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Mathematics > Optimization and Control

arXiv:2108.02703 (math)
[Submitted on 5 Aug 2021]

Title:PI controllers for the general Saint-Venant equations

Authors:Amaury Hayat
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Abstract: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.
Comments: 32 pages
Subjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP)
MSC classes: 93D15, 35B35, 93D05, 93D09, 93D20, 93D25
Cite as: arXiv:2108.02703 [math.OC]
  (or arXiv:2108.02703v1 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2108.02703

Submission history

From: Amaury Hayat [view email]
[v1] Thu, 5 Aug 2021 16:12:41 UTC (499 KB)
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