systems By a comparison principle, some exponential stability criteria are obtained A BohlPerron-type theorem is also given to characterize the input-solution relation of non-homogeneous equations Finally, the problem of robust stability under restricted structured perturbations is investigated Also using the comparison principle, an explicit bound for perturbations under which the systems preserve their exponential stability is obtained i z Tóm tắt Trong cơng trình chúng tơi nghiên cứu phương trình sai phân suy biến tuyến tính cấp cấp hai Đối với phương trình sai phân suy biến tuyến tính cấp 1, chúng tơi sử dụng cách tiếp cận phép chiếu đưa kết như: đặc trưng hóa tính ổn định hệ tác động nhiễu; thiết lập mối quan hệ tính ổn định mũ/ ổn định hệ tính chất nghiệm hệ khơng nhất; mở rộng khái niệm số mũ Bohl cho hệ sai phân suy biến số tính chất Đối với phương trình sai phân suy biến cấp hai, sử dụng cách tiếp cận dùng số lạ Dưới giả thiết số lạ khơng, chúng tơi nghiên cứu tính giải toán giá trị ban đầu điều kiện đầu tương thích, mối quan hệ tập nghiệm hệ ban đầu tập nghiệm hệ đưa dạng quy Bằng cách sử dụng nguyên lý so sánh, tiêu chuẩn cho ổn định mũ thiết lập Một định lý dạng Bohl-Perron đưa nhằm đặc trưng mối quan hệ đầu vào-nghiệm hệ không Cuối cùng, tốn tính ổn định vững tác động nhiễu có cấu trúc Tiếp tục sử dụng nguyên lý so sánh lần nữa, đưa chặn cho nhiễu để hệ bị nhiễu bảo toàn số tính ổn định mũ ii z List of Notations the set of real (complex) numbers (C) R N the set of natural numbers N(n0 ) the set of integers that are greater than or equal to a given integer K R C R or C d the d− dimensional complex vector space d the d− dimensional Euclidean space Cd,d the set of d×d matrices with entries in GL(K ) the set of d×d invertible matrices with entries in kerE the kernel space of E imE the image space of E rank E the rank of matrix E kxk norm of vector k∆k norm of matrix B(0, 1) unit disk on the complex plane det A the determinant of matrix A the matrix tuple rK the (structured) stability radius d H n0 C K x ∆ A (A, B, C) A the conjugate transpose of matrix lp (n0 ) the space of sequences A {qn }n>n0 ⊂ Kd such that P kqn kp < ∞, p > n>n0 spectral radius of matrix ρ(A) diag(σ1 , · · · , σp ) the matrix in C m,n A whose u>0 each component of vector u>v means that ii u u−v >0 iii z entry is σi is positive for any i = 1, , p and the others are zero Contents Page Abstract i Tóm tắt ii List of Notations iii Introduction Chapter Preliminaries 1.1 1.2 Linear singular dierence equations by tractability-index approach 11 1.1.1 De nition of index-1 systems and their properties 11 1.1.2 Solutions of Cauchy problem 13 Linear singular dierence equations by strangeness-index approach 15 1.2.1 De nition of strangeness index and Brüll's results 15 1.2.2 The equivalence between two types of index de nitions 21 1.2.3 Linear time-invariant singular dierence equations of sec 22 ond order 1.3 11 Further auxiliary results iv z 25 Chapter Singular systems of rst-order dierence equations 28 2.1 Stability notions for singular dierence equations 28 2.2 Stability of perturbed equations 32 2.2.1 The case of one-sided perturbation 33 2.2.2 The case of two-sided perturbation 38 41 2.3 2.4 2.5 Bohl-Perron-type stability theorems 2.3.1 Boundedness of solutions of nonhomogenous equations 41 2.3.2 Bohl-Perron-type theorems 46 Bohl exponents and exponential stability 55 2.4.1 Bohl exponents and their basic properties 55 2.4.2 Robustness of Bohl exponents 60 The case of unbounded canonical projector function 65 2.5.1 Uniform stability and exponential stability of perturbed equations 2.5.2 2.6 65 Bolh exponent of solutions and Bohl exponent of the system 67 Conclusion 71 Chapter Singular systems of second-order dierence equations 72 3.1 Initial value problems 72 3.2 Exponential stability 83 3.2.1 Notion of exponential stability 83 3.2.2 Criteria for exponential stability 85 3.2.3 Bohl-Perron theorem 91 3.3 Robust stability 94 3.4 Conclusion 109 v z ... Ngơ Thị Thanh Nga TÍNH ỔN ĐỊNH VÀ ỔN ĐỊNH VỮNG CỦA PHƯƠNG TRÌNH SAI PHÂN TUYẾN TÍNH SUY BIẾN Chun ngành: Phương trình Vi phân Tích phân Mã số: 62 46 01 03 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng... to stability and robust stability of singular discrete-time systems with or without delay, see [ 7, 3 3, 3 4, 4 4, 5 8, 6 3, 7 4, 75] For example, in [7] the authors investigated the stability of linear. .. exponential stability is obtained i z Tóm tắt Trong cơng trình chúng tơi nghiên cứu phương trình sai phân suy biến tuyến tính cấp cấp hai Đối với phương trình sai phân suy biến tuyến tính cấp 1, sử
Ngày đăng: 08/03/2023, 17:45
Xem thêm:
