H. T. Banks and K. Ito, Approximation in LQR problems for infinite dimensional systems with unbounded input operators, J. Math. Systems Estim. Control, vol.7, issue.1, 1997.

H. T. Banks and K. Kunisch, The Linear Regulator Problem for Parabolic Systems, SIAM Journal on Control and Optimization, vol.22, issue.5, pp.684-698, 1984.
DOI : 10.1137/0322043

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

J. Bramble, A. Shatz, V. Thomee, and L. Wahlbin, Some Convergence Estimates for Semidiscrete Galerkin Type Approximations for Parabolic Equations, SIAM Journal on Numerical Analysis, vol.14, issue.2, pp.218-241, 1977.
DOI : 10.1137/0714015

C. Castro and S. Micu, Boundary controllability of a linear semidiscrete 1-D wave equation derived from a mixed finite element method, 2005.

J. S. Gibson, The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces, SIAM Journal on Control and Optimization, vol.17, issue.4, pp.537-565, 1979.
DOI : 10.1137/0317039

L. F. Ho, Observabilitéfrontì ere de l'´ equation des ondes, C. R. Acad. Sci. Paris, vol.302, pp.443-446, 1986.

O. Yu, Imanuvilov, Controllability of parabolic equations, Sb. Math, vol.186, issue.6, pp.879-900, 1995.

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

F. Kappel and D. Salamon, An Approximation Theorem for the Algebraic Riccati Equation, SIAM Journal on Control and Optimization, vol.28, issue.5, pp.1136-1147, 1990.
DOI : 10.1137/0328061

V. Komornik, Exact controllability and stabilization, the multiplier method, 1994.

S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Systems & Control Letters, vol.55, issue.7, 2005.
DOI : 10.1016/j.sysconle.2006.01.004

I. Lasiecka, Convergence Estimates for Semidiscrete Approximations of Nonselfadjoint Parabolic Equations, SIAM Journal on Numerical Analysis, vol.21, issue.5, pp.894-908, 1977.
DOI : 10.1137/0721058

I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. I. Abstract parabolic systems, Encyclopedia of Mathematics and its Applications, vol.74, 2000.

G. Lebeau and L. Robbiano, Contr??le Exact De L??quation De La Chaleur, Communications in Partial Differential Equations, vol.52, issue.1-2, pp.335-356, 1995.
DOI : 10.1016/0022-0396(87)90043-X

L. Leon 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, pp.827-862, 2002.
DOI : 10.1051/cocv:2002025

J. Lions, Exact Controllability, Stabilization and Perturbations for Distributed Systems, SIAM Review, vol.30, issue.1, pp.1-68, 1988.
DOI : 10.1137/1030001

J. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Recherches en Mathématiques Appliquées, vol.1, 1988.

Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, vol.398, 1999.

A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-d heat equation, Séminaire sur les Equations aux Dérivées Partielles, pp.1-22, 1998.

S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numerische Mathematik, vol.91, issue.4, pp.723-768, 2002.
DOI : 10.1007/s002110100338

V. J. Mizel and T. I. Seidman, Observation and prediction for the heat equation, Journal of Mathematical Analysis and Applications, vol.28, issue.2, pp.303-312, 1969.
DOI : 10.1016/0022-247X(69)90029-8

M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-D wave equation, Systems Control Lett, pp.3-4, 2003.

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

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, 2004.
DOI : 10.1051/cocv:2007020

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

R. Rebarber and G. Weiss, Necessary conditions for exact controllability with a finite-dimensional input space, Systems & Control Letters, vol.40, issue.3, pp.217-227, 2000.
DOI : 10.1016/S0167-6911(00)00029-3

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

T. I. Seidman, Observation and prediction for the heat equation, III, Journal of Differential Equations, vol.20, issue.1, pp.18-27, 1976.
DOI : 10.1016/0022-0396(76)90092-9

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

M. Tucsnak and G. Weiss, Simultaneous Exact Controllability and Some Applications, SIAM Journal on Control and Optimization, vol.38, issue.5, pp.1408-1427, 2000.
DOI : 10.1137/S0363012999352716

M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. I. Well-posedness and energy balance, ESAIM Cont. Optim. Calc. Var, vol.9, pp.247-274, 2003.

M. Tucsnak and G. Weiss, How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability, SIAM Journal on Control and Optimization, vol.42, issue.3, pp.907-935, 2003.
DOI : 10.1137/S0363012901399295

G. Weiss, Admissible observation operators for linear semigroups, Israel Journal of Mathematics, vol.15, issue.1, pp.17-43, 1989.
DOI : 10.1007/BF02788172

G. Weiss, Admissibility of Unbounded Control Operators, SIAM Journal on Control and Optimization, vol.27, issue.3, pp.527-545, 1989.
DOI : 10.1137/0327028

E. Zuazua, Boundary observability for the finite-difference space semidiscretizations of the 2-D wave equation in the square, J. Math. Pures Appl, vol.9, issue.5, pp.78-523, 1999.

E. Zuazua, Controllability of partial differential equations and its semidiscrete approximations, Discrete Contin, Dyn. Syst, vol.8, issue.2, pp.469-513, 2002.

E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1-D wave equation, Rendiconti di Matematica, SIAM Review, vol.47, issue.2, pp.197-243, 2005.

E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference method