Lp-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

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

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

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping with an L p functional framework, p ∈ [2, ∞]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results are proved by considering an appropriate functional of the energy in the desired functional spaces introduced by Haraux in [11]. Asymptotic behavior analysis is based on an attractivity result on a trajectory of an infinite-dimensional linear time-varying system with a special structure, which relies on the introduction of a suitable Lyapunov functional. Note that some of the results of this paper apply for a large class of nonmonotone dampings.

Domaines: Equations aux dérivées partielles [math.AP]

Yacine Chitour, Swann Marx, Christophe Prieur. Lp-asymptotic stability analysis of a 1D wave equation with a nonlinear damping. 2019. ⟨hal-02193922v1⟩