Approximation problems for dynamic equations on time scales = Bài toán xấp xỉ cho phương trình động lực trên thang thời gian. Luận án TS. Toán học: 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 ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN Nguyễn Thu Hà BÀI TOÁN 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 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 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 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 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 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 v 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 tốn 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, chúng tơi 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 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 Proposition 3.3.4 Assume that the equation (3.30) is uniformly exponentially stable and lim Tn = T Then n→∞ lim r(A, B; D, E; Tn ) = r(A, B; D, E; T) n→∞ Proof Let ∞ UTcm = lim inf UTcm , Vn = V = m→∞ n=1 m n UTcm , gn = sup G(λ) λ∈Vn m n c By Corollary 3.1.9, UT \ R ⊂ V ⊂ UTc , which follows r−1 (A, B; D, E; T) = sup G(λ) = sup G(λ) λ∈UTc Since Vn ⊃ Vn+1 for all n, gn λ∈V gn+1 Therefore, there exists lim gn n→∞ G(λn ) + n1 If the By definition of gn , for each n we can find λn ∈ Vn such that gn sequence {λn } is not bounded then there exists {nk } such that λnk → ∞ as k → ∞ Since G(λ) is a rational function, there exists the limit G(∞) = lim G(λ) λ→∞ Moreover, since the equation (3.16) is uniformly exponentially stable then UT ⊂ C− Therefore, if n > then n ∈ UTc Thus, r−1 (A, B; D, E; T) G(∞) = lim G(n) n→∞ = lim G(λnk ) = lim gnk = lim gn k→∞ n→∞ k→∞ In the case the sequence {λn } is bounded, there exist λ ∈ C and a sequence {nk } such that λnk → λ as k → ∞ We will prove that λ ∈ UTc Indeed, assume that λ ∈ UT Then, by Proposition 3.1.6, there is δ > such that B(λ, δ) ⊂ UTn for all n large enough This contradicts to λnk ∈ UTcnk for all k and lim λnk = λ Thus, k→∞ r−1 (A, B; D, E; T) G(λ) = lim G(λnk ) = lim gnk = lim gn k→∞ k→∞ n→∞ On the other hand, since UTcn ⊂ Vn , gn sup λ∈UTcn G(λ) = r−1 (A, B; D, E; Tn ) for all n ∈ N Thus, we get r−1 (A, B; D, E; T) lim sup r−1 (A, B; D, E; Tn ) = lim inf r(A, B; D, E; Tn ) n→∞ n→∞ or equivalently, r(A, B; D, E; T) lim inf r(A, B; D, E; Tn ) n→∞ 85 −1 , Combining with (3.39) yields lim r(A, B; D, E; Tn ) = r(A, B; D, E; T) n→∞ The proof is complete When the coefficients of the systems (3.35) vary in n, even they have index 1, the inverse relation of (3.39) is not true as shown in the following example Example 3.3.5 Let Tn = R for all n, we consider equation (3.37) under structured perturbations of the form (3.36) with       −1 −1 0         A =  0 ; Bn =  −2 n1 − 1  ; B = −2 −1  ;   −1 0 −n +     0 n1 0 −1         Dn =  −1  ; En1 =  0  ; En2 =     −1 1 0 −n n  It is easy to verify that ind{A, B} = and    −1 −1 0 t+2 t+2       G(t) =   ; Gn (t) =     0 Thus, r(An , Bn ; Dn , En ; Tn ) = (supt∈iR Gn (t) ) −1       t n −t+n n     3+t n(t+2) = does not tend to −1 r(A, B; D, E; T) = sup G(t) = as n → ∞ t∈iR 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) = 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 sup λ∈UTcn Gn (λ) < ∞ (3.41) On the other hand, Gn (λ) = Enλ (λAn − Bn )−1 Dn = Enλ P (λAn − Bn )−1 Dn + Enλ Q(λAn − Bn )−1 Dn = Enλ P (λAn − Bn )−1 Dn + (λEn1 − En2 )Q(λAn − Bn )−1 Dn 86 Thus, by a similar way as Lemma3.2.10 we see that (3.41) holds if En1 Q = in the case Ind(A, B) = 1, and En1 Q = En2 Q = in the case Ind(A, B) > Theorem 3.3.6 Let