Optimal control in first-order Sobolev spaces with inequality constraints
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In this paper, an elliptic optimal control problem with controls from \(H^1(\varOmega )\) which have to satisfy standard box constraints is considered. Thus, Lagrange multipliers associated with the box constraints are, in general, elements of \(H^1(\varOmega )^\star \) as long as the lower and upper bound belong to \(H^1(\varOmega )\) 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.
KeywordsControl constraints Optimal control Optimality conditions Penalty method Semismooth Newton method
Mathematics Subject Classification49K20 49M05 49M25 49M37
The authors sincerely thank two anonymous referees whose valuable comments were essential in order to enhance the paper’s overall quality. This work is partially supported by the DFG Grant Analysis and Solution Methods for Bilevel Optimal Control Problems within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).
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