  In this paper we consider an optimal control problem (OCP) for the coupled
system of a nonlinear monotone Dirichlet problem with matrix- valued non-smooth
controls in coefficients and a nonlinear equation of Ham- merstein type. Since
problems of this type have no solutions in general, we make a special
assumption on the coefficients of the state equation and in- troduce the class
of so-called solenoidal admissible controls. Using the direct method in
calculus of variations, we prove the existence of an optimal control. We also
study the stability of the optimal control problem with respect to the domain
perturbation. In particular, we derive the sufficient conditions of the
Mosco-stability for the given class of OCPs.
