Boundary stabilization of star-shaped Saint-Venant networks with combined subcritical and supercritical channels

Abstract

In this work, we consider the boundary stabilization of a star-shaped water flow network composed by $n$ ($n\ge 3$) channels. Each channel is modeled by Saint-Venant equations with arbitrary friction and slope. Among which, two channels are in supercritical regime, while the remaining $n-2$ channels are in subcritical regime. We show that in this case, one only needs to apply a static feedback control at the inlet of a supercritical channel to achieve the exponential stability of the non-uniform steady-states in the $H^2$ norm. The main tool we employ is the Lyapunov approach. To validate our theoretical results, a numerical illustration is also given.