Optimal control of elliptic equations with positive measures
1 Faculty of Mathematics, University Duisburg-Essen, 45117
2 Institute of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany.
Revised: 28 July 2015
Accepted: 18 September 2015
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.
Mathematics Subject Classification: 49J20 / 49K20 / 49N15
Key words: Optimal control / measure control / control constraints / Fenchel duality / unbounded operators
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