December  2015, 5(4): 743-760. doi: 10.3934/mcrf.2015.5.743

Exact controllability for the Lamé system

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

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.
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.

show all references

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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