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June  2009, 24(2): 273-313. doi: 10.3934/dcds.2009.24.273

Control and stabilization of a family of Boussinesq systems

Sorin Micu 1, , Jaime H. Ortega 2, , Lionel Rosier 3, and  Bing-Yu Zhang 4,

1. 

Facultatea de Matematica si Informatica, Universitatea din Craiova, 200585, Romania
2. 

Depto. Ingeniería Matemática, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile, Blanco Encalada 2120, Casilla 170-3, Santiago, Chile
3. 

Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France
4. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025

Received  February 2008 Revised  October 2008 Published  March 2009

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.
Keywords: exact controllability, smoothing effect, Boussinesq system, stabilizability..
Mathematics Subject Classification: Primary: 93B05, 93D15; Secondary: 35Q5.
Citation: Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273
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