Sufficient optimality conditions for the optimal control problem

The main purpose of this paper is to establish the sufficient optimality conditions for the optimal controls under some convexity assumptions. For the Bolza problem, under concavity of the Hamiltonian and convexity of the cost functional, the extreme control must be optimal control. However, for the Mayer problem, the cost functional can be relaxed to quasi-convex and pseudo-convex. Finally, some examples illustrate these theoretical results and some counter examples show that the convexity assumptions of these results cannot be further weakened in some sense.