Well-posedness and stability of a 1D wave equation with saturating distributed input

Christophe Prieur (1) , Sophie Tarbouriech (2) , Joâo Manoel Gomes da Silva Gomes da Silva (3)

1 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
3 UFRGS - Departamento de Engenharia Eletrica

In this paper, it is considered a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection. The slope has a finite length and is attached at both boundaries. It is equipped with a distributed actuator subject to a saturation. By closing the loop with a saturating input proportional to the speed of the deformation, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation. The well-posedness is proven by using nonlinear semigroups technics. The asymptotic stability of the closed-loop system, when the tuning parameter has a suitable sign, is proven by Lyapunov technics and a sector condition describing the saturating input.

Domaines: Automatique / Robotique

Christophe Prieur, Sophie Tarbouriech, Joâo Manoel Gomes da Silva Gomes da Silva. Well-posedness and stability of a 1D wave equation with saturating distributed input. CDC 2014 - 53rd IEEE Conference on Decision and Control, Dec 2014, Los Angeles, United States. pp. 2846-2851, ⟨10.1109/CDC.2014.7039826⟩. ⟨hal-01235418⟩