Stabilizability properties of a linearized water waves system

TL;DR

This paper demonstrates the strong stabilization of a linearized water waves system in a rectangular domain using boundary control, achieving polynomial decay of the system's energy through a carefully designed feedback law.

Contribution

It introduces a novel boundary feedback stabilization method for water wave equations, with explicit decay rates and detailed operator analysis.

Findings

Existence of a bounded feedback law ensuring strong stability.

Polynomial decay rate of the solution norm as (1+t)^(-1/6).

Application of advanced operator theory to water wave stabilization.

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 $\zeta$ along the top boundary, and its time derivative $\dot\zeta$. 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).