Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves - Inria - Institut national de recherche en sciences et technologies du numérique
Preprints, Working Papers, ... (Preprint) Year : 2024

Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves

Abstract

Fredholm-type backstepping transformation, introduced by Coron and Lü, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators of the form |Dx| α for α ∈ (1, 3/2]. We present here a new compactness/duality method which hinges on Fredholm's alternative to overcome the α = 3/2 threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operator verifying α > 1, a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for α > 3/2. The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water wave equation exhibiting an operator of critical order α = 3/2.
Fichier principal
Vignette du fichier
Linear_water_waves_stabilization_2.pdf (747.14 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03892656 , version 1 (09-12-2022)
hal-03892656 , version 2 (12-01-2024)

Identifiers

  • HAL Id : hal-03892656 , version 2

Cite

Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, Christophe Zhang. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves. 2024. ⟨hal-03892656v2⟩
149 View
98 Download

Share

More