Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.