Stabilizability properties of a linearized water waves system

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 ζ̇. 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)^−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).