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 ,
Observers for Lipschitz non-linear systems, International Journal of Control, vol.75, issue.3, pp.75-204, 2002. ,
DOI : 10.1109/9.53535
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
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
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
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
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
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
Admissibility and observability of observation operators for semilinear problems, Integral Equations Operator Theory, pp.1-20, 2009. ,
Étude asymptotique du système de Maxwell avec conditions aux limites absorbantes, 1993. ,
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. ,
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. ,
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
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
On Invariant Observers, Lecture Notes in Control and Inform. Sci, vol.322, pp.53-65, 2005. ,
DOI : 10.1007/11529798_4
Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise . [Collection of Applied Mathematics for the Master's Degree, Théorie et applications. [Theory and applications], 1983. ,
State and parameter estimation for nonlinear systems, Applied Mathematics & Optimization, vol.79, issue.1, pp.133-161, 1982. ,
DOI : 10.1007/BF01460122
Introduction aux problèmes d'évolution semi-linéaires, of Mathématiques & Applications (Paris) [Mathematics and Applications], 1990. ,
Mathematical methods in electromagnetism, of Series on Advances in Mathematics for Applied Sciences Linear theory and applications, 1996. ,
DOI : 10.1142/2938
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
An approximation method for exact controls of vibrating systems, SIAM J. Control Optim, 2011. ,
Inverse acoustic and electromagnetic scattering theory, of Applied Mathematical Sciences, 1998. ,
Nonlinear observers in reflexive Banach spaces, ESAIM: Control, Optimisation and Calculus of Variations, vol.9, pp.67-104, 2003. ,
DOI : 10.1007/BF01211519
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
Problèmes inverses et contrôlabilité avec applications en élasticité et IRM, 2010. ,
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. ,
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
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. ,
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
A Course in Robust Control Theory, of Texts in Applied Mathematics, 2000. ,
DOI : 10.1007/978-1-4757-3290-0
Stabilization of heterogeneous maxwell's equations by linear or nonlinear boundary feedback, Electron. J. Differential Equations, vol.26, issue.21, p.pp. (electronic), 2002. ,
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
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
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
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
Time-reversed acoustics, Rep. Prog. Phys, vol.63, pp.1933-1995, 2000. ,
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
The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation, Math. Comp, vol.44, pp.379-390, 1985. ,
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
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
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
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
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
Fixed point theory An introduction, of Mathematics and its Applications, 1981. ,
A time reversal based algorithm for solving initial data inverse problems, Discrete Contin, Dyn. Syst. Ser. S, vol.4, pp.641-652, 2011. ,
Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal, Numér, vol.15, pp.41-78, 1981. ,
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
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
On the exact internal controllability of a petrowsky system, J. Math. Pures Appl, issue.9, pp.71-331, 1992. ,
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
Mathematics of thermoacoustic tomography, European Journal of Applied Mathematics, vol.11, issue.02, pp.191-224, 2008. ,
DOI : 10.1007/BF02921656
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
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
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
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
Handbook of integral equations, 2008. ,
Finite element methods for Maxwell's equations, Numerical Mathematics and Scientific Computation, 2003. ,
DOI : 10.1093/acprof:oso/9780198508885.001.0001
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
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
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
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. ,
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
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
Recovering the initial state of an infinitedimensional system using observers, Automatica, pp.46-1616, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-00529834
Introduction à l'analyse numérique des équations aux dérivées partielles, 1998. ,
Optimizability and estimatability for infinitedimensional linear systems, SIAM J. Control Optim, vol.39, pp.1204-1232, 2000. ,
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. ,
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
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
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
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
A Physically Motivated Class of Scattering Passive Linear Systems, SIAM Journal on Control and Optimization, vol.50, issue.5, 2011. ,
DOI : 10.1137/110846403
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
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
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. ,
Birkhäuser Advanced Texts : Basel Textbooks, 2009. ,
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
Exact Internal Controllability of Maxwell's Equations, Applied Mathematics and Optimization, vol.41, issue.2, pp.155-170, 2000. ,
DOI : 10.1007/s002459911009
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
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
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 ,