Strong stabilization of small water waves in a pool

This paper is about the strong stabilization of small amplitude gravity water waves in a vertical rectangle. The control imposes the horizontal acceleration of the water along one vertical boundary segment, as a multiple of a scalar input function u, times a function h of the position along the active boundary. The state 2: of the system is a vector with two components: one is the water level £ along the top boundary, and the other is its time derivative We prove that for suitable functions ht there exists a bounded feedback functional F such that the feedback u = Fz leads to a strongly stable closed-loop system. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1 4-Our approach uses careful estimates on certain partial Dirichlet to Neumann and Neumann to Neumann operators associated to the rectangular domain, as well as non-uniform stabilization results due to Chill, Paunonen, Seifert, Stahn and Tomilov (2019).