Ind(A, B) = and equation (3.30) be uniformly exponentially stable Assume that limn→∞ (An , Bn ; Dn , En ; Tn ) = (A, B; D, E; T) and (An −A)Q = En1 Q = for all n ∈ N Then, we have r(A, B; D, E; T) = lim inf r(An , Bn ; Dn , En ; Tn ) n→∞ (3.42) 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 σ(An , Bn ) ⊂ S for all n > N Since En1 Q = for all n ∈ N and limn→∞ En = E, it implies that En1 = En1 P and E = E P for all n ∈ N Therefore, Gn (λ) − G(λ) = Enλ (λAn − Bn )−1 − (λA − B)−1 Dn + (Enλ − E λ )(λA − B)−1 Dn + E λ (λA − B)−1 (Dn − D) = En1 λP (λAn − Bn )−1 − (λA − B)−1 Dn + (En1 − E )λP (λA − B)−1 Dn + E λP (λA − B)−1 (Dn − D) − En2 (λAn − Bn )−1 − (λA − B)−1 Dn − (En2 − E )(λA − B)−1 Dn − E (λA − B)−1 (Dn − D) Since limn→∞ (En , Dn ) = (E, D), sup En1 < ∞, sup En2 < ∞, sup Dn < ∞ n∈N n∈N n∈N Therefore, by Lemma 3.2.10 and Proposition 3.2.16, we imply that there exists a constant C > and N ∗ > N such that Gn (λ) G(λ) + C( An − A + Bn − B + Dn − D + En − E ) for all λ ∈ S c , n > N ∗ Since UTcn ⊂ S c , r−1 (An , Bn ; Dn , En ; Tn ) = sup λ∈UTcn sup λ∈UTcn G(λ) + C Gn (λ) An − A + Bn − B + Dn − D + En − E = r−1 (A, B; D, E; Tn ) + C An − A + Bn − B + Dn − D + En − E , for all n > N ∗ Combining this with Proposition 3.3.4, we have lim sup r−1 (An , Bn ; Dn , En ; Tn ) n→∞ lim sup r−1 (A, B; D, E; Tn ) n→∞ = lim r−1 (A, B; D, E; Tn ) = r−1 (A, B; D, E; T), n→∞ 87 or equivalently lim inf r(An , Bn ; Dn , En ; Tn ) n→∞ r(A, B; D, E; T) 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    + n1 −1 0 −1      0 0 1  ; Bn =  n An = A =   −2 −1 0    −1 + n1 −1     −1 1     0  0  −1  ; Dn = D =   ; En1 B= −2    0  −1  −1 1 −1   −1 ; 0  = 0; En2 = I, for all n ∈ N It is easy to see that ind{A, B} = and     1 0 t−2  t−2  t2    −1  n(t−2) −1  ; Gn (t) =   G(t) =     t  0    n(t−2)  0 n(t−2) We see that (An −A)Q = En1 Q = meanwhile r(An , Bn ; Dn , En ; Tn ) = sup Gn (t) t∈iR −1 = for any n ∈ N and r(A, B; D, E; T) = sup G(t) = t∈iR 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 lim (An , Bn ; Dn , En ; Tn ) = (A, B; D, E; T) n→∞ and (An − A)Q = (Bn − B)Q = En1 Q = En2 Q = for all n ∈ N Then, we have r(A, B; D, E; T) = lim inf r(An , Bn ; Dn , En ; Tn ) n→∞ 88 −1 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., Bn = Bn + Dn Σn En ), 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→∞ lim r(A, Bn ; Dn , En ; Tn ) = r(A, B; D, E; T) n→∞ 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 A x((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) = Bn x(t), t ∈ R, n ∈ N, then we obtain a dynamic equation Ax∆ (t) = Bn x(t) on the time scale Tn = n N (3.43) 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→∞ r(A, B; D, E; R) provided Ind(A, B) = and lim (Bn , Dn , En ) = (B, D, E) n→∞ Example 3.3.11 (See [34]) We consider the implicit system described by linear differential equations with structured perturbation Aε x∆ = (B + DΣE)x; 89 (3.44) where Aε = A + εF Assume that Ind(A, B) = and the system Aε x∆ = Bx is exponentially stable when ε = Let Q be a projection on Ker A Suppose further that F Q = With this assumption, the condition (A − Aε )Q = is satisfied Therefore, we can apply Theorem 3.3.6 to obtain lim r(A + εF, B; D, E) = r(A, B; D, E) ε↓0 In case F Q = 0, which implies that (A − Aε )Q = 0, we not expect the above limit is true In fact, in [34] they have proved that lim r(A + εF, B; D, E) = min{r(A, B; D, E), r(F22 , B22 ; D2 , E2 )} ε→0 Where, A= A11 0 ; F = F11 F12 F21 F22 ; D = (D1 , D2 ) ; E = (E1 , E2 ) −1 and A11 , F22 are nonsingular, F22 x∆ = B22 x, A11 x∆ = (B11 − B12 B22 B21 )x are 