Finite-time stabilization of hyperbolic systems over a bounded interval

We investigate the finite-time boundary stabilization of a 1-D first order quasilinear hyperbolic system of diagonal form on [0,1]. The dynamics of both boundary controls are governed by a finite-time stable ODE. The solutions of the closed-loop system issuing from small initial data in Lip([0,1]) are shown to exist for all times and to reach the null equilibrium state in finite time. A finite-time stabilization is also shown to occur when using only one boundary control. The above feedback strategy is then applied to the Saint-Venant system for the regulation of water flows in a canal. Some explicit numerical scheme is provided to numerically investigate the finite-time stabilization of the system with one (resp. two) boundary control(s).