Strong stationarity for optimal control of the obstacle problem with control constraints
G Wachsmuth - SIAM Journal on Optimization, 2014 - SIAM
SIAM Journal on Optimization, 2014•SIAM
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≤u_b quasi-everywhere in Ω
in the case of an upper obstacle y≤ψ), we can answer this question in the affirmative. We …
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≤u_b quasi-everywhere in Ω
in the case of an upper obstacle y≤ψ), we can answer this question in the affirmative. We …
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 quasi-everywhere in in the case of an upper obstacle ), we can answer this question in the affirmative. We also present counterexamples showing that strong stationarity may not hold if or are violated. (An erratum is attached.)
Society for Industrial and Applied Mathematics