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 (3.45) on both the coefficients {A, B} and time scale T Since the exponential stability of this dynamic equation related to the spectrum of the matrices pencil {A, B}, it is worth to consider the dependence of the spectrum when the pair {A, B} varies Next, it is meaningful to investigate the relation of the stability regions for a sequence of time scales {Tn }∞ 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 datadependence of stability regions, exponential stability and stability radii for implicit 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 {Tn }∞ 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 practical 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 datadependence of stability regions, exponential stability and stability radii for implicit linear dynamic equations", Math Control Signals Systems, 28(2), pp 13-28 (Chapter 3) 93 Bibliography [*] Tiếng Việt [1] E.A Barbasin (1973), Mở đầu lý thuyết ổn định, Nhà xuất Khoa học Kỹ thuật, Hà Nội [2] Nguyễn Thế Hồn, Phạm Phu (2003), Cơ sở phương trình vi phân lý thuyết ổn định, Nhà xuất Giáo dục, Hà Nội [3] I.G Malkin (1980), Lý thuyết ổn định chuyển động, Nhà xuất Đại học Trung học chuyên nghiệp, Hà Nội [4] Nguyễn Xuân Liêm (1996), Giải tích hàm, Nhà xuất Giáo dục, Hà Nội [5] Hồng Tụy (1979), Giải tích đại, Tập 1, 2, 3, Nhà xuất Giáo dục, Hà Nội [*] 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, D O’Regan, A Peterson (2002), "Dynamic equations on time scales: a survey", J Comput Appl Math., 141, pp 1-26 [8] H Attouch, R Lucchetti, R.J.B Wets (1991), "The topology of ρ-Hausdorff distance", Ann Mat Pur Appl., 160(4), pp 303-320 [9] H Attouch, R.J.B Wets (1991), "Quantitative stability of variational systems: I The epigraphical distance", Trans Amer Math Soc., 3, pp 695-729 [10] Z Bartosiewicz, E Piotrowska (2013), "On stability of nonlinear systems on time scales", Int J Control, 86(1), pp 139-145 [11] Z Bartosiewics, E Piotrowska and M Wyrwas (2007), "Stability, Stabilization and observers of linear control systems on time scales", Proceeding of the 46th 94 IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec., pp 12-14 [12] M Bohner and A Peterson (2001), Dynamic equations on time scales: An introduction with applications, Birkhăauser, Boston [13] M Bohner and A Peterson (2003), Advances in dynamic equations on time scales, Birkhăauser, Boston [14] M Bohner (2005), "The logarithm on time scales", J Differ Equations Appl., 11(15), pp 1305-1306 [15] M Bracke (2000), On stability radii of parametrized linear differential-algebraic systems, Ph.D thesis, University of Kaiserslautern [16] K.E Brenan, S.L Campbell, L.R Petzold (1996), Numerical solution of initial value problems in differential algebraic equations, SIAM, Philadelphia [17] A.V Bulatov and P.Diamond (2002), "Real structural stability radius of infinite-dimensional linear systems: Its estimate", Automation and Remote Control, 63( 5), pp 713-722 [18] D Burago, Y Burago, S Ivanov (2001), A Course in metric geometry, AMS Graduate Studies in Math, vol 33 Amer Math Soc., Provindence [19] R.L Burden, J.D Fairs (2000), Numerical Analysis, 7th Edition, Brooks Cole [20] R Byers, N Nichols (1993), "On the stability radius of a generalized state-space system", Linear Algebra Appl 189, pp 113-134 [21] K.E Brenan, S.L Campbell, L.R Petzold (1989), Numerical Solution of initial value problems in differential-algebraic equations, North-Holland, New York [22] J.C Butcher (2008), Numerical Methods for Ordinary Differential Equations, Second Edition, John Wiley & Sons