State Feedback Stabilization of the Linearized Bilayer Saint-Venant Model

TL;DR

This paper introduces a novel Lyapunov-based control method to stabilize the linearized bilayer Saint-Venant model, a coupled PDE system, with numerical verification of the stabilization effectiveness.

Contribution

First application of a specific Lyapunov function to stabilize the bilayer Saint-Venant PDE model, providing a new approach for coupled hyperbolic PDE stabilization.

Findings

Successful stabilization of the linearized bilayer Saint-Venant model.

Numerical simulations confirm the theoretical stabilization results.

Abstract

We consider the problem of stabilizing the bilayer \textit{Saint-Venant} model, which is a coupled system of two rightward and two leftward convecting transport partial differential equations (PDEs). In the stability proofs, we employ a Lyapunov function in which the parameters need to be successively determined. To the best of the authors' knowledge, this is the first time this kind of Lyapunov function is employed, and this result is the first one on the stabilization of the linearized bilayer \textit{Saint-Venant} model. Numerical simulations of the bilayer \textit{Saint-Venant} problem are also provided to verify the result.