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

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 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.

Mots clés: Partial differential equation; saturation control; nonlinear control system; Lyapunov function

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

Swann Marx, Eduardo Cerpa, Christophe Prieur, Vincent Andrieu. Global stabilization of a Korteweg-de Vries equation with a distributed control saturated in L 2 -norm. NOLCOS 2016 - 10th IFAC Symposium on Nonlinear Control Systems, Aug 2016, Monterey, CA, United States. ⟨10.1016/j.ifacol.2016.10.150⟩. ⟨hal-01360576v2⟩