Abstract We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function u , times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ζ . We prove that for suitable functions h , there exists a bounded feedback functional F such that the feedback u = F z renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like ( 1 + t ) − 1 6 . Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill et al. (2019).

中文翻译：

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线性化水波系统的稳定性特性

摘要 我们考虑二维矩形域中小振幅重力水波的强稳定性。控制作用于一个横向边界，通过沿该边界施加水的水平加速度，作为标量输入函数 u 乘以沿活动边界高度的给定函数 h 的倍数。系统的状态 z 由两个函数组成：沿顶部边界的水位 ζ 及其时间导数 ζ 。我们证明对于合适的函数 h ，存在一个有界反馈函数 F 使得反馈 u = F z 使闭环系统非常稳定。此外，对于半群生成器域中的初始状态，解的范数像 ( 1 + t ) − 1 6 一样衰减。我们的方法详细分析了与矩形域某些边缘相关的部分 Dirichlet 到 Neumann 和 Neumann 到 Neumann 算子，以及 Chill 等人最近的抽象非均匀稳定结果。(2019)。