Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback

Lucie Baudouin; Emmanuelle Crépeau; Julie Valein

doi:10.1109/TAC.2018.2849564

Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback

Lucie Baudouin ¹ Emmanuelle Crépeau ² Julie Valein ^{3, 4}

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]

2 LMV - Laboratoire de Mathématiques de Versailles

3 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization

Inria Nancy - Grand Est

4 IECL - Institut Élie Cartan de Lorraine

Abstract : This article concerns the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV equation) and an observability inequality approach, with a contradiction argument (for any non critical lengths but without estimation on the decay rate). Some numerical simulations are given to illustrate the results.

Keywords : time-delay Keyword: Korteweg-de Vries non-linear equation Lyapunov functional exponential stability

Type de document :

Article dans une revue

IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, In press, 〈10.1109/TAC.2018.2849564〉

Domaine :

Mathématiques [math] / Equations aux dérivées partielles [math.AP]

Sciences de l'ingénieur [physics] / Automatique / Robotique

Liste complète des métadonnées

https://hal.laas.fr/hal-01643321
Contributeur : Lucie Baudouin <>
Soumis le : lundi 27 novembre 2017 - 13:50:59
Dernière modification le : lundi 11 mars 2019 - 09:09:24

Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback

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Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback

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