Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints

Anton Schiela; Daniel Wachsmuth

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 3, page 771-787
  • ISSN: 0764-583X

Abstract

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In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained, which turn out to be sharp. These rates coincide with rates obtained by numerical experiments, which are included in the paper.

How to cite

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Schiela, Anton, and Wachsmuth, Daniel. "Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.3 (2013): 771-787. <http://eudml.org/doc/273350>.

@article{Schiela2013,
abstract = {In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained, which turn out to be sharp. These rates coincide with rates obtained by numerical experiments, which are included in the paper.},
author = {Schiela, Anton, Wachsmuth, Daniel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {variational inequalities; optimal control; control constraints; regularization; C-stationarity; path-following},
language = {eng},
number = {3},
pages = {771-787},
publisher = {EDP-Sciences},
title = {Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints},
url = {http://eudml.org/doc/273350},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Schiela, Anton
AU - Wachsmuth, Daniel
TI - Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 3
SP - 771
EP - 787
AB - In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained, which turn out to be sharp. These rates coincide with rates obtained by numerical experiments, which are included in the paper.
LA - eng
KW - variational inequalities; optimal control; control constraints; regularization; C-stationarity; path-following
UR - http://eudml.org/doc/273350
ER -

References

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