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Optimal control in first-order Sobolev spaces with inequality constraints

Author

Listed:
  • Yu Deng

    (Technische Universität Bergakademie Freiberg)

  • Patrick Mehlitz

    (Brandenburgische Technische Universität Cottbus-Senftenberg)

  • Uwe Prüfert

    (Technische Universität Bergakademie Freiberg)

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.

Suggested Citation

  • Yu Deng & Patrick Mehlitz & Uwe Prüfert, 2019. "Optimal control in first-order Sobolev spaces with inequality constraints," Computational Optimization and Applications, Springer, vol. 72(3), pages 797-826, April.
    • Handle: RePEc:spr:coopap:v:72:y:2019:i:3:d:10.1007_s10589-018-0053-8
    DOI: 10.1007/s10589-018-0053-8
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    Cited by:

    1. Lam Quoc Anh & Tran Quoc Duy & Le Dung Muu & Truong Van Tri, 2021. "The Tikhonov regularization for vector equilibrium problems," Computational Optimization and Applications, Springer, vol. 78(3), pages 769-792, April.

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