Towards M-stationarity for Optimal Control of the Obstacle Problem with Control Constraints

We consider an optimal control problem, whose state is given as the solution of the obstacle problem. The controls are not assumed to be dense in H−1(Ω). Hence, local minimizers may not be strongly stationary. By a non-smooth regularization technique similar to the virtual control regularization, we prove a system of C-stationarity using only minimal regularity requirements. We show that even a system of M-stationarity is satisfied under the assumption that the regularized adjoint states converge in capacity. We also give a counterexample, showing that this additional assumption might be crucial.