Optimization of Elliptic Type Differential Inclusions with Mixed Boundary Conditions

This paper first deals with the Neumann problem for discrete, discrete-approximate problems and elliptic differential inclusions. Then using the obtained results for discrete inclusions in the form of Euler-Lagrange inclusions necessary and sufficient conditions for optimality are derived for the problems under consideration. Further, these problems are generalized to mixed problems, where the Neumann and Dirichlet conditions are satisfied separately in different parts of the set-valued mapping domain. As an example, a linear optimal control problem is considered, from which the Weierstrass-Pontryagin maximum condition follows. Thus, relying only on the previous auxiliary results, we can obtain the main results of this paper for mixed problems. In turn, the Neumann problem is generalized to the multidimensional case with a second order elliptic operator.