Stabilization of a linear Korteweg-de Vries equation with a saturated internal control

TL;DR

This paper designs saturated control strategies for a linear Korteweg-de Vries PDE, proving well-posedness and asymptotic stability of the nonlinear closed-loop system using nonlinear semigroup theory and Lyapunov functions.

Contribution

It introduces a method for stabilizing a Korteweg-de Vries equation with saturated internal control, ensuring well-posedness and stability.

Findings

Well-posedness established via nonlinear semigroup theory.

Asymptotic stability proven using Lyapunov functions.

Control saturation leads to nonlinear system stabilization.

Abstract

This article deals with the design of saturated controls in the context of partial differential equations. It is focused on a linear Korteweg-de Vries equation, which is a mathematical model of waves on shallow water surfaces. In this article, we close the loop with a saturating input that renders the equation nonlinear. The well-posedness is proven thanks to the nonlinear semigroup theory. The proof of the asymptotic stability of the closed-loop system uses a Lyapunov function.