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

Author

Listed:
  • Eduardo Casas

    (Universidad de Cantabria)

  • Mariano Mateos

    (Universidad de Oviedo)

  • Arnd Rösch

    (Universtät Duisburg-Essen)

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.

Suggested Citation

  • Eduardo Casas & Mariano Mateos & Arnd Rösch, 2018. "Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity," Computational Optimization and Applications, Springer, vol. 70(1), pages 239-266, May.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-018-9979-0
    DOI: 10.1007/s10589-018-9979-0
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    References listed on IDEAS

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    1. Christian Clason & Karl Kunisch, 2012. "A measure space approach to optimal source placement," Computational Optimization and Applications, Springer, vol. 53(1), pages 155-171, September.
    2. Eduardo Casas & Fredi Tröltzsch, 2012. "A general theorem on error estimates with application to a quasilinear elliptic optimal control problem," Computational Optimization and Applications, Springer, vol. 53(1), pages 173-206, September.
    3. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
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