Solvability of an Optimal Control Problem in Coefficients for Ill-Posed Elliptic Boundary Value Problems

We study an optimal control problem (OCP) associated to a linear elliptic equation −div (A(x)grad y + C(x)grad y) = f. The characteristic feature of this control object is the fact that the matrix C(x) is skew-symmetric and belongs to L^2-space (rather than L^infinity). We adopt a symmetric positive defined matrix A(x) as control in L^infinity(Ω;R_NxN). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, we prove that the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions provided a special choice of skew-symmetric L^2-matrix C. The main trick we apply to the proof of the existence result is the approximation of the original OCP by regularized OCPs in perforated domains with fictitious boundary controls on the holes. As a consequence, we show that some of optimal solutions to the original problem can be attainable by solutions of regularized OCPs.