PI controllers for the general Saint-Venant equations
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DOI10.5802/JEP.210zbMATH Open1501.93121arXiv2108.02703OpenAlexW4301594904MaRDI QIDQ2087374
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Publication date: 27 October 2022
Published in: (Search for Journal in Brave)
Abstract: We study the exponential stability in the 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.
Full work available at URL: https://arxiv.org/abs/2108.02703
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