Optimal control of differential inclusions with endpoint constraints and duality

The article considers a high-order optimal control problem and its dual problems described by high-order differential inclusions. In this regard, the established Euler–Lagrange type inclusion, containing the Euler–Poisson equation of the calculus of variations, is a sufficient optimality condition for a differential inclusion of a higher order. It is shown that the adjoint inclusion for the first-order differential inclusions, defined in terms of a locally adjoint mapping, coincides with the classical Euler–Lagrange inclusion. Then the duality theorems are proved.