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A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems

Author

Listed:
  • Francesco Ludovici

    (Technische Universität Darmstadt)

  • Ira Neitzel

    (Rheinische Friedrich-Wilhelms-Universität Bonn)

  • Winnifried Wollner

    (Technische Universität Darmstadt)

Abstract

We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In order to control the feasibility violation induced by the discretization, error estimates for the semilinear partial differential equation are derived. Based upon these estimates, it can be shown that any local minimizer of the semilinear parabolic optimization problems satisfying a weak second-order sufficient condition can be approximated by the discretized problem. Rates for this convergence in terms of temporal and spatial discretization mesh sizes are provided. In contrast to other results in numerical analysis of optimization problems subject to semilinear parabolic equations, the analysis can work with a weak second-order condition, requiring growth of the Lagrangian in critical directions only. The analysis can then be conducted relying solely on the resulting quadratic growth condition of the continuous problem, without the need for similar assumptions on the discrete or time semidiscrete setting.

Suggested Citation

  • Francesco Ludovici & Ira Neitzel & Winnifried Wollner, 2018. "A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 317-348, August.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:2:d:10.1007_s10957-018-1311-8
    DOI: 10.1007/s10957-018-1311-8
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    References listed on IDEAS

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    1. M. Hinze & C. Meyer, 2012. "Stability of semilinear elliptic optimal control problems with pointwise state constraints," Computational Optimization and Applications, Springer, vol. 52(1), pages 87-114, May.
    2. Dmitriy Leykekhman & Dominik Meidner & Boris Vexler, 2013. "Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints," Computational Optimization and Applications, Springer, vol. 55(3), pages 769-802, July.
    3. Wei Gong & Michael Hinze, 2013. "Error estimates for parabolic optimal control problems with control and state constraints," Computational Optimization and Applications, Springer, vol. 56(1), pages 131-151, September.
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