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

¹ Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustraße 7, 14195 Berlin-Dahlem, Germany
schiela@zib.de; http://www.zib.de/schiela
² 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

Received: 20 June 2011
Revised: 11 September 2012

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.