Dynamic programming and optimal control problems on manifolds

Dynamic Programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi Equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimizers. In continuous time optimal control, the dynamic programming methodology has been used for problems with slate space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite dimension C^∞ manifold. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.