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
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 : exponential stability Lyapunov functional Keyword: Korteweg-de Vries non-linear equation time-delay
Document type :
Journal articles
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https://hal.laas.fr/hal-01643321
Contributor : Lucie Baudouin <>
Submitted on : Monday, November 27, 2017 - 1:50:59 PM
Last modification on : Monday, February 10, 2020 - 6:13:49 PM

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Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2019, 64 (4), pp.1403-1414. ⟨10.1109/TAC.2018.2849564⟩. ⟨hal-01643321⟩

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