H. Abou-kandil, G. Freiling, V. Ionescu, and G. Jank, Matrix Riccati Equations in Control and Systems Theory, 2003.
DOI : 10.1007/978-3-0348-8081-7

J. I. Allen, M. Eknes, and G. Evensen, An Ensemble Kalman Filter with a complex marine ecosystem model: hindcasting phytoplankton in the Cretan Sea, Annales Geophysicae, vol.21, issue.1, pp.1-13, 2002.
DOI : 10.5194/angeo-21-399-2003

URL : https://hal.archives-ouvertes.fr/hal-00316987

B. D. Anderson, Stability properties of Kalman-Bucy filters, Journal of the Franklin Institute, vol.291, issue.2, pp.137-144, 1971.
DOI : 10.1016/0016-0032(71)90016-0

J. L. Anderson, An Ensemble Adjustment Kalman Filter for Data Assimilation, Monthly Weather Review, vol.129, issue.12, pp.2884-2903, 2001.
DOI : 10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.5.9952

J. L. Anderson, A Local Least Squares Framework for Ensemble Filtering, Monthly Weather Review, vol.131, issue.4, pp.634-642, 2003.
DOI : 10.1175/1520-0493(2003)131<0634:ALLSFF>2.0.CO;2

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.6543

W. Auzinger, R. Frank, and G. Kirlinger, Modern convergence theory for stiff initial-value problems, Journal of Computational and Applied Mathematics, vol.45, issue.1-2, pp.5-16, 1993.
DOI : 10.1016/0377-0427(93)90260-I

URL : http://doi.org/10.1016/0377-0427(93)90260-i

J. S. Baras, A. Bensoussan, and M. R. James, Dynamic Observers as Asymptotic Limits of Recursive Filters: Special Cases, SIAM Journal on Applied Mathematics, vol.48, issue.5, pp.1147-1158, 1988.
DOI : 10.1137/0148068

URL : http://www.dtic.mil/get-tr-doc/pdf?AD=ADA187578

T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error Arxiv, pp.1311-1831, 2014.
DOI : 10.1098/rspa.2014.0168

URL : http://rspa.royalsocietypublishing.org/content/royprsa/470/2167/20140168.full.pdf

S. Bittanti, A. J. Laub, and J. C. Willems, The Riccati Equation, Communications and Control Engineering Series, 1991.
DOI : 10.1007/978-3-642-58223-3

F. Bolley, A. Guillin, and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.44, issue.5, pp.867-884, 2010.
DOI : 10.1051/m2an/2010045

URL : https://hal.archives-ouvertes.fr/hal-00392397

P. Bougerol and S. Fakhfakh, A note on the stability of the Kalman-bucy filter with randomly time-varying parameters, Journal of Mathematical Sciences, vol.35, issue.1, pp.28-33, 1996.
DOI : 10.1007/BF02367952

R. W. Brockett, Finite Dimensional Linear Systems, 1970.
DOI : 10.1137/1.9781611973884

R. S. Bucy, Nonlinear filtering theory, IEEE Transactions on Automatic Control, vol.10, issue.2, pp.198-198, 1965.
DOI : 10.1109/TAC.1965.1098109

R. S. Bucy, Global theory of the Riccati equation, Journal of Computer and System Sciences, vol.1, issue.4, pp.349-361, 1967.
DOI : 10.1016/S0022-0000(67)80025-4

G. Burgers, P. J. Van-leeuwen, and G. Evensen, Analysis Scheme in the Ensemble Kalman Filter, Monthly Weather Review, vol.126, issue.6, pp.1719-1724, 1998.
DOI : 10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2

P. Cattiaux, F. Malrieu, and A. Guillin, Probabilistic approach for granular media equations in the non uniformly convex case. Probability Theory and Related Fields, pp.19-40, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00021591

F. M. Dannan, Matrix and operator inequalities, Ineq. Pure. and Appl. Math, vol.2, issue.3, p.34, 2001.

P. , D. Moral, and A. , Doucet Interacting Markov Chain Monte Carlo Methods For Solving Nonlinear Measure-Valued Equations, The Annals of Applied Probability, vol.20, issue.2, pp.593-639, 2008.

P. and D. Moral, Feynman-Kac formula. Genealogical and interacting particle approximations, Series: Probability and Applications, 2004.
URL : https://hal.archives-ouvertes.fr/hal-01289272

P. and D. Moral, Mean field simulation for Monte Carlo integration, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00932211

P. Del-moral and J. Tugaut, Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions. https://hal.archives-ouvertes, p.798813, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00798813

B. Dyda and J. Tugaut, Exponential rate of convergence independent from the dimension in a mean-field system of particles, Probability and Mathematical Statistics, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01119526

D. S. Bernstein, Inequalities for the Trace of Matrix Exponentials, SIAM Journal on Matrix Analysis and Applications, vol.9, issue.2, pp.156-158, 1988.
DOI : 10.1137/0609012

