Luận án: Tính ổn định và ổn định vững của phương trình sai phân tuyến tính suy biến

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 TIEU LUAN MOI download : skknchat@gmail.com 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. VŨ HOÀNG LINH and PROF.

NGUYỄN HỮU DƯ HANOI  2018 TIEU LUAN MOI download : skknchat@gmail.com 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 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: PGS. VŨ HOÀNG LINH GS. NGUYỄN HỮU DƯ HÀ NỘI – 2018 TIEU LUAN MOI download : skknchat@gmail.com 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. Vu Hoang Linh and Prof.

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 1 TIEU LUAN MOI download : skknchat@gmail.com Acknowledgments Firstly, I would like to thank my two supervisors Prof. Nguyễn Hữu Dư and especially Assoc.

Vũ Hoàng Linh for the continuous support of my PhD study and related research; for their patience, motivation and im- mense knowledge. Without their help I could not have overcome the difficulties in research and study. I would like to express sincere thanks to Assoc. 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 Foun- dation 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! TIEU LUAN MOI download : skknchat@gmail.com 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 projector- based approach we characterize the stability of the system under perturba- tions and establish the relation between the boundedness of solutions of non- homogeneous systems and the exponential/ uniform stability of the correspond- ing 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 Bohl- Perron-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 TIEU LUAN MOI download : skknchat@gmail.com Tóm tắt Trong công trình này chúng tôi nghiên cứu về phương trình sai phân suy biến tuyến tính cấp một và cấp hai. Đối với phương trình sai phân suy biến tuyến tính cấp 1, chúng tôi sử dụng cách tiếp cận bằng phép chiếu và đưa ra được các kết quả như: đặc trưng hóa tính ổn định của hệ dưới tác động của nhiễu; thiết lập mối quan hệ giữa tính ổn định mũ/ ổn định đều của hệ thuần nhất và tính chất nghiệm của hệ không thuần nhất; mở rộng khái niệm số mũ Bohl cho hệ sai phân suy biến và chỉ ra một số tính chất. Đối với phương trình sai phân suy biến cấp hai, chúng tôi sử dụng cách tiếp cận dùng chỉ số lạ.

Dưới giả thiết chỉ số lạ bằng không, chúng tôi nghiên cứu tính giải được của bài toán giá trị ban đầu và các điều kiện đầu tương thích, mối quan hệ giữa tập nghiệm của hệ ban đầu và tập nghiệm của hệ được đưa về dạng chính quy. Bằng cách sử dụng nguyên lý so sánh, tiêu chuẩn cho sự ổn định mũ được thiết lập. Một định lý dạng Bohl-Perron được đưa ra nhằm đặc trưng mối quan hệ đầu vào-nghiệm của hệ không thuần nhất. Cuối cùng, bài toán về tính ổn định vững dưới tác động của nhiễu có cấu trúc được chỉ ra.

Tiếp tục sử dụng nguyên lý so sánh một lần nữa, chúng tôi đưa ra được một chặn trên cho nhiễu để hệ bị nhiễu vẫn bảo toàn được chỉ số cũng như tính ổn định mũ. ii TIEU LUAN MOI download : skknchat@gmail.com List of Notations R (C) the set of real (complex) numbers N the set of natural numbers N(n0 ) the set of integers that are greater than or equal to a given integer n0 K R or C d C the d− dimensional complex vector space d R the d− dimensional Euclidean space Cd,d the set of d × d matrices with entries in C d GL(K ) the set of d × d invertible matrices with entries in K kerE the kernel space of E imE the image space of E rank E the rank of matrix E kxk norm of vector x k∆k norm of matrix ∆ B(0, 1) unit disk on the complex plane det A the determinant of matrix A A the matrix tuple (A, B, C) rK the (structured) stability radius H A the conjugate transpose of matrix A d kqn kp < ∞, p > 1 P lp (n0 ) the space of sequences {qn }n>n0 ⊂ K such that n>n0 ρ(A) spectral radius of matrix A m,n diag(σ1 , · · · , σp ) the matrix in C whose ii entry is σi for any i = 1, ., p and the others are zero u>0 each component of vector u is positive u>v means that u − v > 0 iii TIEU LUAN MOI download : skknchat@gmail.com Contents Page Abstract i Tóm tắt ii List of Notations iii Introduction 1 Chapter 1 Preliminaries 11 1.1 Linear singular dierence equations by tractability-index approach 11 1. Denition of index-1 systems and their properties. Solutions of Cauchy problem .2 Linear singular dierence equations by strangeness-index approach 15 1.

Denition of strangeness index and Brüll's results. The equivalence between two types of index denitions. Linear time-invariant singular dierence equations of sec- ond order .3 Further auxiliary results. 25 iv TIEU LUAN MOI download : skknchat@gmail.com Chapter 2 Singular systems of rst-order dierence equations 28 2.1 Stability notions for singular dierence equations .2 Stability of perturbed equations.

The case of one-sided perturbation. The case of two-sided perturbation .3 Bohl-Perron-type stability theorems. Boundedness of solutions of nonhomogenous equations. Bohl-Perron-type theorems .4 Bohl exponents and exponential stability.

Bohl exponents and their basic properties. Robustness of Bohl exponents .5 The case of unbounded canonical projector function. Uniform stability and exponential stability of perturbed equations. Bolh exponent of solutions and Bohl exponent of the system 67 2.

71 Chapter 3 Singular systems of second-order dierence equations 72 3.1 Initial value problems. Notion of exponential stability. Criteria for exponential stability. Bohl-Perron theorem.

109 v TIEU LUAN MOI download : skknchat@gmail.com Conclusion 111 The author's publications related to the thesis 113 Appendix 114 Bibliography 118 vi TIEU LUAN MOI download : skknchat@gmail.com Introduction The evolution of certain phenomena in real-world over the course of time is usually described by dierential and dierence equations. In discrete-time scale, mathematical models lead to dierence equations. Dierence equations play important roles in many areas such as control, biology, economics,. Typ- ically, dierence equations can be described in the form F (x(n + k), x(n + k − 1), · · · , x(n + 1), x(n)) = 0, (0.1) d where k is a xed positive integer; n ∈ N; x : Z −→ K and F : Kd × Kd × · · · × Kd −→ Kd .1) is said to be of rst-order.1) is said to be of high-order.

If the highest order term x(n + k) is solved from (0.1) for each n, then we have explicit dierence equations, which are given in the form x(n + k) = f (x(n + k − 1), · · · , x(n + 1), x(n)).2) Explicit dierence equations and their applications have been extensively in- vestigated in many papers and monographs; see [1], [32] and the references therein. However, in many situations system (0.1) is not solvable for x(n + k). Then, we say (0.1) is an implicit dierence equation (IDE) or a singular dier- ence equation (SDE). The simplest case of SDEs is linear SDEs of rst-order, which are given by En y(n + 1) = An y(n) + qn , n ∈ N(n0 ), (0.3) d×d where En , An ∈ K y(n), qn ∈ Kd , N(n0 ) denotes the set of integers that ; are greater than or equal to a given integer n0 , and the leading matrix En may be singular.

The homogeneous equation associated with (0.4) 1 TIEU LUAN MOI download : skknchat@gmail.com Unlike explicit dierence equations, the analysis of SDEs is more complicated. Even the solvability analysis is not trivial. For example, consider the following system " #" # " #" # 1 0 x(n + 1) 1 0 fn = ,n ∈ N (0.5) is equivalent to x(n + 1) = x(n) + fn , 0 = gn. Then, it is easy to see that solutions of the initial value problem (IVP) of SDE (0.5) either has innitely many solutions if gn ≡ 0 or otherwise no solution.

We give here an example to illustrate applications of singular dierence equations in practical areas (see Example 1-1. The fundamental dynamic Leontief model of economic sys- tems is a singular system. Its description model is ([52]): x(k) = Ax(k) + B[x(k + 1) − x(k)] + d(k) (0.6) where x(k) is the n dimensional production vector of n sectors; A ∈ Rn×n is an input-output (or production) matrix; Ax(k) stands for the fraction of n×n production required as input for the current production, B ∈ R is the capital coecient matrix, and B[x(k + 1) − x(k)] is the amount for capacity expansion, which often appears in the form of capital, d(k) is the vector that includes demand or consumption.6) may be rewritten as Bx(k + 1) = (I − A + B)x(k) − d(k).