Optimal control of differential inclusions with endpoint and state constraints and duality

The paper studies optimal control problem described by higher order evolution differential inclusions (DFIs) with endpoint and state constraints. In the term of Euler-Lagrange type inclusion is derived sufficient condition of optimality for higher order DFIs. It is shown that the adjoint inclusion for the first order DFIs, defined in terms of locally adjoint mapping, coincides with the classical Euler-Lagrange inclusion. Then a duality theorem is proved, which shows that Euler-Lagrange inclusions are "duality relations" for both problems. At the end of the paper duality problems for third order linear and fourth order polyhedral DFIs are considered.