| Popis: |
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 z of the system is a vector with two components: one is the water level ζ along the top boundary and the other is its time derivative ζ. We prove that for suitable functions h, 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 + t) − 1/6. 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). |
