On second-order optimality conditions for optimal control problems governed by the obstacle problem. (3rd October 2021)

• Record Type: Journal Article
• Authors: Christof, Constantin; Wachsmuth, Gerd
• Is Part Of: Optimization. Volume 70:Number 10(2021)
• Page Start: 2247
• Page End: 2287
• Publication Date: 2021-10-03
• Subjects: Obstacle problem -- second-order optimality condition -- optimal control -- strong stationarity -- control constraints; 35J86 -- 49J40 -- 49K21; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2020.1778686
• Languages: English
• ISSNs: 0233-1934

Abstract

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.