Ltd [23] A Cabada and D R Vivero (2006), "Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral; application to the caculus of ∆−antiderivatives", Math Comput Modelling, 43, pp 194-207 [24] S.L Campbell (1982), Singular systems of differential equations, Pitman Advanced Publishing Program, San Francisco-London-Melbourne [25] C.J Chyan, N.H Du, V.H Linh (2008), "On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems", J Differential Equations 245, pp 2078-2102 95 [26] J.J DaCunha (2005), "Stability for time varying linear dynamic systems on time scales", J Comput Appl Math., 176(2), pp 381-410 [27] J.J DaCunha and J.M Davis (2011), "A unified Floquet theory for discrete, continuous, and hybrid linear systems", J Differential Equations, 251, pp 29873027 [28] L Dai (1989), Singular control systems, Springer, Berlin-Heidelberg [29] J.M Davis, I.A Gravagne, B Jackson, R.J Marks (2009), "Controllability, observability, realizability and stability of dynamic linear systems", Electron J Differential Equations, 37, pp 1-32 [30] T.S Doan, A Kalauch, S Siegmund and F Wirth (2010), "Stability radii for positive linear time-invariant systems on time scales", Systems &Control Letter, 59, pp 173-179 [31] V Dragan (1998), "The asymptotic behavior of the stability radius for a singularly perturbed linear system", International Journal of Robust and Nonlinear Control, 8, pp 817-829 [32] N.H Du, V.H Linh (2005), "Implicit-system approach to the robust stability for a class of singularly perturbed linear systems", Systems &Control Letters, 54, pp 33-41 [33] N.H Du, V.H Linh (2006), "Stability radii for linear time-invarying differentialequations with respect to dynamic perturbations", J Differential Equations, 230, pp 579-599 [34] N.H Du, V.H Linh (2006), "On the robust stability of implicit linear systems containing a small parameter in the leading term", IMA Journal of Mathematical Control and Information, 23, pp 67-84 [35] N.H Du (2008), "Stability radii of differential-algebraic equations with structured perturbations", Systems &Control Lett., 57, pp 546-553 [36] N.H Du, D.T Lien and V.H Linh (2003), "On complex stability radii for implicit discrete time systems", Vietnam Journal of Mathematics, 31(4), pp 475-488 [37] N.H Du, V.H Linh, V Mehrmann and D.D Thuan (2013), "Stability and robust stability of linear time-invariant delay differential-algebraic equations", SIAM J Matrix Anal Appl., 34, pp 1631-1654 96 [38] N.H Du, V.H Linh and V Mehrmann (2013), Robust stability of differentialalgebraic equations, in: Surveys in Differential-Algebraic Equations I, DAE Forum, pp 63-95 [39] N.H Du, D.D Thuan and N.C Liem (2011), "Stability radius of implicit dynamic equations with constant coefficients on time scales", Systems &Control Letters, 60, pp 596-603 [40] E Griepentrog, R Măarz (1986), Differential-Algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik, Leipzig [41] T Gard and J Hoffacker (2003), "Asymptotic behavior of natural growth on time scales", Dynamic Systems and Applications, 12(1-2), pp 131-148 [42] Gusein Sh Guseinov (2003), "Integration on time scales", J Math Anal Appl., 285, pp 107-127 [43] S Hilger (1990), "Analysis on measure chains - A unified approach to continuous and discrete calculus", Results Math., 18, pp 18-56 [44] E Hairer, S.P Norsett, G Wanner (1993), Solving ordinary differential equations I nonstiff problems, second revised edition, Springer [45] M Hanke, E.I Macana and R Măarz (1998), "On asymptotics in case of linear index differential algebraic equations", SIAM J Numer Anal., 35(1), pp 1326-1346 [46] M Hanke (1994), "Asymptotic expansions for regularization methods of linear fully implicit differential algebraic equations", Z Anal und ihre Anw, 13, pp 513-535 [47] S Hilger (1988), Ein Makettenkalkă ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D thesis, Universităat Wă urzburg [48] D Hinrichsen and AJ Pritchard (1986), "Stability radii of linear systems", Sys &Contr Let., 7, pp 4-10 [49] D Hinrichsen and AJ Pritchard (1986), "Stability radii for structured perturbations and the algebraic Riccati equations linear systems", Sys &Contr Lett., 8, pp 105-113 [50] D Hinrichsen and N.K Son (1998), "Stability radii of positive discrete-time systems under affine parameter perturbations", International Journal of Robust and Nonlinear Control, 8, pp 1169-1188 97 [51] D Hinrichsen, A Ilchmann, A J Pritchard (1989), "Robustness of stability of time-varying linear systems", J Differential Equations 82, pp 219-250 [52] D Hinrichsen, N.K Son (1989), "The complex stability radius of discrete-time systems and symplectic pencils", Proceedings of the 28th IEEE Conference on Decision and Control 1-3, pp 2265-2270 [53] B.J Jackson (2008), "Adaptive control in the nabla setting", Neural Parallel Sci Comput., 16, pp 253-272 [54] B Jacob (1998), "A formula for the stability radius of time-varying systems", J Differential Equations, 142, pp 167-187 [55] P Kunkel, V Mehrmann (2006), Differential-Algebraic equations Analysis and numerical solution, EMS Publishing House, Ză urich, Switzerland [56] L.C Loi, N.H Du, P.K Anh (2002), "On linear implicit non-autonomous systems of difference equations", J Difference Equ Appl., 8(12) [57] Le Hong Lan and Nguyen Chi Liem (2010), "Stability radius of linear dynamic equations with constant coefficients on time scales", VNU Journal of Science, Mathematics-Physics, 26, pp 163-173 [58] R Măarz (1998), "Extra-ordinary differential equation attempts to an analysis of differential algebraic system", Progress in Mathematics, 168, pp 313-334 [59] F Memoli (2012), "Some properties of Gromov-Hausdorff distance", Discrete Comput Geom., 48, pp 416-440 [60] C Păotzsche (2004), "Exponential dichotomies of linear dynamic equations on measure chains slowly varying coefficients", J Anal Math Appl., 289, pp 317335 [61] C Păotzsche, S Siegmund and F Wirth (2003), "A spectral characterization of exponential stability for linear time-invariant systems on time scales", Discrete and Continuous Dynamical System, 9, pp 1123-1241 [62] P.J Psarrakos and M.J Tsatsomeros (2002) "On the stability radius of matrix polynomials", Linear and Multilinear Algebra, 50(1), pp.151-165 [63] G Salinetti, R.J.B Wets (1979), "On the convergence of sequence of convex sets in finite dimensions", SIAM Rev., 21, pp 16-33 [64] H.H Schaefer (1974), Banach latices and positive operators, Springer -Verlag Berlin Heidelberg New York 98 [65] H.R Sun, W.T Li (2007), "Existence theory for positive solutions to onedimensional p-Laplacian boundary value problems on time scales", J Differential Equations, 240, pp 217-248 [66] F.Z Taousser, M Defoort and M Djemai (2014), "Stability analysis of a class of switched linear systems on non-uniform time domains", Systems &Control Letter, 74, pp 24-31 [67] F.Z Taousser, M Defoort and M Djemai (2015), "Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions", Nonlinear Analysis: Hybrid Systems, 16, pp 13-23 [68] D.D Thuan, N.H Du and N.C Liem (2016), "Stabilizability and robust stabilizability of implicit dynamic equations with constant coefficients on time scales", IMA J Math Control Inform., 33(1), pp 121-136 99