C. 4. Quelques, . Variations, . Des, and . Figure, 16 ? Erreur relative entre le système et son observateur au cours du temps, puis état du système et de l'observateur à l'instant T = 20

C. Aboky, G. Sallet, and J. Vivalda, Observers for Lipschitz non-linear systems, International Journal of Control, vol.75, issue.3, pp.75-204, 2002.
DOI : 10.1109/9.53535

C. Alves, A. L. Silvestre, T. Takahashi, and M. Tucsnak, Solving Inverse Source Problems Using Observability. Applications to the Euler???Bernoulli Plate Equation, SIAM Journal on Control and Optimization, vol.48, issue.3, pp.1632-1659, 2009.
DOI : 10.1137/080725635

URL : https://hal.archives-ouvertes.fr/hal-00283402

W. Arendt, Vector-valued laplace transforms and cauchy problems, Israel Journal of Mathematics, vol.36, issue.2, pp.327-352, 1987.
DOI : 10.1007/978-3-642-96439-8

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, J. Blum, and M. Nodet, Diffusive Back and Forth Nudging algorithm for data assimilation, Comptes Rendus Mathematique, vol.349, issue.15-16, pp.349-849, 2011.
DOI : 10.1016/j.crma.2011.07.004

URL : https://hal.archives-ouvertes.fr/inria-00617571

G. A. Baker, Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations, SIAM Journal on Numerical Analysis, vol.13, issue.4, pp.564-576, 1976.
DOI : 10.1137/0713048

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. Baroun and B. Jacob, Admissibility and observability of observation operators for semilinear problems, Integral Equations Operator Theory, pp.1-20, 2009.

H. Barucq, Étude asymptotique du système de Maxwell avec conditions aux limites absorbantes, 1993.

H. Barucq and B. Hanouzet, Étude asymptotique du système de Maxwell avec la condition aux limites absorbante de Silver-Müller. II, C. R. Acad. Sci. Paris Sér. I Math, pp.316-1019, 1993.

J. Blum, F. Dimet, and I. M. Navon, Data assimilation for geophysical fluids, in Handbook of numerical analysis, XIV. Special volume : computational methods for the atmosphere and the oceans, pp.385-441, 2009.

S. Bonnabel, P. Martin, and P. Rouchon, Non-Linear Symmetry-Preserving Observers on Lie Groups, IEEE Transactions on Automatic Control, vol.54, issue.7, pp.1709-1713, 2009.
DOI : 10.1109/TAC.2009.2020646

S. Bonnabel, M. Mirrahimi, and P. Rouchon, Observer-based Hamiltonian identification for quantum systems, Automatica, vol.45, issue.5, pp.1144-1155, 2009.
DOI : 10.1016/j.automatica.2008.12.007

URL : https://hal.archives-ouvertes.fr/hal-00447800

S. Bonnabel and P. Rouchon, On Invariant Observers, Lecture Notes in Control and Inform. Sci, vol.322, pp.53-65, 2005.
DOI : 10.1007/11529798_4

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

N. Carmichael, A. J. Pritchard, and M. D. Quinn, State and parameter estimation for nonlinear systems, Applied Mathematics & Optimization, vol.79, issue.1, pp.133-161, 1982.
DOI : 10.1007/BF01460122

T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, of Mathématiques & Applications (Paris) [Mathematics and Applications], 1990.

M. Cessenat, Mathematical methods in electromagnetism, of Series on Advances in Mathematics for Applied Sciences Linear theory and applications, 1996.
DOI : 10.1142/2938

G. Ciccarella, M. D. Mora, and A. Germani, A Luenberger-like observer for nonlinear systems, International Journal of Control, vol.6, issue.3, pp.537-556, 1993.
DOI : 10.1016/0167-6911(87)90021-1

N. Cîndea, S. Micu, and M. Tucsnak, An approximation method for exact controls of vibrating systems, SIAM J. Control Optim, 2011.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, of Applied Mathematical Sciences, 1998.

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 : 10.1007/BF01211519

S. Cox and E. Zuazua, The rate at which energy decays in a damped String, Communications in Partial Differential Equations, vol.67, issue.1-2, pp.213-243, 1994.
DOI : 10.1080/03605309408821015

N. Cîndea, Problèmes inverses et contrôlabilité avec applications en élasticité et IRM, 2010.

N. Cîndea and M. Tucsnak, Fast and strongly localized observation for a perturbed plate equation, in Optimal control of coupled systems of partial differential equations, Internat. Ser. Numer. Math, vol.158, pp.73-83, 2009.

M. D. Mora, A. Germani, and C. Manes, A state observer for nonlinear dynamical systems, Proceedings of the Second World Congress of Nonlinear Analysts, pp.4485-4496, 1996.
DOI : 10.1016/S0362-546X(97)00184-3

R. Dautray and J. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques Tome 3, Collection du Commissariat à l'Énergie Atomique : Série Scientifique. [Collection of the Atomic Energy Commission : Science Series, 1985.

S. Dolecki and D. L. Russell, A General Theory of Observation and Control, SIAM Journal on Control and Optimization, vol.15, issue.2, pp.185-220, 1977.
DOI : 10.1137/0315015

G. E. Dullerud and F. Paganini, A Course in Robust Control Theory, of Texts in Applied Mathematics, 2000.
DOI : 10.1007/978-1-4757-3290-0

M. Eller, J. E. Lagnese, and S. Nicaise, Stabilization of heterogeneous maxwell's equations by linear or nonlinear boundary feedback, Electron. J. Differential Equations, vol.26, issue.21, p.pp. (electronic), 2002.

M. M. Eller and J. E. Masters, Exact Boundary Controllability of Electromagnetic Fields in a General Region, Applied Mathematics and Optimization, vol.45, issue.1, pp.99-123, 2002.
DOI : 10.1007/s00245-001-0030-x

S. Ervedoza, C. Zheng, and E. Zuazua, On the observability of time-discrete conservative linear systems, Journal of Functional Analysis, vol.254, issue.12, pp.3037-3078, 2008.
DOI : 10.1016/j.jfa.2008.03.005

URL : https://hal.archives-ouvertes.fr/hal-00681702

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.91-111, 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

F. Thomas and . Wu, Time-reversed acoustics, Rep. Prog. Phys, vol.63, pp.1933-1995, 2000.

I. Y. Gejadze, F. Dimet, and V. Shutyaev, 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

T. Geveci and B. Kok, The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation, Math. Comp, vol.44, pp.379-390, 1985.

B. Guo and Z. Zhao, On the convergence of an extended state observer for nonlinear systems with uncertainty, Systems & Control Letters, vol.60, issue.6, pp.420-430, 2011.
DOI : 10.1016/j.sysconle.2011.03.008

G. Haine and K. Ramdani, Observateurs it??ratifs en horizon fini. Application ?? la reconstruction de donn??es initiales pour des EDP d'??volution, Journal Europ??en des Syst??mes Automatis??s, vol.45, issue.7-10, pp.45-715, 2011.
DOI : 10.3166/jesa.45.715-724

T. Hamatsuka and H. Akashi, Stabilizability and observer design of a class of infinite dimensional linear systems, Lecture Notes in Control and Inform. Sci, vol.44, pp.809-830, 1982.
DOI : 10.1007/BFb0044432

T. Hamatsuka, A. Mo-'omen, and H. Akashi, Observer Design for Linear Contractive Control Systems on Hilbert Spaces, SIAM Journal on Control and Optimization, vol.19, issue.5, pp.586-594, 1981.
DOI : 10.1137/0319036

P. Hébrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Systems & Control Letters, vol.48, issue.3-4, pp.199-209, 2003.
DOI : 10.1016/S0167-6911(02)00265-7

V. I. Istr??escu, Fixed point theory An introduction, of Mathematics and its Applications, 1981.

K. Ito, K. Ramdani, and M. Tucsnak, A time reversal based algorithm for solving initial data inverse problems, Discrete Contin, Dyn. Syst. Ser. S, vol.4, pp.641-652, 2011.

C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal, Numér, vol.15, pp.41-78, 1981.

P. Joly and J. Rodríguez, Optimized higher order time discretization of second order hyperbolic problems: Construction and numerical study, Journal of Computational and Applied Mathematics, vol.234, issue.6, pp.1953-1961, 2010.
DOI : 10.1016/j.cam.2009.08.046

URL : https://hal.archives-ouvertes.fr/hal-00976330

R. Kalman and R. Bucy, New Results in Linear Filtering and Prediction Theory, Journal of Basic Engineering, vol.83, issue.1, pp.83-98, 1961.
DOI : 10.1115/1.3658902

V. Komornik, On the exact internal controllability of a petrowsky system, J. Math. Pures Appl, issue.9, pp.71-331, 1992.

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 : 10.1007/BF02921656

J. E. Lagnese, Exact Boundary Controllability of Maxwell???s Equations in a General Region, SIAM Journal on Control and Optimization, vol.27, issue.2, pp.374-388, 1989.
DOI : 10.1137/0327019

F. Dimet, V. Shutyaev, and I. Gejadze, On optimal solution error in variational data assimilation: theoretical aspects, Russian Journal of Numerical Analysis and Mathematical Modelling, vol.94, issue.4, pp.139-152, 2006.
DOI : 10.1002/qj.49711750206

