Optimization of second-order discrete approximation inclusions

The present article studies the approximation of the Bolza problem of optimal control theory with a fixed time interval given by convex and non-convex second-order differential inclusions (P_C). Our main goal is to derive necessary and sufficient optimal conditions for a Cauchy problem of second-order discrete inclusions (P_D). As a supplementary problem, discrete approximation problem (P_DA) is considered. Necessary and sufficient conditions, including distinctive transversality, are proved by incorporating the Euler-Lagrange and Hamiltonian type of inclusions. The basic concept of obtaining optimal conditions is the locally adjoint mappings (LAM) and equivalence theorems, one of the most characteristic features of such approaches with the second-order differential inclusions that are peculiar to the presence of equivalence relations of LAMs. Furthermore, the application of these results are demonstrated by solving some non-convex problem with second-order discrete inclusions.