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December  2018, 7(4): 545-570. doi: 10.3934/eect.2018026

Observability of wave equation with Ventcel dynamic condition

Imen Benabbas and  Djamel Eddine Teniou ,

Laboratoire AMNEDP, Faculté de mathématiques, USTHB, Alger, Algérie

* Corresponding author: Djamel Eddine Teniou

Received  December 2017 Revised  August 2018 Published  September 2018

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.

Keywords: Ingham, observability, wave equation, Ventcel dynamic condition, Fourier series.
Mathematics Subject Classification: Primary: 35L10, 93B07; Secondary: 35P20.
Citation: Imen Benabbas, Djamel Eddine Teniou. Observability of wave equation with Ventcel dynamic condition. Evolution Equations & Control Theory, 2018, 7 (4) : 545-570. doi: 10.3934/eect.2018026
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.  Google Scholar

[9]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[10]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, and John Wiley & Sons, Chichester, 1994.  Google Scholar

[11]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

K. Lemrabet, Problème aux limites de Ventcel dans un domaine non régulier, CRAS Paris, 300 (1985), 531-534.   Google Scholar

[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. Google Scholar

[17]

K. Lemrabet and D. Teniou, Un problème d'évolution de type Ventcel, Maghreb Math. Rev., 1 (1992), 15-29.   Google Scholar

[18]

J.-L. Lions, Contrôlabilité Exacte Perturbation et Stabilisation De Systémes Distribués I, Masson, Paris, 1988.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[24]

J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133 (1973), 301-312.  doi: 10.1007/BF01177870.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.  Google Scholar

[9]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[10]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, and John Wiley & Sons, Chichester, 1994.  Google Scholar

[11]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

K. Lemrabet, Problème aux limites de Ventcel dans un domaine non régulier, CRAS Paris, 300 (1985), 531-534.   Google Scholar

[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. Google Scholar

[17]

K. Lemrabet and D. Teniou, Un problème d'évolution de type Ventcel, Maghreb Math. Rev., 1 (1992), 15-29.   Google Scholar

[18]

J.-L. Lions, Contrôlabilité Exacte Perturbation et Stabilisation De Systémes Distribués I, Masson, Paris, 1988.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[24]

J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133 (1973), 301-312.  doi: 10.1007/BF01177870.  Google Scholar

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