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

A. V. Manzhirov and A. D. Polyanin, Handbook of integral equations, 2008.

P. Monk, Finite element methods for Maxwell's equations, Numerical Mathematics and Scientific Computation, 2003.
DOI : 10.1093/acprof:oso/9780198508885.001.0001

K. Morris and C. Navasca, Approximation of low rank solutions for linear quadratic control of partial differential equations, Computational Optimization and Applications, vol.1, issue.4, pp.93-111, 2010.
DOI : 10.1137/0317039

A. Münch, P. Pedregal, and F. Periago, Optimal design of the damping set for the stabilization of the wave equation, Journal of Differential Equations, vol.231, issue.1, pp.331-358, 2006.
DOI : 10.1016/j.jde.2006.06.009

S. Nicaise, Exact Boundary Controllability of Maxwell's Equations in Heterogeneous Media and an Application to an Inverse Source Problem, SIAM Journal on Control and Optimization, vol.38, issue.4, pp.1145-1170, 2000.
DOI : 10.1137/S0363012998344373

K. D. Phung, Stabilisation frontière du système de Maxwell avec la condition aux limites absorbante de Silver-Müller, C. R. Acad. Sci. Paris Sér. I Math, vol.320, pp.187-192, 1995.

K. D. 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

J. Puel, Une approche non classique d'un probl??me d'assimilation de donn??es, Comptes Rendus Mathematique, vol.335, issue.2, pp.335-161, 2002.
DOI : 10.1016/S1631-073X(02)02390-7

K. Ramdani, M. Tucsnak, and G. Weiss, Recovering the initial state of an infinitedimensional system using observers, Automatica, pp.46-1616, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00529834

P. Raviart and J. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles, 1998.

R. Rebarber and G. Weiss, Optimizability and estimatability for infinitedimensional linear systems, SIAM J. Control Optim, vol.39, pp.1204-1232, 2000.

D. L. Russell and R. I. , Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, in Differential games and control theory, Proc. NSF?CBMS Regional Res. Conf., Univ. Rhode Island Lecture Notes in Pure Appl. Math, vol.10, pp.291-319, 1973.

D. L. Russell and G. Weiss, A General Necessary Condition for Exact Observability, SIAM Journal on Control and Optimization, vol.32, issue.1, pp.1-23, 1994.
DOI : 10.1137/S036301299119795X

V. P. Shutyaev and I. Y. 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.26-179, 2011.
DOI : 10.1515/rjnamm.2011.010

M. Slemrod, The linear stabilization problem in Hilbert space, Journal of Functional Analysis, vol.11, issue.3, pp.334-345, 1972.
DOI : 10.1016/0022-1236(72)90073-0

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, A Physically Motivated Class of Scattering Passive Linear Systems, SIAM Journal on Control and Optimization, vol.50, issue.5, 2011.
DOI : 10.1137/110846403

J. J. Teng, G. Zhang, and S. X. Huang, Some theoretical problems on variational data assimilation, Applied Mathematics and Mechanics, vol.23, issue.1, pp.581-591, 2007.
DOI : 10.1034/j.1600-0870.1992.t01-3-00002.x

R. Triggiani, On the stabilizability problem in Banach space, Journal of Mathematical Analysis and Applications, vol.52, issue.3, pp.383-403, 1975.
DOI : 10.1016/0022-247X(75)90067-0

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 Control Optim. Calc. Var, vol.9, pp.247-274, 2003.

B. Lehrbücher, Birkhäuser Advanced Texts : Basel Textbooks, 2009.

J. M. Urquiza, Rapid Exponential Feedback Stabilization with Unbounded Control Operators, SIAM Journal on Control and Optimization, vol.43, issue.6, pp.2233-2244, 2005.
DOI : 10.1137/S0363012901388452

X. Zhang, Exact Internal Controllability of Maxwell's Equations, Applied Mathematics and Optimization, vol.41, issue.2, pp.155-170, 2000.
DOI : 10.1007/s002459911009

Q. Zhou, Exact internal controllability of Maxwell???s equations, Japan Journal of Industrial and Applied Mathematics, vol.24, issue.2, pp.245-256, 1997.
DOI : 10.1007/BF03167266

E. Zuazua, Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods, SIAM Review, vol.47, issue.2, pp.47-197, 2005.
DOI : 10.1137/S0036144503432862

. Résumé, état initial (ou final) d'un système infini-dimensionnel (typiquement un système gouverné par une Équation aux Dérivées Partielles (EDP) d'évolution) à partir de la connaissance partielle du système sur un intervalle de temps limité. Un champ d'applications dans lequel apparaît fréquemment ce type de problème d'identification est celui de la médecine