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VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE NgÙ Thi Thanh Nga STABILITY AND ROBUST STABILITY OF SINGULAR LINEAR DIFFERENCE EQUATIONS THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI 2018 VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE NgÙ 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 VÔ HO NG LINH and PROF DR NGUY N HÚU Dì HANOI 2018 I HC QUăC GIA H NáI TRìNG I H¯C KHOA H¯C TÜ NHI N Ngæ Thà Thanh Nga T NH ˚N ÀNH V ˚N ÀNH VÚNG CÕA PH×ÌNG TR NH SAI PH N TUY N T NH SUY BI N Chuyản ng nh: Phữỡng trnh Vi phƠn v Tch phƠn M s: 62 46 01 03 LU N NTI NS TO NHC Ngữới hữợng dÔn khoa hồc: PGS.TSKH VƠ HO NG LINH GS.TS NGUY N HÚU D× H N¸I 2018 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: Ngæ Thà Thanh Nga Acknowledgments Firstly, I would like to thank my two supervisors Prof.Dr Nguy„n Hœu D÷ and especially Assoc.Prof.Dr.habil Vơ Ho ng 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.Prof.Dr L¶ V«n Hi»n and Dr Nguy„n Trung Hi‚u for their useful comments and suggestions that led to the improvement of the thesis I would also like to thank Dr Ø øc Thu“n for his collaboration in research My deepest appreciation goes to Prof Ph⁄m Ký 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! 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 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 Under 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 Tâm t›t Trong cỉng tr…nh n y chóng tổi nghiản cứu vã phữỡng trnh sai phƠn suy bin tuyn tnh cĐp mt v cĐp hai i vợi phữỡng trnh sai phƠn suy bin tuyn tnh cĐp 1, chúng tỉi sß dưng c¡ch ti‚p c“n b‹ng ph†p chi‚u v ÷a ÷æc c¡c k‚t qu£ nh÷: °c tr÷ng hâa tnh n nh ca hằ dữợi tĂc ng ca nhiu; thi‚t l“p mŁi quan h» giœa t‰nh Œn ành mô/ n nh ãu ca hằ thun nhĐt v tnh chĐt nghi»m cıa h» khỉng thuƒn nh§t; mð rºng kh¡i ni»m sŁ mơ Bohl cho h» sai ph¥n suy bi‚n v ch mt s tnh chĐt i vợi phữỡng trnh sai phƠn suy bin cĐp hai, chúng tổi sò dửng cĂch tip cn dũng ch s l Dữợi giÊ thit ch¿ sŁ l⁄ b‹ng khỉng, chóng tỉi nghi¶n cøu t‰nh gi£i ÷ỉc cıa b i to¡n gi¡ trà ban ƒu v cĂc iãu kiằn u tữỡng thch, mi quan hằ giœa t“p nghi»m cıa h» ban ƒu v t“p nghi»m ca hằ ữổc ữa vã dng chnh quy Bng cĂch sò dửng nguyản lỵ so sĂnh, tiảu chu'n cho sỹ n nh mụ ữổc thit lp Mt nh lỵ dng Bohl-Perron ÷ỉc ÷a nh‹m °c tr÷ng mŁi quan h» ƒu v o-nghi»m cıa h» khỉng thuƒn nh§t CuŁi cịng, b i toĂn vã tnh n nh vng dữợi tĂc ng ca nhiu cõ cĐu trúc ữổc ch Tip tửc sò dửng nguyản lỵ so sĂnh mt ln na, chúng tổi ữa ữổc mt chn trản cho nhiu hằ b nhiu vÔn bÊo to n ữổc ch sŁ cơng nh÷ t‰nh Œn ành mơ ii List of Notations R (C) N the set of natural numbers N(n0) the set of integers that are greater than or equal to a given integer n0 the set of real (complex) numbers K R or C Cd Rd Cd;d GL(Kd) kerE imE rank E kxk kk B(0; 1) det A A rK AH l p(n 0) (A) diag( 1; ; p) u>0 u>v iii Contents Abstract Tâm t›t List of Notations Introduction Chapter 1.1 