论文标题
线性水波系统的稳定性特性
Stabilizability properties of a linearized water waves system
论文作者
论文摘要
我们认为在二维矩形结构域中，小振幅重力水波的强稳定。该控制作用在一个侧向边界上，通过沿该边界施加水的水平加速度，作为标量输入函数的倍数$ u $，times沿活动边界的高度的给定函数$ h $。系统的状态$ z $由两个功能组成：顶部边界的水位$ζ$以及其时间派生$ \dotζ$。我们证明，对于合适的功能$ h $，存在有界的反馈功能$ f $，因此反馈$ u = fz $使闭环系统非常稳定。此外，对于Semigroup Generator域中的初始状态，解决方案的标准如$（1+t）^{ - \ frac {1} {6}} $。我们的方法对诺伊曼（Neumann）和诺伊曼（Neumann）的部分迪里奇（Neumann）和诺伊曼（Neumann）运算符（Neumann Operators）进行了详细分析，以及与矩形结构域某些边缘相关的诺伊曼（Neumann）操作员，以及最近由Chill，Paunonen，Seifert，Stahn和Tomilov（2019）的近期抽象的非均匀稳定稳定结果。
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 $\dotζ$. We prove that for suitable functions $h$, there exists a bounded feedback functional $F$ such that the feedback $u=Fz$ 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 $(1+t)^{-\frac{1}{6}}$. 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, Paunonen, Seifert, Stahn and Tomilov (2019).
