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 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 I HC QUăC GIA H NáI TRìNG I HC KHOA HC TÜ NHI N Nguy„n Thu H B I TO N X P X CHO PHìèNG TR NH áNG LĩC TR N THANG THI GIAN 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: GS TS NGUY N HU Dì H NáI 2017 Contents Abstract Tâm t›t List of Figures List of Notations Introduction Chapter 1.1 Definition and example 1.2 Differentiation 1.2.1 1.2.2 Delta derivative 1.2.3 Nabla derivative 1.3 Delta and nabla integration 1.3.1 1.3.2 1.3.3 1.3.4 1.4 Exponential function 1.4.1 1.4.2 1.4.3 1.5 Exponential stability of dynamic equ i 1.5.1 1.5.2 Exponential stability of linear 1.6 Hausdorff distance Chapter On the convergence of solutions for dynamic equations on time scales 2.1 Time scale theory in view of approxi 2.2 Convergence of solutions for -dynam 2.2.1 2.2.2 2.2.3 2.3 On the convergence of solutions for n scales 2.3.1 Nabla exponential function 2.3.2 2.3.3 2.3.4 2.4 Approximation of implicit dynamic eq Chapter On data-dependence of implicit dynamic equations on time scales 3.1 Region of the uniformly exponential 3.1.1 3.1.2 3.2 Data-dependence of spectrum and e dynamic equations 3.2.1 3.2.2 Solution of linear implicit dyna 3.2.3 Spectrum of linear implicit dy ii 3.3 Data-dependence of stability radii 3.3.1 Stability radius of linear implicit dynamic equations 3.3.2 Data-dependence of stability radii Conclusion The author’s publications related to the thesis Bibliography 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 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 numer-ical 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 approxima-tion 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 fT ng 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 v Tâm tt c trững ca giÊi tch trản thang thới gian l thng nhĐt v m rng cĂc nghiản cứu  t ữổc i vợi giÊi tch trản thới gian liản tưc ho°c thíi gian ríi r⁄c Trong c¡c th“p ni¶n va qua, viằc nghiản cứu lỵ thuyt giÊi tch trản thang thíi gian cho ta nhi•u k‚t qu£ tŒng qu¡t v cõ nhiãu ứng dửng v o cĂc lắnh vỹc kh¡c Mºt nhœng b i to¡n quan trång cıa gi£i t‰ch tr¶n thang thíi gian l nghi¶n cøu t nh chĐt nh tnh v nh lữổng ca phữỡng tr…nh ºng lüc Trong lu“n ¡n n y, chóng tỉi mun nghiản cứu lỵ thuyt giÊi tch trản thang thới gian theo c¡ch ti‚p c“n mỵi â l gi£i t‰ch trản thang thới gian cặn l b i toĂn xĐp x¿ Cư th” hìn, chóng tỉi x†t kho£ng c¡ch giœa c¡c nghi»m cıa c¡c h» ºng lüc kh¡c v s‡ nghi¶n cøu sü phư thuºc li¶n tưc cıa mºt sŁ °c tr÷ng cıa ph÷ìng tr…nh ºng lüc theo dœ liằu ca phữỡng trnh Lun Ăn bao gỗm hai phn chnh Trữợc ht, chúng tổi xt b i toĂn xĐp x nghiằm ca phữỡng trnh ng lỹc trản thang thới gian v chứng minh ữổc dÂy cĂc nghiằm xn(t) ca phữỡng trnh x = f(t; x) trản dÂy thang thíi gian t÷ìng øng fT ng n=1 s‡ hºi tư n nghiằm x(t) ca phữỡng trnh n y trản thang thới gian T nu nhữ dÂy thang thới gian n y hºi tư v• thang thíi gian T theo kho£ng c¡ch Hausdorff Hìn nœa, chóng tỉi cơng ¡nh gi¡ ÷ỉc tŁc º hºi tö cıa c¡c nghi»m theo tŁc º hºi tư cıa d¢y thang thíi gian h m f thọa mÂn iãu kiằn Lipschitz theo cÊ hai bin Ti‚p theo, ta nghi¶n cøu sü phư thuºc theo tham sŁ v theo sü bi‚n thi¶n cıa thang thíi gian cıa mºt sŁ °c tr÷ng cıa ph÷ìng tr…nh ºng lüc 'n tuyn tnh Bữợc u tiản, ta thit lp mi liản hằ gia cĂc miãn n nh tữỡng ứng ca d¢y c¡c thang thíi gian d¢y thang thíi gian n y hºi tư theo tỉ pỉ Hausdorff CuŁi cịng, chóng ta nghi¶n cøu sü hºi tư cıa b¡n k‰nh Œn ành cıa ph÷ìng tr…nh ºng lüc 'n tuy‚n t‰nh chu nhiu cĐu trúc cÊ hai v ca phữỡng tr…nh c£ h» sŁ v thang thíi gian •u bi‚n thi¶n vi 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 vii It is easy to verify that indfA; Bg = and G(t) = Thus, r(An; Bn; Dn; En; Tn) = (supt2iR kGn(t)k) r(A; B; D; E; T) = sup kG(t)k = as n ! 