Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
An optimal control problem for a semilinear heat equation with distributed control is discussed, where two-sided pointwise box constraints on the control and two-sided pointwise mixed control-state constraints are given. The objective functional is the sum of a standard quadratic tracking type part and a multiple of the L1-norm of the control that accounts for sparsity. Under a certain structural condition on almost active sets of the optimal solution, the existence of integrable Lagrange multipliers is proved for all inequality constraints. For this purpose, a theorem by Yosida and Hewitt is used. It is shown that the structural condition is fulfilled for all sufficiently large sparsity parameters. The sparsity of the optimal control is investigated. Eventually, higher smoothness of Lagrange multipliers is shown up to Hölder regularity.
Keywords: Semilinear heat equation, optimal control, Sparse control, mixed control-state constraints, regular Lagrange multipliers
@article{COCV_2021__27_1_A4_0,
author = {Casas, Eduardo and Tr\"oltzsch, Fredi},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Sparse optimal control for a semilinear heat equation with mixed control-state constraints {\textendash} regularity of {Lagrange} multipliers},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020084},
mrnumber = {4201972},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020084/}
}
TY - JOUR AU - Casas, Eduardo AU - Tröltzsch, Fredi ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020084/ DO - 10.1051/cocv/2020084 LA - en ID - COCV_2021__27_1_A4_0 ER -
%0 Journal Article %A Casas, Eduardo %A Tröltzsch, Fredi %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020084/ %R 10.1051/cocv/2020084 %G en %F COCV_2021__27_1_A4_0
Casas, Eduardo; Tröltzsch, Fredi. Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 2. doi: 10.1051/cocv/2020084
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Cité par Sources :
The first author was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under research project MTM2017-83185-P.
Dedicated to Prof. Dr. Enrique Zuazua on the occasion of his 60th birthday.
