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Exact rate of decay for solutions to damped second order ODE's with a degenerate potential
Observability of wave equation with Ventcel dynamic condition
Laboratoire AMNEDP, Faculté de mathématiques, USTHB, Alger, Algérie |
The main purpose of this work is to prove a new variant of Mehrenberger's inequality. Subsequently, we apply it to establish several observability estimates for the wave equation subject to Ventcel dynamic condition.
References:
[1] |
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), 1024-1065.
doi: 10.1137/0330055. |
[2] |
P. Binding, P. J. Browne and K. Seddighi,
Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh, 37 (1993), 57-72.
doi: 10.1017/S0013091500018691. |
[3] |
M. Cavalcanti, V. Domingos Cavalcanti, R. Fukuoka and D. Toundykov,
Stabilization of the damped wave equation with Cauchy-Ventcel boundary conditions, J. Evol. Equ., 9 (2009), 143-169.
doi: 10.1007/s00028-009-0002-1. |
[4] |
M. Cavalcanti, A. Khemmoudj and M. Medjden,
Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328 (2007), 900-930.
doi: 10.1016/j.jmaa.2006.05.070. |
[5] |
C. Fulton,
Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh, 77A (1977), 293-308.
doi: 10.1017/S030821050002521X. |
[6] |
C. Fulton,
Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh, 87A (1980), 1-34.
doi: 10.1017/S0308210500012312. |
[7] |
C. Gal and L. Tebou,
Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), 324-364.
doi: 10.1137/15M1032211. |
[8] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. |
[9] |
A. E. Ingham,
Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.
doi: 10.1007/BF01180426. |
[10] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, and John Wiley & Sons, Chichester, 1994. |
[11] |
V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. |
[12] |
V. Komornik and P. Loreti,
Observability of rectangular membranes and plates on small sets, EECT., 3 (2014), 287-304.
doi: 10.3934/eect.2014.3.287. |
[13] |
V. Komornik and B. Miara,
Cross-like internal observability of ractangular membranes, EECT., 3 (2014), 135-146.
doi: 10.3934/eect.2014.3.135. |
[14] |
V. Komornik and P. Loreti,
Observability of square membranes by Fourier series methods, Bulletin of the south Ural state university, Series "Mathematical modelling, programming and computer software", 8 (2015), 127-140.
doi: 10.14529/mmp150308. |
[15] |
K. Lemrabet,
Problème aux limites de Ventcel dans un domaine non régulier, CRAS Paris, 300 (1985), 531-534.
|
[16] |
K. Lemrabet, Etude de Divers Problèmes Aux Limites de Ventcel D'origine Physique ou Mécanique dans des Domaines non Réguliers, Ph.D thesis, USTHB, Algiers, 1987. |
[17] |
K. Lemrabet and D. Teniou,
Un problème d'évolution de type Ventcel, Maghreb Math. Rev., 1 (1992), 15-29.
|
[18] |
J.-L. Lions, Contrôlabilité Exacte Perturbation et Stabilisation De Systémes Distribués I, Masson, Paris, 1988. |
[19] |
T. Masrour,
The wave equation with dynamic Wentzell boundary condition in polygonal and polyhedral domains: Observation and exact controllability, International Journal of Partial Differential Equations and Applications, 2 (2014), 13-22.
|
[20] |
M. Mehrenberger,
An Ingham type proof for the boundary observability of a N-d wave equation, C. R. Math. Acad. Sci. Paris, 347 (2009), 63-68.
doi: 10.1016/j.crma.2008.11.002. |
[21] |
S. Nicaise and K. Laoubi,
Polynomial stabilization of the wave equation with Ventcel's boundary conditions, Math. Nachr., 283 (2010), 1428-1438.
doi: 10.1002/mana.200710162. |
[22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[23] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[24] |
J. Walter,
Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133 (1973), 301-312.
doi: 10.1007/BF01177870. |
show all references
References:
[1] |
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), 1024-1065.
doi: 10.1137/0330055. |
[2] |
P. Binding, P. J. Browne and K. Seddighi,
Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh, 37 (1993), 57-72.
doi: 10.1017/S0013091500018691. |
[3] |
M. Cavalcanti, V. Domingos Cavalcanti, R. Fukuoka and D. Toundykov,
Stabilization of the damped wave equation with Cauchy-Ventcel boundary conditions, J. Evol. Equ., 9 (2009), 143-169.
doi: 10.1007/s00028-009-0002-1. |
[4] |
M. Cavalcanti, A. Khemmoudj and M. Medjden,
Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328 (2007), 900-930.
doi: 10.1016/j.jmaa.2006.05.070. |
[5] |
C. Fulton,
Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh, 77A (1977), 293-308.
doi: 10.1017/S030821050002521X. |
[6] |
C. Fulton,
Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh, 87A (1980), 1-34.
doi: 10.1017/S0308210500012312. |
[7] |
C. Gal and L. Tebou,
Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), 324-364.
doi: 10.1137/15M1032211. |
[8] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. |
[9] |
A. E. Ingham,
Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.
doi: 10.1007/BF01180426. |
[10] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, and John Wiley & Sons, Chichester, 1994. |
[11] |
V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. |
[12] |
V. Komornik and P. Loreti,
Observability of rectangular membranes and plates on small sets, EECT., 3 (2014), 287-304.
doi: 10.3934/eect.2014.3.287. |
[13] |
V. Komornik and B. Miara,
Cross-like internal observability of ractangular membranes, EECT., 3 (2014), 135-146.
doi: 10.3934/eect.2014.3.135. |
[14] |
V. Komornik and P. Loreti,
Observability of square membranes by Fourier series methods, Bulletin of the south Ural state university, Series "Mathematical modelling, programming and computer software", 8 (2015), 127-140.
doi: 10.14529/mmp150308. |
[15] |
K. Lemrabet,
Problème aux limites de Ventcel dans un domaine non régulier, CRAS Paris, 300 (1985), 531-534.
|
[16] |
K. Lemrabet, Etude de Divers Problèmes Aux Limites de Ventcel D'origine Physique ou Mécanique dans des Domaines non Réguliers, Ph.D thesis, USTHB, Algiers, 1987. |
[17] |
K. Lemrabet and D. Teniou,
Un problème d'évolution de type Ventcel, Maghreb Math. Rev., 1 (1992), 15-29.
|
[18] |
J.-L. Lions, Contrôlabilité Exacte Perturbation et Stabilisation De Systémes Distribués I, Masson, Paris, 1988. |
[19] |
T. Masrour,
The wave equation with dynamic Wentzell boundary condition in polygonal and polyhedral domains: Observation and exact controllability, International Journal of Partial Differential Equations and Applications, 2 (2014), 13-22.
|
[20] |
M. Mehrenberger,
An Ingham type proof for the boundary observability of a N-d wave equation, C. R. Math. Acad. Sci. Paris, 347 (2009), 63-68.
doi: 10.1016/j.crma.2008.11.002. |
[21] |
S. Nicaise and K. Laoubi,
Polynomial stabilization of the wave equation with Ventcel's boundary conditions, Math. Nachr., 283 (2010), 1428-1438.
doi: 10.1002/mana.200710162. |
[22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[23] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[24] |
J. Walter,
Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133 (1973), 301-312.
doi: 10.1007/BF01177870. |
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