In this manuscript, we consider a control system governed by a general ordinary differential equation on a Riemannian manifold M, with its endpoints satisfying some inequalities and its control constrained to a closed convex set. We obtain a second-order necessary optimality condition through the convex variation for an optimal control problem of this system (i.e., Theorem 2.2). It has the following characteristics. First, comparing to the second-order necessary optimality condition obtained by the needle variation (e.g., [Deng and Zhang, J. Differential Equations, 272 Theorem 2.2]) it works for a different family of candidates for optimal pairs. Second, any condition of controllability type on the linearized system is not required. Third, Theorem 2.2 involves the second-order adjacent subset, which plays roles especially in the case where an admissible control stays at the boundary of the control set. Actually, we give an example (Example 2.6) to support the above features. The curvature tensor also arises in Theorem 2.2.
展开▼