Linear singular dierence equations by tractability-index approach 11 1.1.1 1.1.2 Solutions of Cauchy problem 1.2 Linear singular dierence equations by strangeness-index approach 15 1.2.1 1.2.2 1.2.3 Linear time-invariant singular dierence equations of sec1.3 Further auxiliary results iv Chapter Singular systems of rst-order dierence equations 2.1 Stability notions for singular die 2.2 Stability of perturbed equation 2.2.1 2.2.2 2.3 Bohl-Perron-type stability theo 2.3.1 2.3.2 Bohl-Perron-type theor 2.4 Bohl exponents and exponent 2.4.1 2.4.2 2.5 The case of unbounded canonic 2.5.1 Uniform stability and expo 2.5.2 Bolh exponent of solutions a 2.6 Conclusion Chapter Singular systems of second-order dierence equations 72 3.1 Initial value problems 72 3.2 Exponential stability 3.2.1 83 Notion of exponential stability 83 3.2.2 Criteria for exponential stability 85 3.2.3 Bohl-Perron theorem 91 3.3 Robust stability 3.4 Conclusion 109 94 Calculating similarly as above, we obtain 1 QnGn AnQn = QnGn AnTnTn Qn 1 = QnGn AnTnQnTn Qn 1 = QnGn (En AnTnQn) + En Tn Qn 1 1 = QnGn GnTn Qn + QnGn EnQnTn Qn = Tn Qn 1 1 + PnPnQnTn Qn 1(Gn En = Pn, according to (i)) 1 Tn Qn + = Tn Qn (since PnQn = 0): = Thus, 1 QnGn An = QnGn An(Pn + Qn 1) = QnGn AnPn 1 Tn Qn 1: The second equality of (ii) is proven (iii) To prove that Qen = is a projector onto Nn along Sn we need Qe2n Qen 1, imQen = kerEn = Nn and kerQen = Sn Firstly, we prove that Qe2n = Qen Indeed, = TnQnGn An( Q2 e =T Q G n n n 1 1 = TnQnGn GnGn An + TnQnGn EnGn An 1 = TnQnGn An + TnQnPnGn An = TnQnGn An = Qen Consider En 1Qen = En 1TnQnGn 1An = (since En 1TnQn = 0) Hence, imQen kerEn Let x be an arbitrary element in ker En 1, from the property that Qn is a projector onto kerEn we imply Qn 1x = x We obtain = + Qn 1x = x: From the fact that, Qen 1x = x, we have x imQen 1, i.e kerEn imQen Thus, kerEn = imQen Using Lemma 1.1.1, we have Nn 1 Sn = K d To prove that the projector Qen along Sn onto Nn 1, we will show that if x is an arbitrary element n 116 in Sn, then Qen 1x = Since x Sn, there exists z Kd Anx = Enz Then Qen 1x = TnQnGn Anx = TnQnGn Enz = such that TnQnPnz = 0: We have proven (iii) C Proof of Lemma 1.1.4 Proof Let Qn; Q respectively isomorphisms between Nn and Nn Then, 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Uncertain Discrete Singular Systems", J Dyn Sys., Meas., Control, ;135(3):031018-031018-6 doi:10.1115/1.4023213 125 ... 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. .. other research groups, as well There have been a number of papers that are closely related to the topic of this thesis, e.g ., see [ 7, 3 3, 3 4, 3 5, 4 4, 4 5, 4 6, 5 7, 5 8, 6 3, 7 4, 75] Particular attention... NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE NgÙ Thi Thanh Nga STABILITY AND ROBUST STABILITY OF SINGULAR LINEAR DIFFERENCE EQUATIONS Speciality: Dierential and Integral Equations Speciality
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