Z. Abbas and S. Nicaise, The Multidimensional Wave Equation with Generalized Acoustic Boundary Conditions II: Polynomial Stability, SIAM Journal on Control and Optimization, vol.53, issue.4
DOI : 10.1137/140971348

F. Abdallah, Stabilisation et approximation de certains systèmes distribués par amortissement dissipative et de signe indéfini, p.86, 2013.

F. Abdallah, S. Nicaise, J. Valein, and A. Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications. ESAIM: Control, Optimisation and Calculus of Variations, pp.844-887, 2013.
DOI : 10.1051/cocv/2012036

URL : https://hal.archives-ouvertes.fr/hal-00755744

N. Aissa and D. Hamroun, Stabilization of a wave-wave system, Port. Math. (N.S.), vol.61, issue.89, pp.147-159, 2004.

F. Alabau, Stabilisation fronti??re indirecte de syst??mes faiblement coupl??s, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.328, issue.11, pp.1015-1020, 1999.
DOI : 10.1016/S0764-4442(99)80316-4

F. Alabau, Observabilit?? fronti??re indirecte de syst??mes faiblement coupl??s, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.333, issue.7, pp.645-650, 2001.
DOI : 10.1016/S0764-4442(01)02076-6

F. Alabau, P. Cannarsa, and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, Journal of Evolution Equations, vol.2, issue.2, pp.127-150, 2002.
DOI : 10.1007/s00028-002-8083-0

K. Ammari, A. Henrot, and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise BIBLIOGRAPHY stabilization of a string, Asymptot. Anal, vol.28, pp.3-4215, 2001.

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential Integral Equations, vol.17, issue.1112, pp.1395-1410, 2004.

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, vol.52, issue.4, pp.327-343, 2007.
DOI : 10.1007/s10492-007-0018-1

K. Ammari, M. Jellouli, and M. Khenissi, Stabilization of Generic Trees of Strings, Journal of Dynamical and Control Systems, vol.111, issue.2, pp.177-193, 2005.
DOI : 10.1007/s10883-005-4169-7

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks

J. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, vol.256, pp.5042-5044, 1963.

S. A. Avdonin and S. A. Ivanov, Families of exponentials The method of moments in controllability problems for distributed parameter systems, Translated from the Russian and revised by the authors, p.27, 1995.

H. T. Banks, K. Ito, and C. Wang, Exponentially stable approximations of weakly damped wave equations, Vorau Internat . Ser. Numer. Math, vol.100, issue.142, pp.1-33, 1990.
DOI : 10.1007/978-3-0348-6418-3_1

C. Bardos, G. Lebeau, and J. Rauch, Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary, SIAM Journal on Control and Optimization, vol.30, issue.5, pp.1024-1065, 1992.
DOI : 10.1137/0330055

M. Bassam, D. Mercier, and A. Wehbe, Optimal energy decay rate of Rayleigh beam equation with only one boundary control force, Evol. Equ. Control Theory, vol.4, issue.5, pp.21-38, 2015.

C. D. Benchimol, A Note on Weak Stabilizability of Contraction Semigroups, SIAM Journal on Control and Optimization, vol.16, issue.3, pp.373-379, 1978.
DOI : 10.1137/0316023

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, vol.42, issue.2, pp.455-478, 2010.
DOI : 10.1007/s00208-009-0439-0

URL : https://hal.archives-ouvertes.fr/hal-01257778

H. Brezis, Analyse fonctionnelle Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Théorie et applications, p.35, 1983.

C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method, Numerische Mathematik, vol.102, issue.3, pp.413-462, 2006.
DOI : 10.1007/s00211-005-0651-0

C. Castro, S. Micu, and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square, IMA Journal of Numerical Analysis, vol.28, issue.1, pp.186-214, 2008.
DOI : 10.1093/imanum/drm012

URL : https://hal.archives-ouvertes.fr/hal-00484055

M. M. Cavalcanti, I. Lasiecka, and D. Toundykov, Geometrically constrained stabilization of wave equations with Wentzell boundary conditions, Applicable Analysis, vol.94, issue.1, pp.1427-1452, 2012.
DOI : 10.1137/1104014

G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl, vol.58, issue.93, pp.249-273, 1979.

