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Journal of Optimization Theory and Applications

, Volume 178, Issue 2, pp 317–348 | Cite as

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

  • Francesco Ludovici
  • Ira Neitzel
  • Winnifried Wollner
Article
  • 128 Downloads

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.

Keywords

Optimal control Semilinear parabolic PDE State constraints Pointwise in time constraints Space-time a priori error estimates 

Mathematics Subject Classification

49M25 65M12 65M15 65M60 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institut für Numerische SimulationRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

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