High-order homogenization in optimal control by the Bloch wave method^*

Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Straße 9, 45127 Essen, Germany.

^** Corresponding author: irwin.yousept@uni-due.de

Received: 9 November 2020
Accepted: 2 September 2021

Abstract

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^ε. 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^M), 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.