In this work, the least pointwise upper and/or lower bounds on the state variable on a specified subdomain of a control system under piecewise constant control action are sought. This results in a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosida regularization of the state constraints, the problem can be solved using a superlinearly convergent semi-smooth Newton method. Optimality conditions are derived, convergence of the Moreau-Yosida regularization is proved, and well-posedness and superlinear convergence of the Newton method is shown. Numerical examples illustrate the features of this problem and the proposed approach.
Mots-clés : optimal control, optimal L∞ state constraint, semi-smooth Newton method
@article{M2AN_2011__45_3_505_0,
author = {Clason, Christian and Ito, Kazufumi and Kunisch, Karl},
title = {Minimal invasion: {An} optimal $L^\infty $ state constraint problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {505--522},
publisher = {EDP-Sciences},
volume = {45},
number = {3},
year = {2011},
doi = {10.1051/m2an/2010064},
zbl = {1269.65060},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2010064/}
}
TY - JOUR AU - Clason, Christian AU - Ito, Kazufumi AU - Kunisch, Karl TI - Minimal invasion: An optimal $L^\infty $ state constraint problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 505 EP - 522 VL - 45 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2010064/ DO - 10.1051/m2an/2010064 LA - en ID - M2AN_2011__45_3_505_0 ER -
%0 Journal Article %A Clason, Christian %A Ito, Kazufumi %A Kunisch, Karl %T Minimal invasion: An optimal $L^\infty $ state constraint problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 505-522 %V 45 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2010064/ %R 10.1051/m2an/2010064 %G en %F M2AN_2011__45_3_505_0
Clason, Christian; Ito, Kazufumi; Kunisch, Karl. Minimal invasion: An optimal $L^\infty $ state constraint problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 505-522. doi : 10.1051/m2an/2010064. http://archive.numdam.org/articles/10.1051/m2an/2010064/
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