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

Eduardo Casas (), Mariano Mateos () and Arnd Rösch ()
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Eduardo Casas: Universidad de Cantabria
Mariano Mateos: Universidad de Oviedo
Arnd Rösch: Universtät Duisburg-Essen

Computational Optimization and Applications, 2018, vol. 70, issue 1, No 9, 239-266

Abstract: 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; Primary 49K20; 35K58; 65M15; Secondary 49J52 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10589-018-9979-0

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