Stabilizability properties of a linearized water waves system

Pei Su (1) , Marius Tucsnak (1) , George Weiss (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 TAU - Tel Aviv University

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).

Mots clés: 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

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Dynamique des Fluides [physics.flu-dyn]

Pei Su, Marius Tucsnak, George Weiss. Stabilizability properties of a linearized water waves system. Systems and Control Letters, 2020, ⟨10.1016/j.sysconle.2020.104672⟩. ⟨hal-02458379v2⟩