  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.
