Boundary stabilizability of the linearized viscous Saint-Venant system

We consider a shallow water flow in a channel modeled by the Saint-Venant equations with a viscous term. We are interested in the stabilization of the flow at a steady state. We establish that the semi-group of the linearized system is exponentially stable. However when the convection coefficient is dominant, the natural stabilization turns out to be very slow. One way to enhance the stabilization of the system is to use boundary controls by means of a moving device located at the extremities of the channel. We determine, by an extension method due to Fursikov, boundary Dirichlet controls able to accelerate the stabilization of the flow. Numerical experiments illustrate the efficiency of the control.