A Penalty Approach to Optimal Control of Allen-Cahn Variational Inequalities: MPEC-View

Abstract

A scalar Allen-Cahn-MPEC problem is considered and a penalization technique is applied to show the existence of an optimal control. We show that the stationary points of the penalized problems converge to some stationary points of the limit problem, which however are weaker than C-stationarity conditions.

Keywords:

• Allen-Cahn system
• Mathematical programs with complementarity constraints
• MPECs
• Optimality conditions
• Parabolic obstacle problems

Mathematics Subject Classification:

• 34G25
• 35K86
• 35R35
• 49J20
• 65K10

ACKNOWLEDGMENT

The author thanks the anonymous referee for a very careful reading of the manuscript. Moreover, the author wishes to thank Luise Blank and Harald Garcke for many fruitful hints and discussions. This work was supported by the SPP 1253 “Optimization with Partial Differential Equations” of the German Science Foundation (DFG) through the grant GA695/5-2.