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On the solution stability of parabolic optimal control problems

Author

Listed:
  • Alberto Domínguez Corella

    (Vienna University of Technology)

  • Nicolai Jork

    (Vienna University of Technology)

  • Vladimir M. Veliov

    (Vienna University of Technology)

Abstract

The paper investigates stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may nonlinearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper–the metric subregularity of the mapping associated with the system of first order necessary optimality conditions. This property also enables error estimates for approximation methods. A Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.

Suggested Citation

  • Alberto Domínguez Corella & Nicolai Jork & Vladimir M. Veliov, 2023. "On the solution stability of parabolic optimal control problems," Computational Optimization and Applications, Springer, vol. 86(3), pages 1035-1079, December.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:3:d:10.1007_s10589-023-00473-4
    DOI: 10.1007/s10589-023-00473-4
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    References listed on IDEAS

    as
    1. Martin Seydenschwanz, 2015. "Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 61(3), pages 731-760, July.
    2. Nikolaus Daniels, 2018. "Tikhonov regularization of control-constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 70(1), pages 295-320, May.
    3. Alt, Walter & Schneider, Christopher & Seydenschwanz, Martin, 2016. "Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions," Applied Mathematics and Computation, Elsevier, vol. 287, pages 104-124.
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    Cited by:

    1. William W. Hager & R. Tyrrell Rockafellar & Vladimir M. Veliov, 2023. "Preface to Asen L. Dontchev Memorial Special Issue," Computational Optimization and Applications, Springer, vol. 86(3), pages 795-800, December.

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