Global stabilization of a Korteweg-de Vries equation with a distributed control saturated in L 2 -norm

TL;DR

This paper investigates how a distributed control with L2-norm saturation affects the stability of the Korteweg-de Vries equation, demonstrating well-posedness and asymptotic stability through theoretical analysis and simulations.

Contribution

It introduces a novel approach to stabilize a nonlinear PDE with saturated distributed control, ensuring well-posedness and stability using fixed point and Lyapunov methods.

Findings

Well-posedness established via Banach fixed point theorem

Asymptotic stability proven with Lyapunov function and sector condition

Numerical simulations confirm theoretical stability results

Abstract

This article deals with the design of saturated controls in the context of partial differential equations. It is focused on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. The aim of this article is to study the influence of a saturating in L 2-norm distributed control on the well-posedness and the stability of this equation. The well-posedness is proven applying a Banach fixed point theorem. The proof of the asymptotic stability of the closed-loop system is tackled with a Lyapunov function together with a sector condition describing the saturating input. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation.