Volume 24, Number 2, April–June 2018
|Page(s)||811 - 834|
|Published online||13 June 2018|
Regularization and discretization error estimates for optimal control of ODEs with group sparsity
Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Professur Mathematische Optimierung,
2 Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), 09107 Chemnitz, Germany
Revised: 21 February 2017
Accepted: 29 June 2017
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.
Mathematics Subject Classification: 49K15 / 49J15 / 49M15 / 49M25 / 65K15
Key words: Optimal control / group sparsity / directional sparsity / bang-bang principle / stability analysis / discretization error estimates
© EDP Sciences, SMAI 2018
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.