In this talk we examine a linear-quadratic optimal control problem in which the cost functional and the elliptic 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 and prove a corrector result which allows to approximate the original optimal solution 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 solutions via the corresponding variational inequalities.