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

Abstract

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.

Comparison of optimality systems for the optimal control of the obstacle problem

“…This system is satisfied for all local minimizers under very weak assumptions on the data, cf. [10,Lemma 4.4]. Moreover, we mention that the above cone N weak K (ȳ,λ) provides an appropriate generalization of (13b).…”

“…This formulation for M-stationarity can, however, not be transferred to our problem (1). Therefore, we give an alternative description, see [10]: there is a disjoint decomposition of the biactive set I 00 =Î +0 ∪Î 00 ∪Î 0− , such that…”

“…Note that the above system slightly differs from the system of C-stationarity in [8], since therein, higher regularity of the data of (1) has been utilized. However, it has been shown in [10,Lemma 4.6], that the system (3) with N C K coincides with the system of C-stationarity in [8].…”

“…Next, we will state the definition of M-stationarity from [10] which parallels (17) in the finitedimensional case. Let B =Î ∪B ∪Â s be a disjoint decomposition of the biactive set and we defineK…”

“…The task of providing necessary optimality conditions, i.e., conditions which are satisfied for all local minimizers of (1), received great interest in the last forty years, we refer exemplarily to [1][2][3][4][5][6][7][8][9][10][11].…”