Exact controllability for the Lamé system

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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,

1.	Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092 El Manar, Tunisia
2.	Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS, 31062 Toulouse Cedex

Received August 2014 Revised May 2015 Published October 2015

Abstract
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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 and Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

References:

[1]	L. Aloui, Stabilisation Neumann pour l'équation des ondes sur un domaine extêrieur, J. Math. Pures Appl., 81 (2002), 1113-1134. doi: 10.1016/S0021-7824(02)01261-8.
[2]	K. Andersson and R. Melrose, The propagation of singularities along gliding rays, Invent. Math., 41 (1977), 197-232. doi: 10.1007/BF01403048.
[3]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optimization, 30 (1992), 1024-1065. doi: 10.1137/0330055.
[4]	C. Bardos, T. Masrour and F. Tatout, Singularités du problème d'élastodynamique, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1157-1160.
[5]	C. Bardos, T. Masrour and F. Tatout, Condition nécessaire et suffisante pour la controlabilité exacte et la stabilisation du problème de l'élastodynamique, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1279-1281.
[6]	C. Bardos, T. Masrour and F. Tatout, Obseravation and control of elastic waves, in Singularities and Oscillations (eds. J. Rauch, et al.), IMA Vol. Math. Appl., 91, Springer, New York, NY, 1997, 1-16. doi: 10.1007/978-1-4612-1972-9_1.
[7]	M. Bellassoued, Energy decay for the elastic wave equation with a local time-dependant nonlinear damping, Acta Math. Sinica, English Series, 24 (2008), 1175-1192. doi: 10.1007/s10114-007-6468-2.
[8]	N. Burq, Contrôle de l'équation des ondes dans des ouverts comportant des coins, Bull. Soc. Math. France, 126 (1998), 601-637.
[9]	N. Burq and P. Gérard, Condition Nécessaire et suffisante pour la contrôlabilité exacte des ondes, Comptes Rendus de l'Académie des Sciences, Série I, 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.
[10]	N. Burq and G. Lebeau, Mesures de Défaut de compacité, Application au système de Lamé, Ann. Scient. Ec. Norm. Sup. 4 série, 34 (2001), 817-870. doi: 10.1016/S0012-9593(01)01078-3.
[11]	M. Daoulatli, B. Dehman and M. Khenissi, Local energy decay for the elastic system with nonlinear damping in an exterior domain, SIAM J. Control Optim., 48 (2010), 5254-5275. doi: 10.1137/090757332.
[12]	B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé, J. Math. Pures Appl. (9), 72 (1993), 475-492.
[13]	T. Duyckaerts, Thèse de Doctorat, Université de Paris Sud, 2004.
[14]	P. Gérard, Microlocal defect measures, Com.Par. Diff. Eq., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.
[15]	L. Hörmander, The Analysis of Partial Differential Operators, Vol. 3, Springer-Verlag, 1985.
[16]	G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, J. Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.
[17]	J.-L. Lions, Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1, Rech. Math. Appl., 8, Masson, Paris, 1988.
[18]	M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.
[19]	K. Yamamoto, Singularities of solutions to the boundary value problems for elastic and Maxwell's equations, Japan J. Math., 14 (1988), 119-163.
[20]	K. Yamamoto, Exponential energy decay of solutions of elastic wave equations with the Dirichlet condition, Math. Scand., 65 (1989), 206-220.
[21]	K. Yamamoto, Propagation of microlocal regularities in Sobolev spaces to solutions of boundary value problems for elastic equations, Hokkaido Math. Journal, 35 (2006), 497-545. doi: 10.14492/hokmj/1285766414.

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References:

[1]	L. Aloui, Stabilisation Neumann pour l'équation des ondes sur un domaine extêrieur, J. Math. Pures Appl., 81 (2002), 1113-1134. doi: 10.1016/S0021-7824(02)01261-8.
[2]	K. Andersson and R. Melrose, The propagation of singularities along gliding rays, Invent. Math., 41 (1977), 197-232. doi: 10.1007/BF01403048.
[3]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optimization, 30 (1992), 1024-1065. doi: 10.1137/0330055.
[4]	C. Bardos, T. Masrour and F. Tatout, Singularités du problème d'élastodynamique, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1157-1160.
[5]	C. Bardos, T. Masrour and F. Tatout, Condition nécessaire et suffisante pour la controlabilité exacte et la stabilisation du problème de l'élastodynamique, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1279-1281.
[6]	C. Bardos, T. Masrour and F. Tatout, Obseravation and control of elastic waves, in Singularities and Oscillations (eds. J. Rauch, et al.), IMA Vol. Math. Appl., 91, Springer, New York, NY, 1997, 1-16. doi: 10.1007/978-1-4612-1972-9_1.
[7]	M. Bellassoued, Energy decay for the elastic wave equation with a local time-dependant nonlinear damping, Acta Math. Sinica, English Series, 24 (2008), 1175-1192. doi: 10.1007/s10114-007-6468-2.
[8]	N. Burq, Contrôle de l'équation des ondes dans des ouverts comportant des coins, Bull. Soc. Math. France, 126 (1998), 601-637.
[9]	N. Burq and P. Gérard, Condition Nécessaire et suffisante pour la contrôlabilité exacte des ondes, Comptes Rendus de l'Académie des Sciences, Série I, 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.
[10]	N. Burq and G. Lebeau, Mesures de Défaut de compacité, Application au système de Lamé, Ann. Scient. Ec. Norm. Sup. 4 série, 34 (2001), 817-870. doi: 10.1016/S0012-9593(01)01078-3.
[11]	M. Daoulatli, B. Dehman and M. Khenissi, Local energy decay for the elastic system with nonlinear damping in an exterior domain, SIAM J. Control Optim., 48 (2010), 5254-5275. doi: 10.1137/090757332.
[12]	B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé, J. Math. Pures Appl. (9), 72 (1993), 475-492.
[13]	T. Duyckaerts, Thèse de Doctorat, Université de Paris Sud, 2004.
[14]	P. Gérard, Microlocal defect measures, Com.Par. Diff. Eq., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.
[15]	L. Hörmander, The Analysis of Partial Differential Operators, Vol. 3, Springer-Verlag, 1985.
[16]	G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, J. Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.
[17]	J.-L. Lions, Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1, Rech. Math. Appl., 8, Masson, Paris, 1988.
[18]	M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.
[19]	K. Yamamoto, Singularities of solutions to the boundary value problems for elastic and Maxwell's equations, Japan J. Math., 14 (1988), 119-163.
[20]	K. Yamamoto, Exponential energy decay of solutions of elastic wave equations with the Dirichlet condition, Math. Scand., 65 (1989), 206-220.
[21]	K. Yamamoto, Propagation of microlocal regularities in Sobolev spaces to solutions of boundary value problems for elastic equations, Hokkaido Math. Journal, 35 (2006), 497-545. doi: 10.14492/hokmj/1285766414.

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