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 partialdifferential equations. It focuses on a Korteweg–de Vries equation, which is a nonlinear mathematicalmodel 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 thisequation and some properties of the saturation function. The proof of the asymptotic stability of theclosed-loop system is separated in two cases: (i) when the control acts on all the domain, a Lyapunovfunction together with a sector condition describing the saturating input is used to conclude on thestability; (ii) when the control is localized, we argue by contradiction. Some numerical simulationsillustrate the stability of the closed-loop nonlinear partial differential equation.

Mots clés: Korteweg-de Vries equation; nonlinear system; saturating control; distributed control; stabilization

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. SIAM Journal on Control and Optimization, 2017, 55 (3), pp.1452-1480. ⟨10.1137/16M1061837⟩. ⟨hal-01367622v3⟩