It is well known that optimal control problems with L^{1}-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 L^{2}-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.
Accepted:
DOI: 10.1051/cocv/2017049
Keywords: Optimal control, group sparsity, directional sparsity, bang-bang principle, stability analysis, discretization error estimates
@article{COCV_2018__24_2_811_0, author = {Schneider, Christopher and Wachsmuth, Gerd}, title = {Regularization and discretization error estimates for optimal control of {ODEs} with group sparsity}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {811--834}, publisher = {EDP-Sciences}, volume = {24}, number = {2}, year = {2018}, doi = {10.1051/cocv/2017049}, zbl = {1402.49019}, mrnumber = {3816416}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017049/} }
TY - JOUR AU - Schneider, Christopher AU - Wachsmuth, Gerd TI - Regularization and discretization error estimates for optimal control of ODEs with group sparsity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 811 EP - 834 VL - 24 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017049/ DO - 10.1051/cocv/2017049 LA - en ID - COCV_2018__24_2_811_0 ER -
%0 Journal Article %A Schneider, Christopher %A Wachsmuth, Gerd %T Regularization and discretization error estimates for optimal control of ODEs with group sparsity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 811-834 %V 24 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017049/ %R 10.1051/cocv/2017049 %G en %F COCV_2018__24_2_811_0
Schneider, Christopher; Wachsmuth, Gerd. Regularization and discretization error estimates for optimal control of ODEs with group sparsity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 2, pp. 811-834. doi : 10.1051/cocv/2017049. http://archive.numdam.org/articles/10.1051/cocv/2017049/
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