G. Einicke, Continuous-Time Minimum-Variance Filtering, Smoothing, Filtering and Prediction -Estimating The Past, Present and Future, pp.978-953, 2012.
DOI : 10.5772/39251

URL : https://www.intechopen.com/download/pdf/29942

M. Eknes and G. Evensen, An Ensemble Kalman filter with a 1-D marine ecosystem model, Journal of Marine Systems, vol.36, issue.1-2, pp.75-100, 2002.
DOI : 10.1016/S0924-7963(02)00134-3

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.3107

G. Evensen, Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J Geophys Res, vol.9910, issue.C5, pp.143-162, 1994.
DOI : 10.1029/94jc00572

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.930

G. Evensen, The Ensemble Kalman Filter: theoretical formulation and practical implementation, Ocean Dynamics, vol.53, issue.4, pp.343-367, 2003.
DOI : 10.1007/s10236-003-0036-9

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.5.6990

G. Evensen, Data assimilation : The ensemble Kalman filter, 2007.
DOI : 10.1007/978-3-642-03711-5

G. Evensen, J. Hove, H. C. Meisingset, E. Reiso, and K. S. Seim, Using the EnKF for assisted history matching of a North Sea Reservoir Model SPE, p.106184, 2007.

M. Fiedler, Special matrices and their applications in numerical mathematics, 1986.
DOI : 10.1007/978-94-009-4335-3

G. Gottwald and A. J. Majda, A mechanism for catastrophic filter divergence in data assimilation for sparse observation networks, Nonlinear Processes in Geophysics, vol.20, issue.5, pp.705-712, 2013.
DOI : 10.5194/npg-20-705-2013

J. Harlim and B. Hunt, Local Ensemble Transform Kalman Filter: An Efficient Scheme for Assimilating Atmospheric Data, 2005.

P. Houtekamer and H. L. Mitchell, Data Assimilation Using an Ensemble Kalman Filter Technique, Monthly Weather Review, vol.126, issue.3, pp.796-811, 1998.
DOI : 10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2

URL : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.3.1706&rep=rep1&type=pdf

A. Ilchmann, D. H. Owens, and D. , Sufficient conditions for stability of linear time-varying systems, Systems & Control Letters, vol.9, issue.2, pp.157-163, 1987.
DOI : 10.1016/0167-6911(87)90022-3

C. J. Johns and J. Mandel, A two-stage ensemble Kalman filter for smooth data assimilation Conference on New Developments of Statistical Analysis in Wildlife, Fisheries, and Ecological Research, Environmental and Ecological Statistics. Special issue CCM Report, vol.221, 2005.

D. Kelly, A. J. Majda, and X. T. Tong, Concrete ensemble Kalman filters with rigorous catastrophic filter divergence, To apper in Proc. Natl. Acad. Sci, 2016.
DOI : 10.1073/pnas.0602385103

URL : http://www.pnas.org/content/112/34/10589.full.pdf

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Journal of Dynamic Systems, Measurement, and Control, vol.96, issue.3, 1972.
DOI : 10.1115/1.3426828

J. M. Krause and K. S. Kumar, An alternate stability analysis framework for adaptive control, Systems & Control Letters, vol.7, issue.1, pp.19-24, 1986.
DOI : 10.1016/0167-6911(86)90096-4

V. Kresimir, Damped Oscillations of Linear Systems A Mathematical Introduction, Lecture Notes in Mathematics, vol.2023, 2011.

G. Kreisselmeier, An approach to stable indirect adaptive control, Automatica, vol.21, issue.4, pp.425-431, 1985.
DOI : 10.1016/0005-1098(85)90078-0

H. J. Kushner, On the differential equations satisfied by conditional probability densities of Markov processes, with applications, J. SIAM Control Ser. A, vol.2, issue.1, pp.106-119, 1964.

P. Lancaster and L. Rodman, The Algebraic Riccati Equation, 1995.

K. J. Law, H. Tembine, and R. Tempone, Deterministic mean field ensemble Kalman filtering. Arxiv 1409, p.628, 2015.
DOI : 10.1137/140984415

URL : http://arxiv.org/pdf/1409.0628

F. Le-gland, V. Monbet, and V. D. Tran, Large sample asymptotics for the ensemble Kalman filter The Oxford Handbook of Nonlinear Filtering, pp.598-631, 2011.

