On Second-Order Optimality Conditions for Optimal Control Problems Governed by the Obstacle Problem

TL;DR

This paper investigates second-order optimality conditions for Tikhonov regularized control problems governed by the obstacle problem, providing new conditions, reformulations, and counterexamples to deepen understanding of optimality in such problems.

Contribution

It introduces novel second-order conditions, links obstacle problems to state-constrained control problems, and presents counterexamples highlighting complexities in the analysis.

Findings

Optimal controls can be characterized on the active set.

Reformulation as state-constrained problems for the Poisson equation.

Existence of unique solutions for certain obstacle problems.

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.