PI CONTROLLERS FOR THE GENERAL SAINT-VENANT EQUATIONS

We study the exponential stability in the H² norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single proportional-integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain of the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the input-to-state stability (ISS) of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.