# PI controllers for the general Saint-Venant equations

arXiv: https://arxiv.org/abs/2108.02703 · HTML: https://mlsanity.com/paper/2108.02703/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2021.

## Built on by (most cited first, 1 of 1)

- [Backstepping-Guided Reinforcement Learning for Wide-Range Saint-Venant Canal Regulation](https://mlsanity.com/paper/2608.20089/) (2026)

## Abstract (arXiv)

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 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.