Asymptotic behaviour of a linearized water waves system in a rectangle

Abstract

We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a scalar input function u. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ∂ζ ∂t . We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a special change of variables and a scattering semigroup, which provide the possiblity to apply the Trotter-Kato approximation theorem. Moreover, we use a detailed analysis of Fourier series for the dimensionless version of the partial Dirichlet to Neumann and Neumann to Neumann operators introduced in [1].

Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves

“…We may refer to the asymptotic stabilization and exponential stabilisation results of water waves systems [1,2] that are based on external "damping" forces and "observability" of the closed-loop systems. Alternatively, stabilizability properties of linearized water waves systems with controls acting on the boundary have been recently studied in [32,33]. Our contribution, as a direct consequence of the Fredholm backstepping transformation we obtain a rapid stabilisation result, that is exponential stability with arbitrarily decay rate.…”

Shallow water waves generated by a floating object: a control theoretical perspective

“…This could lead, in particular, to a description of the reachable space for nonlinear systems in which the fluid is modeled by the nonlinear shallow water equations. Finally, let us mention that an interesting question could be to consider the corresponding boundary control problems, in the spirit of [33] (or a short version [34]), [31] and [32].…”