D. Auroux and J. Blum, Back and forth nudging algorithm for data assimilation problems, Comptes Rendus Mathematique, vol.340, issue.12, pp.873-878, 2005.
DOI : 10.1016/j.crma.2005.05.006
URL : https://hal.archives-ouvertes.fr/inria-00189644

D. Auroux and J. Blum, A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm, Nonlinear Processes in Geophysics, vol.15, issue.2, pp.305-319, 2008.
DOI : 10.5194/npg-15-305-2008
URL : https://hal.archives-ouvertes.fr/inria-00327422

D. Auroux, J. Blum, and M. Nodet, Diffusive Back and Forth Nudging algorithm for data assimilation, Comptes Rendus Mathematique, vol.349, issue.15-16, pp.15-16849, 2011.
DOI : 10.1016/j.crma.2011.07.004
URL : https://hal.archives-ouvertes.fr/inria-00617571

J. Baras and A. Bensoussan, On observer problems for systems governed by partial differential equations. No. SCR TR 86?47, pp.1-26, 1987.

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. Blum, L. Dimet, F. Navon, and I. , Data Assimilation for Geophysical Fluids, XIV. Special volume: computational methods for the atmosphere and the Oceans, pp.385-441, 2009.
DOI : 10.1016/S1570-8659(08)00209-3
URL : https://hal.archives-ouvertes.fr/inria-00391892

H. Brezis, Functional analysis. Sobolev spaces and partial differential equations, 2011.
DOI : 10.1007/978-0-387-70914-7

D. Chapelle, N. Cîndea, D. Buhan, M. Moireau, and P. , Exponential Convergence of an Observer Based on Partial Field Measurements for the Wave Equation, Mathematical Problems in Engineering, vol.2012, p.12, 2012.
DOI : 10.1016/j.matpur.2008.09.002
URL : https://hal.archives-ouvertes.fr/inria-00619504

J. Couchouron and P. Ligarius, Nonlinear observers in reflexive Banach spaces, ESAIM: Control, Optimisation and Calculus of Variations, vol.9, pp.67-104, 2003.
DOI : 10.1051/cocv:2003001

R. Curtain and G. Weiss, Well, posedness of triples of operators (in the sense of linear systems theory) In: Control and estimation of distributed parameter systems, Vorau Internat. Ser. Numer. Math. Birkhäuser, vol.91, pp.41-59, 1988.

R. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM Journal on Control and Optimization, vol.45, issue.1, pp.273-297, 2006.
DOI : 10.1137/040610489

P. Dular and C. Geuzaine, GetDP reference manual: the documentation for GetDP, a general environment for the treatment of discrete problems

S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems, Journal de Math??matiques Pures et Appliqu??es, vol.91, issue.1, pp.20-48, 2009.
DOI : 10.1016/j.matpur.2008.09.002
URL : https://hal.archives-ouvertes.fr/hal-00681693

M. Fink, Time reversal of ultrasonic fields. I. Basic principles, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol.39, issue.5, pp.555-556, 1992.
DOI : 10.1109/58.156174

M. Fink, D. Cassereau, A. Derode, C. Prada, O. Roux et al., Time-reversed acoustics, Reports on Progress in Physics, vol.63, issue.12, pp.1933-1995, 2000.
DOI : 10.1088/0034-4885/63/12/202

G. García and T. Takahashi, Numerical observers with vanishing viscosity for the 1d wave equation, Advances in Computational Mathematics, vol.47, issue.2, pp.1-35, 2013.
DOI : 10.1007/s10444-013-9320-5

B. Gebauer and O. Scherzer, Impedance-Acoustic Tomography, SIAM Journal on Applied Mathematics, vol.69, issue.2, pp.565-576, 2008.
DOI : 10.1137/080715123

I. Gejadze, L. Dimet, F. Shutyaev, and V. , On optimal solution error covariances in variational data assimilation problems, Journal of Computational Physics, vol.229, issue.6, pp.2159-2178, 2010.
DOI : 10.1016/j.jcp.2009.11.028
URL : https://hal.archives-ouvertes.fr/hal-00787292

C. Geuzaine and J. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, vol.69, issue.4, pp.1309-1331, 2009.
DOI : 10.1002/nme.2579

B. Guo and Z. Shao, Well-posedness and regularity for non-uniform Schr??dinger and Euler-Bernoulli equations with boundary control and observation, Quarterly of Applied Mathematics, vol.70, issue.1, pp.111-132, 2012.
DOI : 10.1090/S0033-569X-2011-01243-0

B. Guo and X. Zhang, The Regularity of the Wave Equation with Partial Dirichlet Control and Colocated Observation, SIAM Journal on Control and Optimization, vol.44, issue.5, pp.1598-1613, 2005.
DOI : 10.1137/040610702

B. Guo and Z. Zhang, On the well-posedness and regularity of the wave equation with variable coefficients, ESAIM: Control, Optimisation and Calculus of Variations, vol.13, issue.4, pp.776-792, 2007.
DOI : 10.1051/cocv:2007040

K. Ito, K. Ramdani, and M. Tucsnak, A time reversal based algorithm for solving initial data inverse problems, Discrete and Continuous Dynamical Systems - Series S, vol.4, issue.3, pp.641-652, 2011.
DOI : 10.3934/dcdss.2011.4.641
URL : https://hal.archives-ouvertes.fr/hal-00586314

