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Polynomial stabilization of some dissipative hyperbolic systems
1. | UR Analyse et Contrôle des Edp (05/UR/15-01), Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019 Monastir, Tunisia |
2. | Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1 |
3. | Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9 |
References:
[1] |
G. Allaire, Homogenization of the Navier-Stokes equations and derivation of Brinkman's law,, In Mathématiques appliquées aux sciences de l'ingénieur (Santiago, (1989), 7.
|
[2] |
K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks,, ESAIM Control Optim. Calc. Var., 6 (2001), 361.
doi: 10.1051/cocv:2001114. |
[3] |
P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows,, Numer. Math., 81 (1999), 497.
doi: 10.1007/s002110050401. |
[4] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837.
doi: 10.1090/S0002-9947-1988-0933321-3. |
[5] |
C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques,, Rend. Sem. Mat. Univ. Politec. Torino 1988, (1989), 11.
|
[6] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.
doi: 10.1137/0330055. |
[7] |
A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.
doi: 10.1002/mana.200410429. |
[8] |
C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765.
doi: 10.1007/s00028-008-0424-1. |
[9] |
A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.
doi: 10.1007/s00208-009-0439-0. |
[10] |
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math., 46 (1989), 245.
|
[11] |
J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978).
|
[12] |
M. J. Lighthill, On sound generated aerodynamically. I. General theory,, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564.
doi: 10.1098/rspa.1952.0060. |
[13] |
M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound,, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1.
doi: 10.1098/rspa.1954.0049. |
[14] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics],, Masson, (1988).
|
[15] |
K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control Optim., 35 (1997), 1574.
doi: 10.1137/S0363012995284928. |
[16] |
Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.
doi: 10.1007/s00033-004-3073-4. |
[17] |
K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92.
doi: 10.1016/j.jde.2007.05.016. |
[18] |
L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Math. Acad. Sci. Paris, 350 (2012), 57.
doi: 10.1016/j.crma.2011.12.001. |
[19] |
E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asymptotic Anal., 1 (1988), 161.
|
[20] |
E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures Appl. (9), 70 (1991), 513.
|
show all references
References:
[1] |
G. Allaire, Homogenization of the Navier-Stokes equations and derivation of Brinkman's law,, In Mathématiques appliquées aux sciences de l'ingénieur (Santiago, (1989), 7.
|
[2] |
K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks,, ESAIM Control Optim. Calc. Var., 6 (2001), 361.
doi: 10.1051/cocv:2001114. |
[3] |
P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows,, Numer. Math., 81 (1999), 497.
doi: 10.1007/s002110050401. |
[4] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837.
doi: 10.1090/S0002-9947-1988-0933321-3. |
[5] |
C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques,, Rend. Sem. Mat. Univ. Politec. Torino 1988, (1989), 11.
|
[6] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.
doi: 10.1137/0330055. |
[7] |
A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.
doi: 10.1002/mana.200410429. |
[8] |
C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765.
doi: 10.1007/s00028-008-0424-1. |
[9] |
A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.
doi: 10.1007/s00208-009-0439-0. |
[10] |
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math., 46 (1989), 245.
|
[11] |
J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978).
|
[12] |
M. J. Lighthill, On sound generated aerodynamically. I. General theory,, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564.
doi: 10.1098/rspa.1952.0060. |
[13] |
M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound,, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1.
doi: 10.1098/rspa.1954.0049. |
[14] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics],, Masson, (1988).
|
[15] |
K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control Optim., 35 (1997), 1574.
doi: 10.1137/S0363012995284928. |
[16] |
Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.
doi: 10.1007/s00033-004-3073-4. |
[17] |
K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92.
doi: 10.1016/j.jde.2007.05.016. |
[18] |
L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Math. Acad. Sci. Paris, 350 (2012), 57.
doi: 10.1016/j.crma.2011.12.001. |
[19] |
E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asymptotic Anal., 1 (1988), 161.
|
[20] |
E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures Appl. (9), 70 (1991), 513.
|
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