Control and stabilization of a family of Boussinesq systems

Abstract

This paper studies the internal controllability and stabilizabilityof a family of Boussinesq systems recently proposed by J. L. Bona,M. Chen and J.-C. Saut to describe the two-way propagation of smallamplitude gravity waves on the surface of water in a canal. Thespace of the controllable data for the associated linear system isdetermined for all values of the four parameters. As an applicationof this newly established exact controllability, some simplefeedback controls are constructed such that the resultingclosed-loop systems are exponentially stable. When the parametersare all different from zero, the local exact controllability andstabilizability of the nonlinear system are also established.