A Survey of Optimal Control Problems Evolved on Riemannian Manifolds (CSIAM Transactions on Applied Mathematics, 2022)
In this paper, we present our optimality results on optimal control problemsfor ordinary differential equations on Riemannian manifolds. For the problems withfree states at the terminal time, we obtain the first and second-order necessary conditions, dynamical programming principle, and their relations. Then, we consider theproblems 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 ofthe above results concerning second-order optimality conditions, the curvature tensorof the underlying manifold plays a crucial role.