Volume 27, 2021
|Number of page(s)||28|
|Published online||21 October 2021|
High-order homogenization in optimal control by the Bloch wave method*
Universität Duisburg-Essen, Fakultät für Mathematik,
** Corresponding author: firstname.lastname@example.org
Accepted: 2 September 2021
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 Aε. 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(εM), 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.
Mathematics Subject Classification: 35B27 / 35P05 / 49J20
Key words: Optimal control / periodic homogenization / Bloch analysis
© The authors. Published by EDP Sciences, SMAI 2021
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