OPTIMAL L2 -CONTROL PROBLEM IN COEFFICIENTS FOR A LINEAR ELLIPTIC EQUATION

Thierry Horsin (1) , Peter Kogut (2)

1 M2N - Modélisation mathématique et numérique
2 Department of Differential equations

In this paper we study an optimal control problem (OCP) asso- ciated to a linear elliptic equation on a bounded domain Ω. The matrix- valued coefficients A of such systems is our control in Ω and will be taken in L2 (Ω; RN ×N ) 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 L∞ -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.

Mots clés: Control in coefficients; non-variational solutions; variational conver- gence; fictitious control

Domaines: Optimisation et contrôle [math.OC]

Thierry Horsin, Peter Kogut. OPTIMAL L2 -CONTROL PROBLEM IN COEFFICIENTS FOR A LINEAR ELLIPTIC EQUATION. 2013. ⟨hal-00824943⟩