OPTIMIZATION OF THE BOLZA PROBLEM WITH HIGHER-ORDER DIFFERENTIAL INCLUSIONS AND INITIAL POINT AND STATE CONSTRAINTS

This paper is devoted to the duality of the Bolza problem with higher order differential inclusions and constraints on the initial point and state, which can make a significant contribution to the theory of optimal control. To this end in the form of Euler-Lagrange type inclusions and transversality conditions, sufficient optimality conditions are derived. It is remarkable that in a particular case the Euler-Lagrange inclusion coincides with the classical Euler-Poisson equation of the Calculus of Variations. The main idea of obtaining optimal conditions is locally conjugate mappings. It turns out that inclusions of the Euler-Lagrange type for Both direct and dual problems are "duality relations". To implement this approach, sufficient optimality conditions and duality theorems are proved in the Mayer problem with a second-order linear optimal control problem and third-order polyhedral differential inclusions, reflecting the special features of the variational geometry of polyhedral sets.