  This paper on the whole concerns with the duality of Mayer problem for k-th
order differential inclusions, where k is an arbitrary natural number. Thus,
this work for constructing the dual problems to differential inclusions of any
order can make a great contribution to the modern development of optimal
control theory. To this end in the form of Euler-Lagrange type inclusions and
transversality conditions the sufficient optimality conditions are derived. The
principal idea of obtaining optimal conditions is locally adjoint mappings. It
appears that the Euler-Lagrange type inclusions for both primary and dual
problems are "duality relations". To demonstrate this approach, some semilinear
problems with k-th order differential inclusions are considered. Also, the
optimality conditions and the duality theorem in problems with second order
polyhedral differential inclusions are proved. These problems show that
maximization in the dual problems are realized over the set of solutions of the
Euler-Lagrange type differential inclusions/equations.
