Boundary feedback control of linear hyperbolic systems: Application to the Saint-Venant-Exner equations

Distributed parameter systems modeled by hyperbolic partial differential equations are considered in this paper. The dynamic models include a source term and heterodirectional velocities. A boundary control problem is introduced and it is first shown that it is well-posed (in the sense of Tucsnak and Weiss), under appropriate assumptions coupling the boundary conditions and the source term. Then a sufficient exponential stability condition is derived using operator theory. This condition is written in terms of Linear Matrix Inequalities that are numerically tractable and that allow an optimization program. Connections with another classical stability condition are given. This approach is applied to the Saint-Venant-Exner equation describing the dynamics of the water level, of the water flow and of the sediment inside of a channel. The effect of the friction and of the slope are taken into account in the application model.