  The paper is devoted to the optimization of a first mixed boundary value
problem for parabolic differential inclusions (DFIs) with Laplace operator. For
this, a problem with a parabolic discrete inclusion is defined, which is the
main auxiliary problem. With the help of locally adjoint mappings, necessary
and sufficient conditions for the optimality of parabolic discrete inclusions
are proved. Then, using the method of discretization of parabolic DFIs and the
already obtained optimality conditions for discrete inclusions, the necessary
and sufficient conditions for the discrete-approximate problem are formulated
in the form of the Euler-Lagrange type inclusion. Thus, using specially proved
equivalence theorems, without which it would hardly be possible to obtain the
desired result for the problem posed, we establish sufficient optimality
conditions for a parabolic DFIs. To demonstrate the above approach, some linear
problems and polyhedral optimization with inclusions of parabolic type are
investigated.
