On second-order optimality conditions for optimal control problems governed by the obstacle problem

Abstract

This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature and also covers those situations where the problem at hand involves additional box-constraints on the control. As a byproduct, our approach shows in particular that Tikhonov regularized optimal control problems for the obstacle problem can be reformulated as state-constrained optimal control problems for the Poisson equation and that problems involving a subharmonic obstacle and a convex objective function are uniquely solvable. The paper concludes with three counterexamples which illustrate that rather peculiar effects can occur in the analysis of second-order optimality conditions for optimal control problems governed by the obstacle problem and that necessary second-order conditions for such problems may be hard to derive.

Keywords:

• Obstacle problem
• second-order optimality condition
• optimal control
• strong stationarity
• control constraints

AMS CLASSIFICATIONS:

• 35J86
• 49J40
• 49K21

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the German Research Foundation (DFG) under grant number WA 3636/4-1 within the priority program ‘Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization’ (SPP 1962). The first author gratefully acknowledges the support by the International Research Training Group Munich-Graz, IGDK 1754, funded by the German Research Foundation (DFG) and the Austrian Science Fund (FWF) under Project Number 188264188/GRK1754.