Strong stabilization of a linearized gravity-capillary water waves system in a tank

We consider the strong stabilization of a gravity-capillary water waves in a rectangular tank. The control acts on one lateral boundary, by producing the horizontal acceleration of the water, as a multiple of a scalar input function u, times a given function h of the height along the active boundary. We first establish the well-posedness of the whole water waves system by formulating this system as an abstract linear control system with the state z given in terms of the water level ζ. Then we show that for suitable functions h, there exists a functional F such that the closed-loop system, with the state feedback u = Fz, is strongly stable. Moreover, for the initial data in the domain of the semigroup generator, the energy of the control system decays like ${(1 + t)^{ - \frac{3}{2}}}$. Our approach is based on a detailed construction of the Dirichlet to Neumann and Neumann to Neumann operators associated with certain edges of the domain, as well as a general non-uniform stabilization result by Ammari and Tucsnak (2001).