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 an 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 et al. (J Eur Math Soc 20(3), 657–745, 2018 ) 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.

中文翻译：

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三维水波控制

我们通过施加到自由表面的局部外部压力研究具有表面张力的空间周期性水波的精确可控性。我们证明，在任何维度上，只要控制区域满足几何控制条件，对于足够小的和规则的数据，精确可控性在任意短的时间内保持不变。该结果先前由 Alazard 等人获得。（J Eur Math Soc 20(3), 657–745, 2018 ）用于二维水波。我们的证明结合了迭代方案，该方案将原始拟线性方程的可控性降低到线性方程序列的可控性，以及线性控制问题的半经典方法。