PI controller for the general Saint-Venant equations

Amaury Hayat (1, 2, 3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We study the exponential stability in the $H^{2}$ 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 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 of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.

Mots clés: Hyperbolic Equations; Lyapunov Stability; Shallow Water Equations; Saint-Venant Equations; Exponential stability; Input to state stability; Inhomogeneous

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Automatique / Robotique

Amaury Hayat. PI controller for the general Saint-Venant equations. 2021. ⟨hal-01827988v4⟩