Methods and Applications of Analysis

Volume 26 (2019)

Number 2

Special Issue in Honor of Roland Glowinski (Part 1 of 2)

Guest Editors: Xiaoping Wang (Hong Kong University of Science and Technology) and Xiaoming Yuan (The University of Hong Kong)

Exponential stability of PI control for Saint-Venant equations with a friction term

Pages: 101 – 112



Georges Bastin (Department of Mathematical Engineering, Université catholique de Louvain, Louvain-La-Neuve, Belgium)

Jean-Michel Coron (Laboratoire Jacques-Louis Lions, Université Paris-Diderot, Sorbonne Université, Paris,


We consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In this article, our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates.


hyperbolic systems, boundary control, Lyapunov stability, Saint-Venant equations

2010 Mathematics Subject Classification

35F61, 93D15, 93D30

Dedicated to Roland Glowinski, a master and a friend, on the occasion of his 80th birthday

Received 1 March 2018

Accepted 12 April 2019

Published 2 April 2020