Strong Stationarity for Optimal Control of the Obstacle Problem with Control Constraints

We consider the distributed optimal control of the obstacle problem with control constraints. Since Mignot proved in 1976 the necessity of a system which is equivalent to strong stationarity, it has been an open problem whether such a system is still necessary in the presence of control constraints. Using moderate regularity of the optimal control and an assumption on the control bounds (which is implied by ua < 0 ≤ ub quasi-everywhere (q.e.) in Ω in the case of an upper obstacle y ≤ ψ), we can answer this question in the afrmative. We also present counterexamples showing that strong stationarity may not hold if ua < 0 or 0 ≤ ub are violated.