Optimal control in first-order Sobolev spaces with inequality constraints

Yu Deng (), Patrick Mehlitz () and Uwe Prüfert ()
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Yu Deng: Technische Universität Bergakademie Freiberg
Patrick Mehlitz: Brandenburgische Technische Universität Cottbus-Senftenberg
Uwe Prüfert: Technische Universität Bergakademie Freiberg

Computational Optimization and Applications, 2019, vol. 72, issue 3, No 10, 797-826

Abstract: Abstract In this paper, an elliptic optimal control problem with controls from $$H^1(\varOmega )$$ H 1 ( Ω ) which have to satisfy standard box constraints is considered. Thus, Lagrange multipliers associated with the box constraints are, in general, elements of $$H^1(\varOmega )^\star $$ H 1 ( Ω ) ⋆ as long as the lower and upper bound belong to $$H^1(\varOmega )$$ H 1 ( Ω ) as well. If these bounds possess less regularity, the overall existence of a Lagrange multiplier is not even guaranteed. In order to avoid the direct solution of a not necessarily available KKT system, a penalty method is suggested which finds the minimizer of the control-constrained problem. Its convergence properties are analyzed. Furthermore, some numerical strategies for the computation of optimal solutions are suggested and illustrated.

Keywords: Control constraints; Optimal control; Optimality conditions; Penalty method; Semismooth Newton method; 49K20; 49M05; 49M25; 49M37 (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10589-018-0053-8

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