  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.
