Sufficient conditions for a minimum of the free-final-time optimal control problem (Q911026)

Sufficient conditions are derived for a minimum of the free-final-time optimal control problem. First, the second variation is obtained for an admissible comparison path from a perturbed initial state to the final constraint manifold. Next, equations for the neighboring extremal from the perturbed initial state to the final constraint manifold are derived and used to write the second variation as a perfect square. Then, it is shown that, if the sufficient conditions are satisfied, the second variation is positive for all admissible comparison paths, that is, those paths which can be obtained from the minimal path by a system of weak variations. Two example problems are solved to illustrate the use of these conditions.