1: t2iR We see in the above example that even (An; Bn; Dn; En) ! (A; B; D; E) and Ind(A; B) = 1, the stability radii r(An; Bn; Dn; En; Tn) may be equal to for all n N while r(A; B; D; E; T) 6= Thus, for the lower semi-continuity of the stability radii, we need to add some further assumptions In order our problem has a practical significance, we suppose r(An; Bn; Dn; En; Tn) > for all n N, which follows from the formula (3.38) that On the other hand, Gn( ) = E ( An n = EnP ( An b 86 Thus, by a similar way as Lemma3.2.10 we see that (3.41) holds if En Q = in the case Ind(A; B) = 1, and En Q = En Q = in the case Ind(A; B) > b b Theorem 3.3.6 Let Ind(A; B) = and equation (3.30) be uniformly exponentially stable Assume that limn!1(An; Bn; Dn; En; Tn) = (A; B; D; E; T) and (An En Q = for all n N Then, we have r(A; B; D; E; T) = lim inf r(An; Bn; Dn; En; Tn): n!1 Proof Let S be defined in (3.27) Then S is open and there exists an integer number N > such that S UT \ UTn and (An; Bn) S for all n > N: Since En Qb = for all n N and limn!1 En = E, it implies that Since limn!1(En; Dn) = (E; D), sup kEn k < 1; Therefore, by Lemma 3.2.10 and Proposition 3.2.16, we imply that there exists a constant C > and N > N such that kGn( )k kG( )k + C(kAn c for all S ; n > N : Since UTn r c (An; Bn; Dn; En; Tn) = sup kGn( )k 21 T sup Uc n kG( )k + C kA = r (A; B; D; E; Tn) + C An k for all n > N : Combining this with Proposition 3.3.4, we have 1 lim sup r (An; Bn; Dn; En; Tn) lim sup r (A; B; D; E; Tn) n!1 87 n or equivalently lim inf r(An; Bn; Dn; En; Tn) > r(A; B; D; E; T): n!1 Now, by Proposition 3.3.3, we obtain (3.42) The proof is complete To develop this theorem, we consider the case where Ind(A; B) > The following example shows that assumptions in Theorem 3.3.6 perhaps is not enough to the convergence of the stability radii as is shown by the following example Example 3.3.7 Let us consider the stability radii of (3.35) under structured perturbations of the form (3.36) with Tn = R for all n N and An = A = B= for all n N: It is easy to see that indfA; Bg = and G (t) = B n n(tt B B B @ We see that (A = for any n N and r(A; B; D; E; T) = Thus, to obtain a similarly result we have to give some additional assumptions on the coefficients Theorem 3.3.8 Let Ind(A; B) > Assume that and (An A)Q = (Bn B)Q = E b n!1 88 Proof This theorem can be proved by a similar way as it has done in the proof of Theorem (3.3.6) by using Proposition 3.2.17 If we restrict that the structured perturbations act only on the right-hand side (i.e., Ben = Bn + Dn nEn), we have the following corollary Corollary 3.3.9 Let Ind(A; B) = and equation (3.30) be uniformly exponentially stable Assume that lim (Bn; Dn; En; Tn) = (B; D; E; T) Then, we have n!1 lim r(A; Bn; Dn; En; Tn) = r(A; B; D; E; T): n!1 From the corollary, in the following example we will show that the stability radii of implicit difference equations obtained from LIDEs by the Euler methods, will tend to the stability radius of LIDEs when the mesh step tends to zero Example 3.3.10 Consider the equation Ax (t) = Bx(t); t R: Applying the explicit Euler method with step size h > to this equation we have Ax((m + 1)h) x(mh) = Bx(mh); m N: h This equation can be considered as a dynamic equation Ax (t) = Bx(t) on the time scale Th = hN It is easily seen that lim Th = R Therefore, by Proposih!0 tion 3.3.4, lim r(A; B; D; E; Th) = r(A; B; D; E; R): Moreover, if we use the explicit h!0 Euler method with the mesh step h = n to Ax (t) = Bnx(t); t R; n N; then we obtain a dynamic equation Ax (t) = Bnx(t) on the time scale Tn = n N Let r(A; Bn; Dn; En; Tn) be the stability radius of equation (3.43) on Tn By Corollary 3.3.9 we obtain lim r(A; Bn; Dn; En; Tn) = n!1 r(A; B; D; E; R) provided Ind(A; B) = and lim (Bn; Dn; En) = (B; D; E) n!1 Example 3.3.11 (See [34]) We consider the implicit system described by linear differential equations with structured perturbation A"x = (B + D E)x; 89 where A" = A + "F Assume that Ind(A; B) = and the system A "x = Bx is exponentially stable when " = Let Q be a projection on Ker A Suppose further that F Q = With this assumption, the condition (A A ")Q = is satisfied Therefore, we can apply Theorem 3.3.6 to obtain lim r(A + "F; B; D; E) = r(A; B; D; E): "#0 In case F Q 6= 0, which implies that (A A")Q 6= 0, we not expect the above limit is true In fact, in [34] they have proved that lim r(A + "F; B; D; E) = minfr(A; B; D; E); r(F22; B22; D2; E2)g: "!0 Where, A= and A11; F22 uniformly exponentially stable This example shows that the conditions in the Theorem 3.3.6 are not too heavy Conclusion of Chapter 3: In this chapter, we are concerned with the dependence of some characteristics of implicit dynamic equations of the form Ax (t) = Bx(t); t T on both the coefficients fA; Bg and time scale T Since the exponential stability of this dynamic equation related to the spectrum of