K. A. Lisaeter, J. Rosanova, and G. Evensen, Assimilation of ice concentration in a coupled ice?ocean model, using the Ensemble Kalman filter, Ocean Dynamics, vol.53, issue.4, pp.368-388, 2003.
DOI : 10.1007/s10236-003-0049-4

J. Lawson and Y. Lim, A Birkhoff Contraction Formula with Applications to Riccati Equations, SIAM Journal on Control and Optimization, vol.46, issue.3, pp.930-951, 2007.
DOI : 10.1137/050637637

E. Kalnay, Atmospheric modeling, data assimilation, and predictability, 2003.
DOI : 10.1017/CBO9780511802270

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.88.5725

D. G. Luenberger, Observers for multivariable systems, IEEE Transactions on Automatic Control, vol.11, issue.2, pp.190-199, 1966.
DOI : 10.1109/TAC.1966.1098323

C. Moler and C. Van-loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, vol.20, issue.4, 2003.
DOI : 10.1137/1020098

G. Naevdal, L. M. Johnsen, S. I. Aanonsen, and E. H. Vefring, Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter, SPE Journal, vol.10, issue.01, pp.66-74, 2005.
DOI : 10.2118/84372-PA

A. J. Majda and J. Harlim, Filtering complex turbulent systems, 2012.
DOI : 10.1017/CBO9781139061308

F. Malrieu, Logarithmic Sobolev Inequalities for some nonlinear PDE's . Stochastic Processes and their Applications, pp.109-132, 2001.
DOI : 10.1016/s0304-4149(01)00095-3

URL : http://doi.org/10.1016/s0304-4149(01)00095-3

H. P. Mckean-jr, A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS, Proc. Nat. Acad. Sci. USA, pp.1907-1911, 1966.
DOI : 10.1073/pnas.56.6.1907

A. J. Majda and J. Harlim, Catastrophic filter divergence in filtering nonlinear dissipative systems, Comm. Math. Sci, vol.8, pp.27-43, 2008.

S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean- Vlasov and Boltzmann models Probabilistic models for nonlinear partial differential equations, Lecture Notes in Mathematics, pp.1627-1996, 1996.

D. Ocone and E. Pardoux, Asymptotic Stability of the Optimal Filter with Respect to Its Initial Condition, SIAM Journal on Control and Optimization, vol.34, issue.1, pp.226-243, 1996.
DOI : 10.1137/S0363012993256617

E. Ott, B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich et al., A local ensemble Kalman filter for atmospheric data assimilation, Tellus A: Dynamic Meteorology and Oceanography, vol.56, issue.131, pp.415-428, 2004.
DOI : 10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2

M. A. Poubelle, I. R. Petersen, M. R. Gevers, and R. R. Bitmead, A miscellany of results of an equation of Count J. F. Riccati, IEEE Transactions on Automatic Control, vol.31, issue.7, pp.651-654, 1986.
DOI : 10.1109/TAC.1986.1104355

H. H. Rosenbrock, The Stability of Linear Time-dependent Control Systems???, Journal of Electronics and Control, vol.60, issue.1, pp.73-80, 1963.
DOI : 10.1080/00207216308937556

A. Seiler, G. Evensen, J. Skjervheim, J. Hove, and J. G. Vab, Using the EnKF for history matching and uncertainty quantification of complex reservoir models. in Computational Methods for Large Scale Inverse Problems and Quantification of Uncertainty, 2010.

J. Skjervheim, G. Evensen, S. I. Aanonsen, B. O. Ruud, and T. A. Johansen, Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter SPE, p.95789, 2005.
DOI : 10.2118/95789-ms

D. A. Snyder, On the Relation of Schatten Norms and the Thompson Metric, ArXiv, pp.1608-03301, 2016.

J. Sua, B. Lib, and W. H. , Chen On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs, pp.149-154, 2015.

E. D. Sontag, Mathematical Control Theory, 1998.

A. S. Sznitman, Topics in propagation of chaos, course given at the Saint-Flour Probability Summer School, Lecture Notes in Math, pp.164-251, 1464.

X. T. Tong, A. J. Majda, and D. Kelly, Nonlinear stability and ergodicity of ensemble based Kalman filters. Arxiv 1507, p.8307, 2015.

C. Van-loan, The Sensitivity of the Matrix Exponential, SIAM Journal on Numerical Analysis, vol.14, issue.6, pp.971-981, 1977.
DOI : 10.1137/0714065

X. Wen and W. H. Chen, Real-Time Reservoir Model Updating Using Ensemble Kalman Filter SPE-92991-MS, SPE Reservoir Simulation Symposium, 2005.
DOI : 10.2523/92991-ms

W. M. Wonham, On a Matrix Riccati Equation of Stochastic Control, SIAM Journal on Control, vol.6, issue.4, pp.681-697, 1968.
DOI : 10.1137/0306044

X. Yang, A Matrix Trace Inequality, Journal of Mathematical Analysis and Applications, vol.250, issue.1, pp.372-374, 2000.
DOI : 10.1006/jmaa.2000.7068

X. Yang, X. Yand, and K. L. Teo, A Matrix Trace Inequality, Journal of Mathematical Analysis and Applications, vol.263, issue.1, pp.327-333, 2001.
DOI : 10.1006/jmaa.2001.7613

M. I. Zelikin, ON THE THEORY OF THE MATRIX RICCATI EQUATION, Mathematics of the USSR-Sbornik, vol.73, issue.2, pp.970-984, 1991.
DOI : 10.1070/SM1992v073n02ABEH002549

A. D. Ziebur, by Analyzing $e^{At} $, SIAM Review, vol.12, issue.1, pp.98-102, 1970.
DOI : 10.1137/1012005