Optime L2-Control Problem In Coefficients For A Linea Elliptic Equation. II. Approximation Of Solutions And Optimality Conditions

TL;DR

This paper investigates a Dirichlet optimal control problem for a linear elliptic equation with matrix coefficients in L2, exploring solution approximation and optimality conditions, especially addressing non-uniqueness and singular solutions.

Contribution

It introduces a novel approach to approximate solutions of the control problem using perforated domains with boundary controls, addressing non-uniqueness and singularity issues.

Findings

Optimal control problem is well-posed despite non-uniqueness.

Some optimal solutions can be approximated via perforated domain problems.

The skew-symmetric part of the control matrix belongs to L2, allowing for broader control classes.

Abstract

In this paper we study we study a Dirichlet optimal control prob- lem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control A(x) belongs to L2-space (rather than Linfinty). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissi- ble controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.