M. Krstic, B. Guo, and A. Smyshlyaev, Boundary Controllers and Observers for the Linearized Schr??dinger Equation, SIAM Journal on Control and Optimization, vol.49, issue.4, pp.1479-1497, 2011.
DOI : 10.1137/070704290

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European Journal of Applied Mathematics, vol.11, issue.02, pp.191-224, 2008.
DOI : 10.1088/0266-5611/23/5/016

L. Dimet, F. Shutyaev, V. Gejadze, and I. , On optimal solution error in variational data assimilation: theoretical aspects, Russian Journal of Numerical Analysis and Mathematical Modelling, vol.21, issue.2, pp.139-152, 2006.
DOI : 10.1515/156939806776369492
URL : https://hal.archives-ouvertes.fr/inria-00391910

K. Liu, Locally Distributed Control and Damping for the Conservative Systems, SIAM Journal on Control and Optimization, vol.35, issue.5, pp.1574-1590, 1997.
DOI : 10.1137/S0363012995284928

D. Luenberger, Observing the State of a Linear System, IEEE Transactions on Military Electronics, vol.8, issue.2, pp.74-80, 1964.
DOI : 10.1109/TME.1964.4323124

P. Moireau, D. Chapelle, L. Tallec, and P. , Joint state and parameter estimation for distributed mechanical systems, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.6-8, pp.6-8659, 2008.
DOI : 10.1016/j.cma.2007.08.021
URL : https://hal.archives-ouvertes.fr/hal-00175623

K. Phung and X. Zhang, Time Reversal Focusing of the Initial State for Kirchhoff Plate, SIAM Journal on Applied Mathematics, vol.68, issue.6, pp.1535-1556, 2008.
DOI : 10.1137/070684823

K. Ramdani, M. Tucsnak, and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers, Automatica, vol.46, issue.10, pp.1616-1625, 2010.
DOI : 10.1016/j.automatica.2010.06.032
URL : https://hal.archives-ouvertes.fr/hal-00529834

D. Salamon, Realization theory in Hilbert space, Mathematical Systems Theory, vol.15, issue.1, pp.147-164, 1989.
DOI : 10.1007/BF02088011

V. Shutyaev and I. Gejadze, Adjoint to the Hessian derivative and error covariances in variational data assimilation, Russian Journal of Numerical Analysis and Mathematical Modelling, vol.26, issue.2, pp.179-188, 2011.
DOI : 10.1515/rjnamm.2011.010

A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic PDEs, Systems & Control Letters, vol.54, issue.7, pp.613-625, 2005.
DOI : 10.1016/j.sysconle.2004.11.001

O. Staffans and G. Weiss, Transfer functions of regular linear systems. II. The system operator and the Lax-Phillips semigroup, Transactions of the American Mathematical Society, vol.354, issue.08, pp.3229-3262, 2002.
DOI : 10.1090/S0002-9947-02-02976-8

O. Staffans and G. Weiss, Transfer Functions of Regular Linear Systems Part III: Inversions and Duality, Integral Equations and Operator Theory, vol.49, issue.4, pp.517-558, 2004.
DOI : 10.1007/s00020-002-1214-8

J. Teng, G. Zhang, and S. Huang, Some theoretical problems on variational data assimilation, Applied Mathematics and Mechanics, vol.23, issue.1, pp.581-591, 2007.
DOI : 10.1007/s10483-007-0510-2

M. Tucsnak and G. Weiss, Observation and control for operator semigroups Birkhäuser Verlag, Basel 40 Weiss G (1989) The, representation of regular linear systems on Hilbert spaces, Control and estimation of distributed parameter systems (Vorau, pp.401-416, 1988.

G. Weiss, Regular linear systems with feedback, Mathematics of Control, Signals, and Systems, vol.342, issue.1, pp.23-57, 1994.
DOI : 10.1007/BF01211484

G. Weiss, Transfer functions of regular linear systems. I. Characterizations of regularity, Transactions of the American Mathematical Society, vol.342, issue.2, pp.827-854, 1994.
DOI : 10.1090/S0002-9947-1994-1179402-6

G. Weiss and R. Curtain, Exponential Stabilization of a Rayleigh Beam Using Collocated Control, IEEE Transactions on Automatic Control, vol.53, issue.3, pp.643-654, 2008.
DOI : 10.1109/TAC.2008.919849

G. Weiss and R. Rebarber, Optimizability and Estimatability for Infinite-Dimensional Linear Systems, SIAM Journal on Control and Optimization, vol.39, issue.4, pp.1204-1232, 2000.
DOI : 10.1137/S036301299833519X

G. Weiss, O. Staffans, and M. Tucsnak, Well-posed linear systems?a survey with emphasis on conservative systems, Int J Appl Math Comput Sci, vol.11, issue.1, pp.7-33, 2001.

X. Zhang, C. Zheng, and E. Zuazua, Time discrete wave equations: Boundary observability and control, Discrete and Continuous Dynamical Systems, vol.23, issue.1/2, pp.571-604, 2009.
DOI : 10.3934/dcds.2009.23.571