On the control issues for higher-order nonlinear dispersive equations on the circle

Roberto de A. Capistrano–Filho, Chulkwang Kwak, Francisco J. Vielma Leal

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The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of regularity, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space Hs(T) for any s≥0. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.

Original languageEnglish
Article number103695
JournalNonlinear Analysis: Real World Applications
StatePublished - Dec 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Ltd


  • Bourgain spaces
  • Control problems
  • KdV-type equation
  • Propagation of regularity/compactness


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