International Journal of Partial Differential Equations and Applications
ISSN (Print): 2376-9548 ISSN (Online): 2376-9556 Website: http://www.sciepub.com/journal/ijpdea Editor-in-chief: Mahammad Nurmammadov
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International Journal of Partial Differential Equations and Applications. 2014, 2(1), 13-22
DOI: 10.12691/ijpdea-2-1-3
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The Wave Equation with Dynamic Wentzell Boundary Condition in Polygonal and Polyhedral Domains: Observation and Exact Controllability

Tawfik Masrour1,

1Research Laboratory (M2.I), Mathematical Modeling for Analysis and Decision Making Research team (M2APD), Moulay Ismail University ENSAM, Meknes, Morocco

Pub. Date: February 10, 2014

Cite this paper:
Tawfik 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. 2014; 2(1):13-22. doi: 10.12691/ijpdea-2-1-3

Abstract

We study in this article the boundary observability and the exact controllability for a problem of transmission. The system is governed by the wave equation with Wentzell dynamic artificial condition on the boundary. The geometrical domains considered are polyhedrons or polygons.

Keywords:
wave equation Wentzell boundary condition Wentsell Ventcel polygonal domains polyhedral domains observationexact controllability

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References:

[1]  A. Wentzell, «On boundary conditions for multi-dimensional diffusion processes,» Theory Probab. Appl 4, p. 164-177, 1959.
 
[2]  G. R. Goldstein, «Derivation and physical interpretation of general boundary conditions,» Advances in Differential Equations, vol. 11, n 14, pp. 457-480, 2006.
 
[3]  I. Achdou, O. Pironneau et F. Valentin, «Effective boundary conditions for laminar flows over periodic rough boundaries,» Journal of Computational Physics , vol. 147, n 11, pp. 187-218, 1998.
 
[4]  W. Arendt, G. Metafune, D. Pallara et S. Romanell, «The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions,» Semigroup Forum Springer-Verlag., vol. 67, n 12, pp. 247-261, 2003.
 
[5]  Y. AMIRAT, G. A. CHECHKIN et R. R. GADYL’SHIN, «Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary,» Computational Mathematics and Mathematical Physics, vol. 46, n 11, pp. 97-110, 2006.
 
[6]  V. Bonnaillie-Noël, M. Dambrine, F. Hérau et G. Vial, «On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain,» SIAM Journal on Mathematical Analysis, vol. 42, n 12, pp. 931-945., 2010.
 
[7]  M. M. A. Khemmoudj, «Exponential decay for the semilinear damped Cauchy–Ventcel problem,» Bol. Soc. Parana. Mat., p. 97-116, 2004.
 
[8]  K. Lemrabet, «Problème aux limites de Ventcel dans un domaine non régulier,» C. R. Acad. Sci. Paris Sér I Math. 300 (15), p. 531-534, 1985.
 
[9]  T. Masrour, Controlabilité et observation des systèmes distribués, problèmes et méthodes, Paris: Ecole Nationale des Ponts et Chaussées Paris, 1995.
 
[10]  E. Zuazua, «Propagation, observation, and control of waves approximated by finite difference methods,» SIAM review, vol. 47, n 12, pp. 197-243, 2005.
 
[11]  J. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribués tome 1, Paris: Masson, 1988, pp. [9] J.L. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribués, tome 1, Masson, Paris, 1988.
 
[12]  R. B. Melrose et J. Sjöstrand, «Singularities of boundary value problems. I,» Communications on Pure and Applied Mathematics, vol. 31, n 15, pp. 593-617, 1978.
 
[13]  R. B. Melrose et J. Sjöstrand, «Singularities of boundary value problems. II,» Communications on Pure and Applied Mathematics, vol. 35, n 12, pp. 129-168, 1982.
 
[14]  C. Bardos, G. Lebeau et J. Rauch, «Sharp and sufficient conditions for the observation, control, and stabilization of waves from the boundary,» SIAM Journal, Control and Optimisation 30, pp. 1024-1065, 1992.
 
[15]  C. Bardos, T. Masrour et F. Tatout, «Singularités du problème d'élastodynamique,» Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 320 (9), pp. 1157-1160, 1995.
 
[16]  C. Bardos, T. Masrour et F. Tatout, «Condition nécessaire et suffisante pour la contrôlabilité exacte et la stabilisation du problème de l'élastodynamique,» Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 320 (10), pp. 1279-12981, 1995.
 
[17]  L. Tartar, «H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,» Proc. Roy. Soc. Edinburgh Sect. A, vol. 115, n 13-4, pp. 193-230, 1990.
 
[18]  L. Tartar, The general theory of homogenization: a personalized introduction, (Vol. 7). Springer., 2009.
 
[19]  C. Bardos et T. Masrour, «Mesures de défaut: observation et contrôle de plaques,» Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 323 (6), pp. 621-626, 1996.
 
[20]  P. Grisvard, «Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités,» Journal de mathématiques pures et appliquées, vol. 68, n 12, pp. 215-259, 1989.
 
[21]  S. Nicaise, «About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation. I: Regularity of the solutions,» Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, vol. 19 (3), n 13, 1992.
 
[22]  A. Chaïra, «Equation des ondes et régularité sur un ouvert lipschitzien,» Comptes rendus de l'Académie des sciences, Série 1, Mathématique, vol. 316, n 11, pp. 33-36, 1993.
 
[23]  M. Cavalcanti, A. Khemmoudj et M. Medjden, «Uniform stabilization of damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions,» J. Math. Anal. Appl. 328, pp. 900-930, 2007.
 
[24]  M. Cavalcanti et H. Oquendo, «Frictional versus viscoelastic damping in a semilinear wave equation,» SIAM J. Control Optim, vol. 42, n 14, p. 1310-1324, 2003.
 
[25]  A. Heminna, «Exact controllability of the linear elasticity system with evolutive Ventcel conditions,» Portugaliae Mathematica. Nova Série, vol. 58, n 13, pp. 271-315, 2001.
 
[26]  I. Lasiecka et D. Tataru, «Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping,» Differential Integral Equations , vol. 6, p. 507-533, 1993.
 
[27]  I. Lasiecka et R. Triggiani, «Uniform exponential energy decay of wave equations in a bounded region with L^2 (0,∞,L^2 (Γ))-feedback control in the Dirichlet boundary conditions».
 
[28]  I. Lasiecka, R. Triggiani et P. Yao, «Inverse observability estimates for second-order hyperbolic equations with variable coefficients,» J. Math. Anal. Appl. 235 (1), p. 13-57, 1999.
 
[29]  P. Grisvard, Singularities in boundary value problems, Vol. 22. Springer, 1992.
 
[30]  T. Masrour, «Convergence des fonctions propres de troisième espèce pour le laplacien,» Comptes rendus de l'Académie des sciences. Série 1, Mathématique,, pp. 309-312, 1995.
 
[31]  C. Bardos, T. Masrour et F. Tatout, «Observation and control of elastic waves. In Singularities and Oscillations,» Springer New York, pp. 1-16, 1997.