SU, Pei; TUCSNAK, Marius; WEISS, George

doi:10.1016/j.sysconle.2020.104672

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SU, Pei
Institut de Mathématiques de Bordeaux [IMB]

TUCSNAK, Marius
Institut de Mathématiques de Bordeaux [IMB]

WEISS, George
Tel Aviv University [TAU]

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Article de revue

Ce document a été publié dans

Systems and Control Letters. 2020

Elsevier

Résumé en anglais

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).< Réduire

Mots clés en anglais

strong stabilization

state feedback

Neumann to Neumann map

Dirichlet to Neumann map

Linearized water waves equation

Hilbert's inequality

operator semigroup

collocated actuators and sensors

URI

https://oskar-bordeaux.fr/handle/20.500.12278/192345

DOI

http://dx.doi.org/10.1016/j.sysconle.2020.104672

Project ANR

Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure - ANR-18-CE40-0027

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251