Optimal L2-control problem in coefficients for a linear elliptic equation

TL;DR

This paper investigates an optimal control problem for a linear elliptic PDE with matrix coefficients in L2, addressing issues of non-uniqueness and characterizing variational and non-variational solutions.

Contribution

It establishes well-posedness of the control problem and distinguishes between variational and non-variational optimal solutions, including their attainability via boundary control.

Findings

The control problem admits at least one solution.

Optimal solutions may be singular and non-unique.

Some solutions are attainable through boundary control.

Abstract

In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation {on a bounded domain $\Omega$}. The matrix-valued coefficients A of such systems is our control taken in L2 which in particular may comprise som cases of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may face non-uniqueness of weak solutions--- namely, approximable solutions as well as another type of weak solutions that can not be obtained through the bouded-approximation of matrix A. Following the direct method in the calculus of variations, we show that the given OCP is well-posed in the sense that it admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this, we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can be attainable by solutions of special optimal boundary control problems.