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.

Keywords

Optimal control Parabolic equations Directionally sparse solutions Finite element approximation Numerical quadrature Error estimates 

Mathematics Subject Classification

Primary 49K20 35K58 65M15 Secondary 49J52 

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

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain
  2. 2.Departamento de Matemáticas, Campus de GijónUniversidad de OviedoGijónSpain
  3. 3.Fakultät für MathematikUniverstät Duisburg-EssenEssenGermany

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