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Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

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Abstract

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005, Santander, Spain

    Eduardo Casas

  2. Departamento de Matemáticas, Campus de Gijón, Universidad de Oviedo, 33203, Gijón, Spain

    Mariano Mateos

  3. Fakultät für Mathematik, Universtät Duisburg-Essen, 45127, Essen, Germany

    Arnd Rösch

Authors
  1. Eduardo Casas
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  2. Mariano Mateos
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  3. Arnd Rösch
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Corresponding author

Correspondence to Mariano Mateos.

Additional information

Eduardo Casas and Mariano Mateos were partially supported by the Spanish Ministerio de Economía y Competitividad under Projects MTM2014-57531-P and MTM2017-83185-P.

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Casas, E., Mateos, M. & Rösch, A. Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity. Comput Optim Appl 70, 239–266 (2018). https://doi.org/10.1007/s10589-018-9979-0

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  • DOI: https://doi.org/10.1007/s10589-018-9979-0

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