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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 |
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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