Differential-Geometric Methods: A Powerful Set of New Tools for Optimal Control

Diierential-geometric methods can and have been successfully used in control theory, not only to pose and solve new problems, but also to get new insights and prove new results about classical problems of optimal control. In particular, they yield new results on local controllability, continuity properties of value functions, the structure of reachable sets, and the properties of optimal trajectories. For large classes of systems given by controlled diierential equations with real-analytic right-hand sides, these methods make it possible |in conjunction with the theory of subanalytic sets|-to prove piecewise analyticity of the value function and the existence of a piecewise smooth optimal feedback.