Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques

Swann Marx (1) , Yacine Chitour (2) , Christophe Prieur (3)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 L2S - Laboratoire des signaux et systèmes
3 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité

In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymp-totically stable with a linear damping. To do so, we first characterize, in terms of Lyapunov functionals, several types of asymptotic stability for linear infinite-dimensional systems, namely the exponential and the polynomial stability. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term with compensates the nonlinearity. Our results are then applied to the linearized Korteweg-de Vries equation and some wave equations.

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

Swann Marx, Yacine Chitour, Christophe Prieur. Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques. 2018. ⟨hal-01857265v1⟩