PI controllers for the general Saint-Venant equations

We study the exponential stability in H 2 norm of nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single proportional-integral (PI) control at one end channel. Using good but simple Lyapunov function we find explicit condition on gain PI to ensure any steady-states. This is independent slope, coefficient, length river, inflow disturbance and, more surprisingly, can be made steady-state considered. When time-dependent no exist, still have input-to-state (ISS) system, show that changing slightly enables recover slowly varying trajectories.