In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.
Keywords: optimal control problems, semilinear equations, finite element approximations, $$ error estimates
@article{M2AN_2021__55_5_2293_0,
author = {Allendes, Alejandro and Fuica, Francisco and Ot\'arola, Enrique and Quero, Daniel},
title = {\protect\emph{A posteriori} error estimates for semilinear optimal control problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2293--2322},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021033},
mrnumber = {4328495},
zbl = {1485.35200},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021033/}
}
TY - JOUR AU - Allendes, Alejandro AU - Fuica, Francisco AU - Otárola, Enrique AU - Quero, Daniel TI - A posteriori error estimates for semilinear optimal control problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2293 EP - 2322 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021033/ DO - 10.1051/m2an/2021033 LA - en ID - M2AN_2021__55_5_2293_0 ER -
%0 Journal Article %A Allendes, Alejandro %A Fuica, Francisco %A Otárola, Enrique %A Quero, Daniel %T A posteriori error estimates for semilinear optimal control problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2293-2322 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021033/ %R 10.1051/m2an/2021033 %G en %F M2AN_2021__55_5_2293_0
Allendes, Alejandro; Fuica, Francisco; Otárola, Enrique; Quero, Daniel. A posteriori error estimates for semilinear optimal control problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2293-2322. doi: 10.1051/m2an/2021033
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