Global stabilization of a Korteweg-de Vries equation with saturating distributed control

Swann Marx (1) , Eduardo Cerpa (2) , Christophe Prieur (1) , Vincent Andrieu (3)

1 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité
2 UTFSM - Universidad Tecnica Federico Santa Maria [Valparaiso]
3 LAGEP - Laboratoire d'automatique et de génie des procédés

This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability; ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation.

Mots clés: Korteweg-de Vries equation; stabilization; distributed control; saturating control; nonlinear system AMS subject classifications 93C20; 93D15; 35Q53

Domaines: Automatique / Robotique; Equations aux dérivées partielles [math.AP]

Swann Marx, Eduardo Cerpa, Christophe Prieur, Vincent Andrieu. Global stabilization of a Korteweg-de Vries equation with saturating distributed control. 2016. ⟨hal-01367622v2⟩