On the role of regularity in mathematical control theory
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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 tautological 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 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.