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On the Role of Regularity in Mathematical Control Theory

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On the Role of Regularity in Mathematical Control Theory On the role of regularity in mathematical control theory by Saber Jafarpour A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen's University Kingston, Ontario, Canada April 2016 Copyright c Saber Jafarpour, 2016 Abstract In this thesis, we develop a coherent framework for studying time-varying vector fields of different regularity classes and their flows. This setting has the benefit of unifying all classes of regularity. In particular, it includes the real analytic regularity and provides us with tools and techniques for studying holomorphic extensions of real analytic vector fields. We show that under suitable \integrability" conditions, a time-varying real analytic vector field on a manifold can be extended to a time- varying holomorphic vector field on a neighbourhood of that manifold. Moreover, in this setting, the \nonlinear" differential equation governing the flow of a time- varying vector field can be considered as a \linear" differential equation on an infinite dimensional locally convex vector space. We show that, in the real analytic case, the \integrability" of the time-varying vector field ensures convergence of the sequence of Picard iterations for this linear differential equation, giving us a series representation for the flow of a time-varying real analytic vector field. Using the framework we develop in this thesis, we study a parametization- independent model in control theory called tautological control system. In the tauto- logical control system setting, instead of defining a control system as a parametrized family of vector fields on a manifold, it is considered as a subpresheaf of the sheaf of vector fields on that manifold. This removes the explicit dependence of the systems i on the control parameter and gives us a suitable framework for studying regularity of control systems. We also study the relationship between tautological control systems and classical control systems. Moreover, we introduce a suitable notion of trajectory for tautological control systems. Finally, we generalize the orbit theorem of Sussmann and Stefan to the tautological framework. In particular, we show that orbits of a tautological control system are immersed submanifolds of the state manifold. It turns out that the presheaf structure on the family of vector fields of a system plays an important role in characterizing the tangent space to the orbits of the system. In particular, we prove that, for globally defined real analytic tautological control systems, every tangent space to the orbits of the system is generated by the Lie brackets of the vector fields of the system. ii Acknowledgments First, I would like to thank my supervisor, Professor Andrew Lewis. Andrew gave me the opportunity and the courage to do a PhD in mathematics. The conversations we had, about Mathematics in general and about our research in particular, were very instructive for me. I would like to thank Professors Alberto Bressan, Abdol-Reza Mansouri, Martin Guay, and Oleg Bogoyavlenskij for accepting to be in my PhD defence committee. I am also very thankful of Professor Bahman Gharesifard for his valuable advices about my academic career. I want to thank my parents and my sister for all their love and supports. My parents are definitely the most precious things in my life and I humbly dedicate this thesis to them. During my stay in Kingston, I was fortunate enough to have many good friends. I would like to thank Naci Saldi for his kindness and hospitality. I would like to thank Ivan Penev for all discussions we had together and also for his priceless comments on some parts of this thesis. Lastly, I would like to thank Jennifer read for all her helps from the first day of my arrival to Canada to the last day of my PhD here. iii Statement of Originality I herby declare that, except where it is indicated with acknowledgment and citation, the results of this thesis are original. iv Contents Abstract i Acknowledgments iii Statement of Originality iv Contents v List of Figures vii Chapter 1: Introduction 1 1.1 Literature review . 1 1.2 Contribution of thesis . 19 Chapter 2: Mathematical notation and background 27 2.1 Manifolds and mappings . 27 2.2 Complex analysis . 35 2.3 Topological vector spaces . 36 2.3.1 Basic definitions . 36 2.3.2 Locally convex topological vector spaces . 37 2.3.3 Uniformity on topological vector spaces . 39 2.3.4 Dual space and dual topologies . 42 2.3.5 Curves on locally convex topological vector spaces . 47 2.3.6 Inductive limit and projective limit of topological vector spaces 54 2.3.7 Tensor product of topological vector spaces . 