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This article examines a linear-quadratic elliptic optimal control problem inwhich the cost functional and the state equation involve a highly oscillatoryperiodic coefficient $A^\varepsilon$. The small parameter $\varepsilon>0$denotes the periodicity length. We propose a high-order effective controlproblem with constant coefficients that provides an approximation of theoriginal one with error $O(\varepsilon^M)$, where $M\in\mathbb{N}$ is as largeas one likes. Our analysis relies on a Bloch wave expansion of the optimalsolution and is performed in two steps. In the first step, we expand the lowestBloch eigenvalue in a Taylor series to obtain a high-order effective optimalcontrol problem. In the second step, the original and the effective problem arerewritten in terms of the Bloch and the Fourier transform, respectively. Thisallows for a direct comparison of the optimal control problems via thecorresponding variational inequalities.

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