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

TL;DR

This paper develops a high-order homogenization method for linear-quadratic elliptic optimal control problems with highly oscillatory coefficients, using Bloch wave expansion to approximate solutions with controllably small errors.

Contribution

It introduces a novel high-order effective control problem with constant coefficients that accurately approximates the original problem using Bloch wave analysis.

Findings

Achieves an approximation error of O(ε^M) for the control problem

Uses Bloch wave expansion to analyze the optimal solution

Provides a systematic method for high-order homogenization

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 $A^\varepsilon$. The small parameter $\varepsilon>0$ 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 $O(\varepsilon^M)$, where $M\in\mathbb{N}$ 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.