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

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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.

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Acknowledgements

The authors are grateful for the support of their former host institutions. To this end, I. Neitzel acknowledges the support of the Technische Universität München and F. Ludovici and W. Wollner the support of the Universität Hamburg.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Francesco Ludovici & Winnifried Wollner

  2. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Ira Neitzel

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  1. Francesco Ludovici
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  2. Ira Neitzel
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  3. Winnifried Wollner
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Correspondence to Winnifried Wollner.

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Ludovici, F., Neitzel, I. & Wollner, W. A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems. J Optim Theory Appl 178, 317–348 (2018). https://doi.org/10.1007/s10957-018-1311-8

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  • DOI: https://doi.org/10.1007/s10957-018-1311-8

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