This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient A$$. The small parameter ε > 0 denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error O(ε$$), where M ∈ ℕ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities, leading to our main theoretical result on the high-oder approximation.
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Keywords: Optimal control, periodic homogenization, Bloch analysis
@article{COCV_2021__27_1_A102_0,
author = {Lamacz-Keymling, Agnes and Yousept, Irwin},
title = {High-order homogenization in optimal control by the {Bloch} wave method},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021088},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021088/}
}
TY - JOUR AU - Lamacz-Keymling, Agnes AU - Yousept, Irwin TI - High-order homogenization in optimal control by the Bloch wave method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021088/ DO - 10.1051/cocv/2021088 LA - en ID - COCV_2021__27_1_A102_0 ER -
%0 Journal Article %A Lamacz-Keymling, Agnes %A Yousept, Irwin %T High-order homogenization in optimal control by the Bloch wave method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021088/ %R 10.1051/cocv/2021088 %G en %F COCV_2021__27_1_A102_0
Lamacz-Keymling, Agnes; Yousept, Irwin. High-order homogenization in optimal control by the Bloch wave method. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 100. doi: 10.1051/cocv/2021088
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Cité par Sources :
The work of the second author was funded by German Research Foundation (DFG grants YO159/2-2 and YO159/4-1).
