Control of three dimensional water waves

We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained by Alazard, Baldi, and Han-Kwan for 2-D water waves. Our proof combines an iterative scheme, that reduces the controllability of the original quasi-linear equation to that of a sequence of linear equations, with a semiclassical approach for the linear control problems.