B. Dehman, G. Lebeau, and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation. Annales Scientifiques de l'École Normale Supérieure, pp.525-551, 2003.

S. Ervedoza and E. Zuazua, Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves, Numerische Mathematik, vol.78, issue.5, pp.597-634, 2008.
DOI : 10.1007/s00211-008-0153-y

URL : https://hal.archives-ouvertes.fr/hal-00681710

N. Fourrier, Analysis of existence, regularity and stability of solutions to wave equations with dynamic boundary conditions, p.104, 2013.

N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, vol.2, issue.10, pp.631-667, 2013.

C. G. Gal, G. R. Goldstein, and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, Journal of Evolution Equations, vol.3, issue.4, pp.623-635, 2003.
DOI : 10.1007/s00028-003-0113-z

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Transactions of the American Mathematical Society, vol.236, pp.385-394, 1978.
DOI : 10.1090/S0002-9947-1978-0461206-1

S. Gerbi and B. Said-houari, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, vol.74, issue.18, pp.7137-7150, 2011.
DOI : 10.1016/j.na.2011.07.026

URL : https://hal.archives-ouvertes.fr/hal-00339258

J. S. Gibson, A Note on Stabilization of Infinite Dimensional Linear Oscillators by Compact Linear Feedback, SIAM Journal on Control and Optimization, vol.18, issue.3, pp.311-316, 1980.
DOI : 10.1137/0318022

R. Glowinski, Ensuring well-posedness by analogy; Stokes problem and boundary control for the wave equation, Journal of Computational Physics, vol.103, issue.2, pp.189-221, 1992.
DOI : 10.1016/0021-9991(92)90396-G

R. Glowinski, W. Kinton, and M. F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation, International Journal for Numerical Methods in Engineering, vol.14, issue.3, pp.623-635, 1989.
DOI : 10.1002/nme.1620270313

R. Glowinski, C. H. Li, J. Lions, and . Errata, A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods, Japan J. Appl. Math, vol.7, issue.15, p.144, 1990.

R. Glowinski and J. Lions, Exact and approximate controllability for distributed parameter systems, In Acta numerica Acta Numer, vol.15, pp.159-333, 1995.

I. Gohberg and K. , Introduction to the Theory of Linear Nonselfadjoint Operators, p.28

P. J. Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum, vol.37, issue.2, pp.333-365, 2014.
DOI : 10.1007/s00233-013-9534-3

URL : https://hal.archives-ouvertes.fr/hal-00968742

P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol.24, issue.97, pp.94-98, 1985.
DOI : 10.1137/1.9781611972030

B. Guo, Riesz Basis Approach to the Stabilization of a Flexible Beam with a Tip Mass, SIAM Journal on Control and Optimization, vol.39, issue.6, pp.1736-1747, 2001.
DOI : 10.1137/S0363012999354880

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl, vol.68, issue.128, pp.457-465, 1989.

T. Hell, A. Ostermann, and M. Sandbichler, Modification of dimension-splitting methods???overcoming the order reduction due to corner singularities, IMA Journal of Numerical Analysis, vol.35, issue.3, pp.1078-1091, 2015.
DOI : 10.1093/imanum/dru030

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, vol.1, issue.22, pp.43-56, 1985.

J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1 ??? d wave equation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.33, issue.2, pp.407-438, 1999.
DOI : 10.1051/m2an:1999123

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Mathematische Zeitschrift, vol.3, issue.2, pp.367-379, 1936.
DOI : 10.1112/plms/s2-38.1.458

K. Ito and F. Kappel, The Trotter-Kato theorem and approximation of PDEs, Mathematics of Computation of the American Mathematical Society, vol.67, issue.221, pp.21-44, 1998.
DOI : 10.1090/S0025-5718-98-00915-6

T. Kato, Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, p.66, 1976.

V. Komornik, Rapid Boundary Stabilization of the Wave Equation, SIAM Journal on Control and Optimization, vol.29, issue.1, pp.197-208, 1991.
DOI : 10.1137/0329011

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, pp.25-140, 1995.

