  An optimal control problem subject to an elliptic obstacle problem is
studied. We obtain a numerical approximation of this problem by discretising
the PDE obtained via a Moreau--Yosida type penalisation. For the resulting
discrete control problem we provide a condition that allows to decide whether a
solution of the necessary first order conditions is a global minimum. In
addition we show that the corresponding result can be transferred to the limit
problem provided that the above condition holds uniformly in the penalisation
and discretisation parameters. Numerical examples with unique global solutions
are presented.
