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.
This paper is devoted to sufficient condition for Strong Metric sub‐Regularity (SMsR for short) of the set‐valued mapping corresponding to the local description of Pontryagin maximum principle for the Mayer‐type optimal control problems with convexity condition of the Hamiltonian and functional. In particular, stability property of optimal control for the Mayer‐type problem has been established for the occasion of a polyhedral control set and entirely bang‐bang solution structure. Moreover, based on the sufficiency of SMsR and stability property of optimal control, we give the approximate errors of Euler discretization methods utilized to such problems.
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