Minimum time with bounded energy, minimum energy with bounded time

Necessary and sufficient conditions for the regularity of the minimum time function and minimum energy function for a control system with controls in L^p([0,+infinity], R^m) and p greater than or equal to 1 are given in terms of topological properties of the reachable sets. In particular, standard local controllability assumptions are sufficient to yield the continuity of both value functions for linear systems and p greater than or equal to 1 and the Holder continuity of the minimum time function for nonlinear systems and p > 1.