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A nonstandard approach to a data assimilation problem. (Une approche non classique d’un problème d’assimilation de données.) (French. Abridged English version) Zbl 1003.35042

Summary: We consider evolution problems such as diffusion-convection equations or linearized Navier-Stokes system that we would like to “predict” on a time interval \((T_0,T_0+T)\) but for which the initial value of the state variable is unknown. However, “measures” of the solutions are known on a time interval \((0,T_0)\). The classical approach in data assimilation is to look for the initial value at time 0 and this is known to be an ill-posed problem. Here we propose to look for the value of the state variable at time \(T_0\) (the final time of the “measures”) and we prove on some basic examples that this is a well-posed problem. We give a result of exact reconstruction of the value at \(T_0\) which is based on global Carleman inequalities and we give an approximation algorithm which uses classical optimal control auxiliary problems.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35R25 Ill-posed problems for PDEs
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[1] Blum, J.; Le Dimet, F. X., Assimilation de données pour les fluides géophysiques, Matapli, 67 (janvier 2002)
[2] Courtier, Ph.; Talagrand, O., Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory, Q. J. R. Meteorol. Soc., 113, 1311-1328 (1987)
[3] A. Dubova, A. Osses, J.-P. Puel, Exact controllability on trajectories for a transmission parabolic problem, à paraître dans ESAIM: Control Optim. Calc. Var., volume dedicated to the memory of Jacques-Louis Lions; A. Dubova, A. Osses, J.-P. Puel, Exact controllability on trajectories for a transmission parabolic problem, à paraître dans ESAIM: Control Optim. Calc. Var., volume dedicated to the memory of Jacques-Louis Lions
[4] Fursikov, A.; Imanuvilov, O., Controllability of Evolution Equations. Controllability of Evolution Equations, Lecture Notes Series, 34 (1996), RIM-GARC, Seoul National University · Zbl 0862.49004
[5] Imanuvilov, O., On exact controllability for the Navier-Stokes equations, ESAIM: Control Optim. Calc. Var., 3, 97-131 (1998), www.emath.fr/cocv/ · Zbl 1052.93502
[6] Imanuvilov, O., Remarks on exact controllability for Navier-Stokes equations, ESAIM: Control Optim. Calc. Var., 6, 39-72 (2001), www.emath.fr/cocv/ · Zbl 0961.35104
[7] O. Imanuvilov, J.-P. Puel, Global Carleman estimates for linearized Navier-Stokes equations and applications, en préparation; O. Imanuvilov, J.-P. Puel, Global Carleman estimates for linearized Navier-Stokes equations and applications, en préparation
[8] O. Imanuvilov, M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, UTMS 98-46; O. Imanuvilov, M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, UTMS 98-46 · Zbl 1065.35079
[9] F.X. Le Dimet, Une étude générale d’analyse objective variationnelle des champs météorologiques, Rapport scientifique LAMP 28, Université de Clermont II, BP 45, 63170 Aubière, France, 1980; F.X. Le Dimet, Une étude générale d’analyse objective variationnelle des champs météorologiques, Rapport scientifique LAMP 28, Université de Clermont II, BP 45, 63170 Aubière, France, 1980
[10] X Le Dimet, F.; Talagrand, O., Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus, 38A, 97-110 (1986)
[11] Lions, J.-L., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0203.09001
[12] J.-P. Puel, A nonstandard approach to a data assimilation problem, à paraı̂tre; J.-P. Puel, A nonstandard approach to a data assimilation problem, à paraı̂tre
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