Convergence and regularization results for optimal control problems with sparsity functional

¹ Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany.
² Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrae 69, 4040 Linz, Austria. daniel.wachsmuth@ricam.oeaw.ac.at

Received: 27 August 2009
Revised: 9 February 2010
Revised: 18 May 2010

Abstract

Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L¹-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.