American Institute of Mathematical Sciences

Advanced
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

[1]

Nikita Kalinin, Mikhail Shkolnikov. Introduction to tropical series and wave dynamic on them. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2827-2849. doi: 10.3934/dcds.2018120
[2]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[3]

G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509
[4]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[5]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[6]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343
[7]

Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013
[8]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[9]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[10]

Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601
[11]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[12]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191
[13]

Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[14]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[15]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253
[16]

Faustino Maestre, Pablo Pedregal. Dynamic materials for an optimal design problem under the two-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 973-990. doi: 10.3934/dcds.2009.23.973
[17]

Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[18]

Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070
[19]

Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic & Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036
[20]

RazIye Mert, A. Zafer. A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations. Conference Publications, 2011, 2011 (Special) : 1061-1067. doi: 10.3934/proc.2011.2011.1061

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (156)
  • HTML views (534)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences