| This thesis is a contribution to solving problems of extracting optimal controls for complex systems. Its novelty consists of a detailed examination of Finsler geometry based control by connections as proposed by Kohn and Nerode and its relation to Pontryagin's maximum principle. The long term hope is that these methods will form the underpinning of the applications of control of hybrid systems.;Any advances in mathematical and algorithmic techniques for solving such problems would have wide application in business, industry, and science. The widespread use of Bellman's dynamic programming, Dantzig's linear programming, Kalman's optimization with linear quadratic cost functions, demonstrate this. But symbolic and numerical techniques historically have fallen well short of yielding efficient computation procedures to obtain near optimal controls for complex systems. Most investigations have been based on Pontryagin's geometric representation of optimal control and his maximum principle. Kohn and Nerode have proposed a different problem formulation aimed at extracting robust controls as a function of state (synthesis problem with a robustness requirement) in the context of a Finsler geometry corresponding to the optimal control problem. This leads to the study of geometric, symbolic, and numerical methods for solving geodesic equations for connections in Finsler Geometry. A principal result of this thesis is the determination of the relations between the Finsler and Pontryagin formulations of optimal control, and the transformation from one to the other. A second principal result is establishing the relations between robustness and curvature. Curvature is used to quantify the spread of geodesics due to disturbances. Finally, the thesis concludes with numerical integration schemes for computing controls and local connections. |
