Optimal control of elliptic equations with positive measures

¹ Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany.
christian.clason@uni-due.de
² Institute of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany.
anton.schiela@uni-bayreuth.de

Received: 9 March 2015
Revised: 28 July 2015
Accepted: 18 September 2015

Abstract

Optimal control problems without control costs in general do not possess solutions due to the lack of coercivity. However, unilateral constraints together with the assumption of existence of strictly positive solutions of a pre-adjoint state equation, are sufficient to obtain existence of optimal solutions in the space of Radon measures. Optimality conditions for these generalized minimizers can be obtained using Fenchel duality, which requires a non-standard perturbation approach if the control-to-observation mapping is not continuous (e.g., for Neumann boundary control in three dimensions). Combining a conforming discretization of the measure space with a semismooth Newton method allows the numerical solution of the optimal control problem.