Volume 25, 2019
|Number of page(s)||55|
|Published online||25 October 2019|
Control of a Boussinesq system of KdV–KdV type on a bounded interval
Departamento de Matemática, Universidade Federal de Pernambuco (UFPE),
2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária, Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil.
3 Centre Automatique et Systèmes (CAS) and Centre de Robotique (CAOR), MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272 Paris Cedex 06, France.
* Corresponding author: Lionel.Rosier@mines-paristech.fr
Accepted: 10 June 2018
We consider a Boussinesq system of KdV–KdV type introduced by J.L. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley–Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.
Mathematics Subject Classification: 35Q53 / 37K10 / 93B05 / 93D15
Key words: Boussinesq system / KdV–KdV system / exact controllability / stabilization
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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