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