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December 2015, 5(4): 743-760. doi: 10.3934/mcrf.2015.5.743
Exact controllability for the Lamé system
Belhassen Dehman 1, and Jean-Pierre Raymond 2,
Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092 El Manar, Tunisia
Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS, 31062 Toulouse Cedex
Received August 2014 Revised May 2015 Published October 2015
In this article, we prove an exact boundary controllability result for the isotropic elastic wave system in a bounded domain $\Omega$ of $\mathbb{R}^{3}$. This result is obtained under a microlocal condition linking the bicharacteristic paths of the system and the region of the boundary on which the control acts. This condition is to be compared with the so-called Geometric Control Condition by Bardos, Lebeau and Rauch [3]. The proof relies on microlocal tools, namely the propagation of the $C^{\infty}$ wave front and microlocal defect measures.
Keywords: elasticity, Lamé system, microlocal defect measures, geometric control condition, controllability, propagation of wave front..
Mathematics Subject Classification: Primary: 93C20, 93B05; Secondary: 74B05, 35L51, 35L0.
Citation: Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743
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Belhassen Dehman Jean-Pierre Raymond