Boundary feedback stabilization of hydraulic jumps

Georges Bastin (1) , Jean-Michel Coron (2, 3) , Amaury Hayat (2, 3) , Peipei Shang (4)

1 CSAM - Centre for Systems Engineering and Applied Mechanics
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 CaGE - Control And GEometry
4 Tongji University

In an open channel, a hydraulic jump is an abrupt transition between a torrential (super-critical) flow and a fluvial (subcritical) flow. In this article hydraulic jumps are represented by discontinuous shock solutions of hyperbolic Saint-Venant equations. Using a Lyapunov approach, we prove that we can stabilize the state of the system in $H^2$-norm as well as the hydraulic jump location, with simple feedback boundary controls and an arbitrary decay rate, by appropriately choosing the gains of the feedback boundary controls.

Mots clés: Saint-Venant equations; Boundary feedback controls; Hydraulic jump; Lyapunov Functions; Nonlinear EDP

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Georges Bastin, Jean-Michel Coron, Amaury Hayat, Peipei Shang. Boundary feedback stabilization of hydraulic jumps. IFAC Journal of Systems and Control, 2019. ⟨hal-02004457⟩