  We propose and analyze a posteriori error estimators for an optimal control
problem that involves an elliptic partial differential equation as state
equation and a control variable that enters the state equation as a
coefficient; pointwise constraints on the control variable are considered as
well. We consider two different strategies to approximate optimal variables: a
fully discrete scheme in which the admissible control set is discretized with
piecewise constant functions and a semi-discrete scheme where the admissible
control set is not discretized; the latter scheme being based on the so-called
variational discretization approach. We design, for each solution technique, an
a posteriori error estimator and show, in two and three dimensional Lipschitz
polygonal/polyhedral domains (not necessarily convex), that the proposed error
estimator is reliable and efficient. We design, based on the devised
estimators, adaptive strategies that deliver optimal experimental rates of
convergence for the performed numerical examples.
