Mathematical Sciences: The Structure of the Small-Time Reachable Set and Regularity Properties of Optimal Trajectories for Control- Linear Systems

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This project addresses the qualitative structure of small-time reachable sets and piecewise regularity properties of optimal controls for nonlinear systems. For single-input systems it is known that optimal trajectories can exhibit undesirable features such as chattering bang-bang arcs if the dimension of the system becomes too large. This may be due to a lack of possibility to influence the system via the control vector field. To explore that scenario this project seeks to analyze the local structure of small-time reachable sets in dimension n for increasing values of n, in particular, to develop tests for optimality of complex concatenations of bang and singular arcs, and ii) to investigate multi-input systems using higher-order approximating cones which take into account specific structures of trajectories under consideration. The underlying technical framework is provided by differential-geometric descriptions and Lie-algebraic computational techniques. The long term aim of this work is to achieve a qualitative understanding of the local behavior of nonlinear control systems. Aside from its evident theoretical relevance, such a knowledge might also be practically of interest for any possible future approach to design involving nonlinear components.