Second order optimality conditions for the Bolza endpoint constraint control problem evolved on a Riemannian manifold

In this paper, we derive some second order necessary and sufficient conditions of optimal controls for a general control system on a Riemannian manifold. In our control problem, the endpoint is fixed and the control set is open in a Euclidian space. The curvature tensor of the manifold appears explicitly in our results. One of our optimality conditions can be used to recover the following famous geometry result: The shortest geodesic connecting two fixed points on a Riemannian manifold must satisfy the second variation of energy.