the matrices pencil fA; Bg, it is worth to consider the dependence of the spectrum when the pair fA; Bg varies Next, it is meaningful to investigate the relation of the stability regions for a sequence of time scales fT n g n=1 and we consider the continuity of stability radii for the equations (3.45) on the triple (A; B; T) We obtain the main following results: establishing a relationship between the stability regions corresponding to a convergent sequence of time scales; analyzing how the spectrum of matrix pairs (A; B) and the exponential stability of (3.45) depend on data; 90 studying the convergence of the stability radii of equations (3.45) with general structured perturbations on the data These results are significant in practical problems We give conditions to ensure that the running of mechanic systems is stable under small perturbations This chapter is written based on the contents of the paper Nguyen Thu Ha, Nguyen Huu Du and Do Duc Thuan (2016), "On data- dependence of stability regions, exponential stability and stability radii for im-plicit linear dynamic equations", Math Control Signals Systems, 28(2), pp 13-28 91 Conclusion This thesis deals with two main problems The first problem is concerned with the convergence of solutions of dynamic equations x (t) = f(t; x) on time scales fTng n=1 when this sequence converges to the time scale T The convergent rate of solutions is estimated when f satisfies the Lipschitz condition in both variables A similar problem for nabla dynamic equations x (t) = f(t; x) is considered The second problem in this thesis analyzes the data-dependence of the stability regions, spectrum of matrix pair, exponential stability and stability radii for linear implicit dynamic equations of arbitrary index Relevant properties of the stability regions as well as the relation between spectrum of matrix pair and exponential stability have been investigated We have shown that the exponential stability and the stability radii depend continuously on the coefficient matrices and time scales As a prac-tical consequence, the complex stability radius of the linear differential algebraic equations with constant coefficients can be approximated by one of the implicit difference equations, for which it is more easy to compute As a future work, we can deal with the convergence of solutions for implicit dynamic equations on a sequence of time scales or a sequence of the coefficient functions f Also, an analysis of the exponential stability and the stability radii for time-varying implicit dynamic equations on time scales with respect to structured perturbations acting on both sides seems to be an interesting problem Last, the question if the above results are still valid for switching systems is interesting For these problems, we thing that more technical difficulties are expected 92 The author’s publications related to the thesis Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2016), "On the convergence of solutions to dynamic equations on time scales", Qual Theory Dyn Syst., 15(2), pp 453 469 (Chapter 2) Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2015), "On the convergence of solutions to nabla dynamic equations on time scales", Dyn Syst Appl., 24(4), pp 451-465 (Chapter 2) Nguyen Thu Ha, Nguyen Huu Du and Do Duc Thuan (2016), "On data- dependence of stability regions, exponential stability and stability radii for im-plicit linear dynamic equations", Math Control Signals Systems, 28(2), pp 13-28 (Chapter 3) 93 Bibliography [ Ti‚ng Vi»t [1] E.A Barbasin (1973), M u lỵ thuyt n nh, Nh xu§t b£n Khoa håc Kÿ thu“t, H Nºi [2] Nguy„n Th‚ Ho n, Ph⁄m Phu (2003), Cì sð ph÷ìng tr…nh vi phƠn v lỵ thuyt n nh, Nh xuĐt bÊn GiĂo dửc, H Ni [3] I.G Malkin (1980), Lỵ thuyt Œn ành chuy”n ºng, Nh xu§t b£n ⁄i håc v Trung hồc chuyản nghiằp, H Ni [4] Nguyn XuƠn Liảm (1996), Gi£i t‰ch h m, Nh xu§t b£n Gi¡o dưc, H Nºi [5] Ho ng Töy (1979), Gi£i t‰ch hi»n ⁄i, T“p 1, 2, 3, Nh xu§t b£n Gi¡o dưc, H Nºi [ English [6] E Akin Bohner, M Bohner and F Akin (2005), "Pachpatte inequalities on time scales", J Inequalities Pure and Applied Mathematics, 6(1) [7] R Agarwal, M Bohner, 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15(2 ), pp 453 469 Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2015 ), "On the convergence of solutions to nabla dynamic equations