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Article Dans Une Revue Journal de l'École polytechnique — Mathématiques Année : 2022

PI controller for the general Saint-Venant equations

Résumé

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.
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Dates et versions

hal-01827988 , version 1 (02-07-2018)
hal-01827988 , version 2 (11-10-2018)
hal-01827988 , version 3 (25-01-2019)
hal-01827988 , version 4 (05-08-2021)
hal-01827988 , version 5 (20-12-2023)

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Citer

Amaury Hayat. PI controller for the general Saint-Venant equations. Journal de l'École polytechnique — Mathématiques, 2022, ⟨10.5802/jep.210⟩. ⟨hal-01827988v5⟩
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