60 2.3.8 Nuclear spaces . 63 2.4 Sheaves and presheaves . 65 2.5 Groupoids and pseudogroups . 73 Chapter 3: Time-varying vector fields and their flows 77 3.1 Introduction . 77 3.2 Topology on the space Γν(E) ...................... 81 v 3.2.1 Topology on Γ1(E)........................ 82 3.2.2 Topology on Γhol(E) ....................... 82 3.2.3 Topology on Γ!(E)........................ 87 3.3 Cν-vector fields as derivations on Cν(M) . 109 3.4 Cν-maps as unital algebra homomorphisms on Cν(M) . 120 3.5 Topology on L(Cν(M); Cν(N)) . 129 3.6 Curves on the space L(Cν(M); Cν(N)) . 135 3.6.1 Bochner integrable curves . 136 3.6.2 Space L1(T; L(Cν(M); Cν(N))) . 136 3.6.3 Space L1(T; Der(Cν(M))) . 137 3.6.4 Space C0(T; L(Cν(M); Cν(N))) . 138 3.6.5 Absolutely continuous curves . 138 3.6.6 Space AC(T; L(Cν(M); Cν(N))) . 141 3.7 Extension of real analytic vector fields . 142 3.7.1 Global extension of real analytic vector fields . 144 3.7.2 Local extension of real analytic vector fields . 148 3.8 Flows of time-varying vector fields . 154 3.9 The exponential map . 170 Chapter 4: Tautological control systems 181 4.1 Introduction . 181 4.2 Cν-tautological control systems . 185 4.3 Parametrized vector fields of class Cν . 187 4.4 Cν-control systems . 188 4.4.1 Extension of real analytic control systems . 191 4.5 From control systems to tautological control systems and vice versa . 195 4.6 Trajectories of Cν-tautological systems . 196 Chapter 5: The orbit theorem for tautological control systems 202 5.1 Introduction . 202 5.2 Reachable sets of Cν-tautological control systems . 203 5.3 Algebraic structure of orbits . 210 5.4 Geometric structure of orbits . 212 5.4.1 Singular foliations . 212 5.4.2 The orbit theorem . 219 5.4.3 The real analytic case . 227 Chapter 6: Conclusions and future work 238 6.1 Conclusions . 238 6.2 Future work . 239 vi List of Figures 5.1 Privileged Chart . 214 vii 1 Chapter 1 Introduction 1.1 Literature review In the mathematical theory of control, regularity of maps and functions plays a crucial role. Literature of control theory is replete with theories which work for a specific class of regularity but fail in other classes. The most well-known example is stabilizability of nonlinear control systems. While it is clear that for linear control systems con- trollability implies stabilizability, for nonlinear control systems this implication may or may not be true based on the regularity of the stabilizing feedback. In his 1979 paper, Sussmann proved that every real analytic system which satisfies some strong controllability condition is stabilizable by \piecewise analytic" controls [78]. However, some years later, Brockett showed that controllability of a nonlinear system does not necessarily imply stabilizability by \continuous" feedbacks [14]. Another example is the role that real analyticity plays in characterizing the fundamental properties of systems. One can show that the accessibility of real analytic systems at point x0 can be completely characterized using the Lie bracket of their vector fields at point x0 [79]. However, such a characterization is not generally possible for smooth systems 1.1. LITERATURE REVIEW 2 [79]. While in the physical world there are very few, and possibly no, maps which are smooth but not real analytic, in mathematics the gap between these two classes of regularity is huge. This can be seen using the well-known fact that if a real analytic function is zero on an open set, then it is zero everywhere. However, using a parti- tion of unity, one can construct many non-zero smooth functions that are zero on a given open set [49]. Moreover, the techniques and analysis for studying real analytic systems are sometimes completely different from their smooth counterparts. Roughly speaking, a map f is real analytic on a domain D if the Taylor series of f around every point x0 2 D converges to f in a neighbourhood of x0. By definition, for the Taylor series of f on D to exist, derivatives of f of any order should exist and be continuous at every point x0 2 D. This means that all real analytic maps are of class C1. The converse implication is not true, since there are some examples of functions that are C1 but not real analytic. Although, nowadays, these examples of smooth but not real analytic functions are well-known, it is surprising to know that at the beginning stages of theory of real analytic functions, mathematicians had difficulty understanding them. In the nineteenth century, mathematicians started to think more about the natural question of which functions can be expanded in a Taylor series around a point. Lagrange and Hankel believed that the existence of all derivatives of a function implies the convergence of its Taylor series [8].
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