Strong Stationarity for Optimal Control of the Obstacle Problem with Control Constraints
Gerd Wachsmuth · SIAM Journal on Optimization · 2014
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 $u_a < 0 \le u_b$ quasi-everywhere in $\Omega$ in the case of an upper obstacle $y \le \psi$), we can answer this question in the affirmative. We also present counterexamples showing that strong stationarity may not hold if $u_a < 0$ or $0 \le u_b$ are violated. (An erratum is attached.)