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl, vol.69, issue.91, pp.33-54, 1990.

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, Journal of Differential Equations, vol.50, issue.2, pp.163-182, 1983.
DOI : 10.1016/0022-0396(83)90073-6

J. E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.10, issue.7, p.33, 1989.
DOI : 10.1137/1.9781611970821

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Applied Mathematics & Optimization, vol.137, issue.2, pp.189-224, 1992.
DOI : 10.1007/BF01182480

L. León and E. Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation, ESAIM: Control, Optimisation and Calculus of Variations, vol.8, issue.15, pp.827-862, 2002.
DOI : 10.1051/cocv:2002025

J. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations, Recherches en Mathématiques Appliquées . Masson, p.23, 1988.

W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math, vol.59, issue.1, pp.17-34, 1999.

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Annali di Matematica Pura ed Applicata, vol.152, issue.1, pp.281-330, 1988.
DOI : 10.1007/BF01766154

W. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat, vol.48, pp.61-75, 1999.

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Zeitschrift f??r angewandte Mathematik und Physik, vol.56, issue.4, pp.630-644, 2005.
DOI : 10.1007/s00033-004-3073-4

URL : https://hal.archives-ouvertes.fr/hal-00129703

P. Loreti and B. Rao, Optimal Energy Decay Rate for Partially Damped Systems by Spectral Compensation, SIAM Journal on Control and Optimization, vol.45, issue.5, pp.1612-1632, 2006.
DOI : 10.1137/S0363012903437319

URL : https://hal.archives-ouvertes.fr/hal-00129724

A. Marica and E. Zuazua, Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach, Lect. Notes Comput. Sci. Eng, vol.101, issue.140, pp.285-324
DOI : 10.1007/978-3-319-08025-3_9

S. Micu and E. Zuazua, Asymptotics for the Spectrum of a Fluid/Structure Hybrid System Arising in the Control of Noise, SIAM Journal on Mathematical Analysis, vol.29, issue.4, pp.967-1001, 1998.
DOI : 10.1137/S0036141096312349

P. Morse and K. Ingard, Theoretical Acoustics. International series in pure and applied physics, p.88, 1968.
DOI : 10.1115/1.3564682

D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Mathematische Nachrichten, vol.8, issue.3, pp.299-318, 2006.
DOI : 10.1002/mana.200310362

URL : http://arxiv.org/abs/1008.0293

A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.39, issue.2, pp.377-418, 2005.
DOI : 10.1051/m2an:2005012

B. Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space. Universitext, pp.17-37, 2010.

M. Negreanu and E. Zuazua, A 2-Grid Algorithm for the 1-d Wave Equation, Mathematical and numerical aspects of wave propagation?WAVES 2003, pp.213-217, 2003.
DOI : 10.1007/978-3-642-55856-6_34

S. Nicaise and K. Laoubi, Polynomial stabiization of the wave equation with Ventcel's boundary conditions, Mathematische Nachrichten, vol.184, issue.10, pp.1428-1438, 2010.
DOI : 10.1002/mana.200710162

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete and Continuous Dynamical Systems - Series S, vol.9, issue.3, pp.791-813, 2016.
DOI : 10.3934/dcdss.2016029

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, vol.2, issue.3, pp.425-479, 2007.
DOI : 10.3934/nhm.2007.2.425

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol.44, issue.66, pp.33-36, 1983.
DOI : 10.1007/978-1-4612-5561-1

J. Prüss, On the Spectrum of C 0 -Semigroups, Transactions of the American Mathematical Society, vol.284, issue.2, pp.847-857, 1984.
DOI : 10.2307/1999112

