Luận án tiến sĩ approximation problems for dynamic equations on time scales , bài toán xấp xỉ cho phương trình động lực trên thang thời gian 624601

VIETNAM NATIONAL UNIVERSITY, HANOI HANOI UNIVERSITY OF SCIENCE Nguyen Thu Ha APPROXIMATION PROBLEMS FOR DYNAMIC EQUATIONS ON TIME SCALES THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI – 2017 z 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 z ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN Nguyễn Thu Hà BÀI TỐN XẤP XỈ CHO PHƯƠNG TRÌNH ĐỘNG LỰC TRÊN THANG THỜI GIAN 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: GS TS NGUYỄN HỮU DƯ HÀ NỘI – 2017 z Contents Page Abstract v Tóm tắt vi List of Figures vii List of Notations ix Introduction Chapter Preliminary 11 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 i z 1.5.1 Concept of the exponential stability 26 1.5.2 Exponential stability of linear dynamic equations with constant coefficient 28 1.6 Hausdorff distance 31 Chapter 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 On data-dependence of implicit dynamic equations on time scales 3.1 58 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 ii z 3.3 Data-dependence of stability radii 79 3.3.1 Stability radius of linear implicit dynamic equations 80 3.3.2 Data-dependence of stability radii 82 Conclusion 91 The author’s publications related to the thesis 93 Bibliography 94 z Acknowledgments 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 was working on this thesis He has always encouraged me in my work and provided me with 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 they have 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, Mechanics and Informatics, Hanoi University of Science for their teaching, continuous support, tremendous research and study environment they have created I also thank to my classmates for their friendship and suggestion I will never forget their care and 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 what I have accomplished Hanoi, December 27, 2017 PhD student Nguyen Thu Ha iv z 37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.2237.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.66 Abstract The characterization of analysis on time scales is the unification and expansion of results 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 and has many applications in diverse fields One of the most important problems in analysis on time scales is to study the quality and quantity of dynamic equations such as long term behaviour of solutions; controllability; methods for solving numerical solutions In this thesis we want to study the analysis theory on time scales under a new approach That is, the analysis on time scales is also an approximation problem Precisely, we consider the distance between the solutions of different dynamical systems or study the continuous data-dependence of some characters of dynamic 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 of solutions 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 of these time scales tends to the time scale T in Hausdorff topology Moreover, we can compare the convergent rate of solutions with the Hausdorff distance between Tn and T when the function f satisfies the Lipschitz condition in both variables Next, we study the continuous dependence of some characters for the linear implicit dynamic equation on the coefficients as well on the variation of time scales For the first step, we establish relations between the stability regions corresponding a sequence of time scales when this sequence of time scales converges in Hausdorff topology; after, we give some conditions ensuring the continuity of the spectrum of matrix pairs; finally, we study the convergence of the stability radii for implicit dynamic equations with general structured perturbations on the both sides under the variation of the coefficients and time scales 37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.99 v z 37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.2237.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.66 Tóm tắt Đặc trưng giải tích thang thời gian thống mở rộng nghiên cứu đạt giải tích thời gian liên tục thời gian rời rạc Trong thập niên vừa qua, việc nghiên cứu lý thuyết giải tích thang thời gian cho ta nhiều kết tổng quát có nhiều ứng dụng vào lĩnh vực khác Một toán quan trọng giải tích thang thời gian nghiên cứu tính chất định tính định lượng phương trình động lực Trong luận án này, muốn nghiên cứu lý thuyết giải tích thang thời gian theo cách tiếp cận Đó giải tích thang thời gian cịn tốn xấp xỉ Cụ thể hơn, xét khoảng cách nghiệm hệ động lực khác nghiên cứu phụ thuộc liên tục số đặc trưng phương trình động lực theo liệu phương trình Luận án bao gồm hai phần Trước hết, chúng tơi xét tốn xấp xỉ nghiệm phương trình động lực thang thời gian chứng minh dãy nghiệm xn (t) phương trình x∆ = f (t, x) dãy thang thời gian tương ứng {Tn }∞ n=1 hội tụ đến nghiệm x(t) phương trình thang thời gian T dãy thang thời gian hội tụ thang thời gian T theo khoảng cách Hausdorff Hơn nữa, đánh giá tốc độ hội tụ nghiệm theo tốc độ hội tụ dãy thang thời gian hàm f thỏa mãn điều kiện Lipschitz theo hai biến Tiếp theo, ta nghiên cứu phụ thuộc theo tham số theo biến thiên thang thời gian số đặc trưng phương trình động lực ẩn tuyến tính Bước đầu tiên, ta thiết lập mối liên hệ miền ổn định tương ứng dãy thang thời gian dãy thang thời gian hội tụ theo tô pô Hausdorff Cuối cùng, nghiên cứu hội tụ bán kính ổn định phương trình động lực ẩn tuyến tính chịu nhiễu cấu trúc hai vế phương trình hệ số thang thời gian biến thiên 37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.99 vi z 37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.2237.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.66 Declaration This work has been completed at Hanoi University of Science, Vietnam National University under the supervision of Prof Dr Nguyen Huu Du I declare hereby that the results presented in it are new and have never been used in any other thesis Author: Nguyen Thu Ha 37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.99 vii z ζλ µn (γ T,Tn (h)) − ζλ µ(h) ∆h ζλ (µn (h))∆n h − ζλ (µ(h))∆h s s s Z s1 r Z t∧τi X
T,T n ζλ (µ(h)) − ζλ (µn (γ ζλ (µ(h)) − ζλ µn (γ T,Tn (h)) ∆h (h))) ∆h + = s Z si Z t∧σn (si ) t∧τ i
|ζλ (µ(h)) − ζλ µn (γ T,Tn (h)) |∆h + + r−1 Z X i=1 !
ζλ (µ(h)) − ζλ µn (γ T,Tn (h))