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

TL;DR

This paper develops a method for optimizing a mixed boundary value problem for parabolic differential inclusions in three dimensions, using discretization, adjoint mappings, and equivalence theorems to establish optimality conditions.

Contribution

It introduces a novel approach combining discretization and adjoint mappings to derive optimality conditions for parabolic differential inclusions with mixed boundary conditions.

Findings

Established necessary and sufficient optimality conditions for discrete parabolic inclusions

Proved equivalence theorems linking discrete and continuous problems

Demonstrated the approach on linear and polyhedral optimization problems

Abstract

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.