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Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with a Moving Point Control

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Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

Abstract

In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements in space. For this discretization we show optimal order of convergence with respect to the time and the space discretization parameters modulo some logarithmic terms. Error analysis for the same problem was carried out in the recent paper (Gong and Yan, SIAM J Numer Anal 54:1229–1262, 2016), however, the analysis there contains a serious flaw. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, that have an independent interest.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Dmitriy Leykekhman

  2. Chair of Optimal Control, Center for Mathematical Sciences, Technical University of Munich, Boltzmannstraße 3, 85748, Garching by Munich, Germany

    Boris Vexler

Authors
  1. Dmitriy Leykekhman
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  2. Boris Vexler
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Corresponding author

Correspondence to Boris Vexler .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

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Leykekhman, D., Vexler, B. (2017). Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with a Moving Point Control. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_13

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