  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.
