A Survey of Optimal Control Problems Evolved on Riemannian Manifolds

. In this paper, we present our optimality results on optimal control problems for ordinary differential equations on Riemannian manifolds. For the problems with free states at the terminal time, we obtain the first and second-order necessary conditions, dynamical programming principle, and their relations. Then, we consider the problems with the initial and final states satisfying some inequality-type and equality-type constraints, and establish the corresponding first and second-order necessary conditions of optimal pairs in the sense of either spike or convex variations. For each of the above results concerning second-order optimality conditions, the curvature tensor of the underlying manifold plays a crucial role.