We kindly invite you to the talk "Control problem of a water waves system in a tank" by Pei, Su, Apr. 6th, 3:00pm Vienna Time.

Contact to get an invite link.

We first study the stability of a class of skew-adjoint control systems. With the assumptions on the spectrum of the evolution operators involved in the control system, we obtain an explicit non-uniform decay rate of the energy, provided that the initial data is smooth.
Then we consider a small-amplitude water waves system in a rectangular domain, where the control acts on one lateral boundary, by imposing the velocity of the water. The equation is a fully linear and fully dispersive version of the Zakharov-Craig-Sulem formulation. Based on the Dirichlet to Neumann and Neumann to Neumann maps, we establish the well-posedness of the whole system, which is addressed by formulating the equations as an abstract linear control system. Afterwards, there exists a feedback functional, such that the corresponding control system is strongly stable. We obtain the decay rate of the energy by using the above general results.
Finally, we consider the asymptotic behaviour of the above system in the shallow water regime, i.e. the horizontal scale of the domain is much larger than the typical water depth. 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 detailed analysis of the Fourier series and the di- mensionless version of the evolution operators mentioned above, as well as a scattering semigroup and the Trotter-Kato approximation theorem.
This is partially collaborated with M. Tucsnak (Bordeaux) and G. Weiss (Tel Aviv).