Second Order Necessary Conditions and Sensitivity Relations for Optimal Control Problems on Riemannian Manifolds

摘要

In this note, we study the second order optimality conditions of optimal control problems on Riemannian manifolds. On one hand, we give the second order necessary conditions for optimal controls when the initial state is constrained to a closed subset of a Riemannian manifold. On the other hand, we obtain the second order sensitivity relations between the necessary conditions and the value function of an optimal control problem: the first and second order dual variables are related to the superjet of the value function. Both results show that the second order optimality conditions of optimal control problems on a Riemannian manifold are closely related to the curvature tensor of the manifold.