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The paper is devoted to the optimization of a first mixed boundary valueproblem for parabolic differential inclusions (DFIs) with Laplace operator. Forthis, a problem with a parabolic discrete inclusion is defined, which is themain auxiliary problem. With the help of locally adjoint mappings, necessaryand sufficient conditions for the optimality of parabolic discrete inclusionsare proved. Then, using the method of discretization of parabolic DFIs and thealready obtained optimality conditions for discrete inclusions, the necessaryand sufficient conditions for the discrete-approximate problem are formulatedin the form of the Euler-Lagrange type inclusion. Thus, using specially provedequivalence theorems, without which it would hardly be possible to obtain thedesired result for the problem posed, we establish sufficient optimalityconditions for a parabolic DFIs. To demonstrate the above approach, some linearproblems and polyhedral optimization with inclusions of parabolic type areinvestigated.

Optimization of the first mixed boundary value problem for parabolic differential inclusions in a three spatial dimension