Control and stabilization of a family of Boussinesq systems

Sorin Micu; Jaime H. Ortega; Lionel Rosier; Bing-Yu Zhang

doi:10.3934/dcds.2009.24.273

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2009, Volume 24, Issue 2: 273-313. Doi: 10.3934/dcds.2009.24.273

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Control and stabilization of a family of Boussinesq systems

1.
Facultatea de Matematica si Informatica, Universitatea din Craiova, 200585
2.
Depto. Ingeniería Matemática, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile, Blanco Encalada 2120, Casilla 170-3, Santiago
3.
Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex
4.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025

Received: January 31, 2008

Revised: September 30, 2008

Published: May 2009

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Abstract

This paper studies the internal controllability and stabilizability of a family of Boussinesq systems recently proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. The space of the controllable data for the associated linear system is determined for all values of the four parameters. As an application of this newly established exact controllability, some simple feedback controls are constructed such that the resulting closed-loop systems are exponentially stable. When the parameters are all different from zero, the local exact controllability and stabilizability of the nonlinear system are also established.

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Mathematics Subject Classification: Primary: 93B05, 93D15; Secondary: 35Q53.

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