A note on the control and stabilization of a higher-order water wave model

Abstract

<p style='text-indent:20px;'>The paper deals with the internal controllability and stabilizability of a family of Boussinesq systems introduced by J. L. Bona, M. Chen and J.-C. Saut to describe two-way propagation of small amplitude gravity waves on the surface of water in a canal. By applying the moment method, we first obtain the exact controllability of the linearized system in suitable Hilbert spaces, which implies its exponential stabilization by suitable feedback. Then, by using a contraction mapping principle, we establish the local exact controllability and exponential stabilization for the original nonlinear system.</p>