Pré-Publication, Document De Travail Année : 2025

Small-time local controllability of a KdV system for all critical lengths

Jingrui Niu
Shengquan Xiang
  • Fonction : Auteur
  • PersonId : 1211873

Résumé

In this paper, we consider the small-time local controllability problem for the KdV system on an interval with a Neumann boundary control. In 1997, Rosier discovered that the linearized system is uncontrollable if and only if the length is critical, namely $L=2\pi\sqrt{(k^2+ kl+ l^2)/3}$ for some integers $k$ and $l$. Coron and Cr\'epeau (2003) proved that the nonlinear system is small-time locally controllable even if the linearized system is not, provided that $k= l$ is the only solution pair. Later, Cerpa and Crepeau showed that the system is large-time locally controllable for all critical lengths. In 2020, Coron, Koenig, and Nguyen found that the system is not small-time locally controllable if $2k+l\not \in 3\mathbb{N}^*$. We demonstrate that if the critical length satisfies $2k+l \in 3\mathbb{N}^*$ with $k\neq l$, then the system is not small-time locally controllable. This paper, together with the above results, gives a complete answer to the longstanding open problem on the small-time local controllability of KdV on all critical lengths since the pioneer work by Rosier
Fichier principal
Vignette du fichier
main.pdf (577.5 Ko) Télécharger le fichier
Vignette du fichier
mu.pdf (18.03 Ko) Télécharger le fichier
Format Figure, Image
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04908486 , version 1 (23-01-2025)

Licence

Copyright (Tous droits réservés)

Identifiants

  • HAL Id : hal-04908486 , version 1

Citer

Jingrui Niu, Shengquan Xiang. Small-time local controllability of a KdV system for all critical lengths. 2025. ⟨hal-04908486⟩
0 Consultations
0 Téléchargements

Partager

More