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

Published online by Cambridge University Press:  04 March 2013

Anton Schiela  and
Daniel Wachsmuth
Anton Schiela
Affiliation:
Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustraße 7, 14195 Berlin-Dahlem, Germany. schiela@zib.de; http://www.zib.de/schiela
Daniel Wachsmuth
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A–4040 Linz, Austria; daniel.wachsmuth@oeaw.ac.at; http://www.ricam.oeaw.ac.at/people/page/wachsmuth
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Abstract

In the article an optimal control problem subject to a stationary variational inequalityis investigated. The optimal control problem is complemented with pointwise controlconstraints. The convergence of a smoothing scheme is analyzed. There, the variationalinequality is replaced by a semilinear elliptic equation. It is shown that solutions ofthe regularized optimal control problem converge to solutions of the original one. Passingto the limit in the optimality system of the regularized problem allows to proveC-stationarity of local solutions of the original problem. Moreover, convergence rateswith 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 numericalexperiments, which are included in the paper.

Keywords

Variational inequalitiesoptimal controlcontrol constraintsregularizationC-stationaritypath-following
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 771 - 787
Copyright
© EDP Sciences, SMAI, 2013

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