J. P. Quinn and D. L. Russell, Synopsis, Proc. Roy. Soc. Edinburgh Sect. A, pp.97-127, 1977.
DOI : 10.1002/cpa.3160140327

K. Ramdani, T. Takahashi, and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations, ESAIM: Control, Optimisation and Calculus of Variations, vol.13, issue.3, pp.503-527, 2007.
DOI : 10.1051/cocv:2007020

URL : https://hal.archives-ouvertes.fr/hal-00140476

B. Rao, A compact perturbation method for the boundary stabilization of the Rayleigh beam equation, Applied Mathematics & Optimization, vol.105, issue.3, pp.253-264, 1996.
DOI : 10.1007/BF01204704

B. Rao, L. Toufayli, and A. Wehbe, Stability and controllability of a wave equation with dynamical boundary control, Mathematical Control and Related Fields, vol.5, issue.2, pp.305-320, 2015.
DOI : 10.3934/mcrf.2015.5.305

B. Rao and A. Wehbe, Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent, Journal of Evolution Equations, vol.5, issue.2, pp.137-152, 2005.
DOI : 10.1007/s00028-005-0171-5

URL : https://hal.archives-ouvertes.fr/hal-00129704

J. M. Rivera and Y. Qin, Polynomial decay for the energy with an acoustic boundary condition, Applied Mathematics Letters, vol.16, issue.2, pp.249-256, 2003.
DOI : 10.1016/S0893-9659(03)80039-3

D. L. Russell, Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods, Journal of Differential Equations, vol.19, issue.2, pp.344-370, 1975.
DOI : 10.1016/0022-0396(75)90009-1

D. L. Russell, Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, SIAM Review, vol.20, issue.4, pp.639-739, 1978.
DOI : 10.1137/1020095

D. L. Russell, A General Framework for the Study of Indirect Damping Mechanisms in Elastic Systems, Journal of Mathematical Analysis and Applications, vol.173, issue.2, pp.339-358, 1993.
DOI : 10.1006/jmaa.1993.1071

J. Simon, Compact sets in the spaceL p (O,T; B), Annali di Matematica Pura ed Applicata, vol.287, issue.1, pp.65-96, 1987.
DOI : 10.1007/BF01762360

L. R. Tébou and E. Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numerische Mathematik, vol.95, issue.3, pp.563-598, 2003.
DOI : 10.1007/s00211-002-0442-9

L. T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite difference space discretization of the 1???d wave equation, Advances in Computational Mathematics, vol.78, issue.2, pp.337-365, 2007.
DOI : 10.1007/s10444-004-7629-9

J. Valein and E. Zuazua, Stabilization of the Wave Equation on 1-d Networks, SIAM Journal on Control and Optimization, vol.48, issue.4, pp.2771-2797, 2009.
DOI : 10.1137/080733590

C. J. Arendt, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, vol.306, issue.2, pp.837-852, 1988.
DOI : 10.1090/S0002-9947-1988-0933321-3

A. Wehbe, Rational energy decay rate for a wave equation with dynamical control, ESAIM: Proceedings, vol.8, issue.147, pp.357-364, 2003.
DOI : 10.1051/proc:2000012

A. Wehbe, Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls, Bull. Belg. Math. Soc. Simon Stevin, vol.13, issue.62, pp.385-400, 2006.

G. Q. Xu, D. Y. Liu, and Y. Q. Liu, Abstract Second Order Hyperbolic System and Applications to Controlled Network of Strings, SIAM Journal on Control and Optimization, vol.47, issue.4, pp.1762-1784, 2008.
DOI : 10.1137/060649367

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, vol.15, issue.2, pp.205-235, 1990.

E. Zuazua, Uniform Stabilization of the Wave Equation by Nonlinear Boundary Feedback, SIAM Journal on Control and Optimization, vol.28, issue.2, pp.466-477, 1990.
DOI : 10.1137/0328025

E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation, Rend. Mat. Appl, vol.24, issue.140, pp.201-237, 2004.