Regular Article

Regularization and discretization error estimates for optimal control of ODEs with group sparsity

¹ Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Professur Mathematische Optimierung, 07743 Jena, Germany
christopher.schneider@uni-jena.de
² Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), 09107 Chemnitz, Germany

^a Corresponding author: gerd.wachsmuth@mathematik.tu-chemnitz.de, http://www.tu-chemnitz.de/mathematik/part_dgl/people/wachsmuth, http://orcid.org/0000-0002-3098-1503

Received: 15 July 2016
Revised: 21 February 2017
Accepted: 29 June 2017

Abstract

It is well known that optimal control problems with L¹-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L²-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method.