Control Operators and Fundamental Control Functions in Data Assimilation

  • V. Shutyaev
Conference paper
Part of the NATO Science Series book series (NAIV, volume 26)


Consider mathematical model of a physical pro cess that is described by the evolution problem
$$ \left\{ {\begin{array}{*{20}c} {\frac{{d\phi }} {{dt}} + A(t)\phi = f,t \in (0,T)} \\ {\phi \left| {_{t = 0} = u,} \right.} \\ \end{array} } \right. $$
where ϕ = ϕ(t) is the unknown function belonging for any A(t) is an operatior (generally, non linear) acting for each t in the Hilbert space X, uX, and f = f(t) is a prescribed function.


Data Assimilation Control Operator Regul Ariz Ation Method Kronecker Delta Approximate Cont Rollability 
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  1. Agoshkov, V.I. and Marchuk, G.I. (1993) On solvability and numerical solution of data assimilation problems, Russ. J. Numer. Anal. Math. Modelling 8, 1–16.CrossRefGoogle Scholar
  2. Bellman, R. (1957) Dynamic Programming, Princeton Univ. Press, New Jersey.Google Scholar
  3. Glowinski, R. and J.-L. Lions, J.-L. (1994) Exact and approximate controllability for distributed parameter systems, Acta Numerica 1, 269–378.CrossRefGoogle Scholar
  4. Kravaris, C. and Seinfeld, J.H. (1985) Identification of parameters in distributed parameter systems by regularization, SIAM J. Control and Optimization 23, 217.CrossRefGoogle Scholar
  5. Kurzhanskii, A.B. and Khapalov, A.Yu. (1991) An observation theory for distributed-parameter systems, J. Math. Syst. Estimat. Control 1, 389–440.Google Scholar
  6. Le Dimet, F.-X. and Talagrand, O. (1986) Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus 38A, 97–110.CrossRefGoogle Scholar
  7. Lions, J.-L. (1968) Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris.Google Scholar
  8. Lions, J.-L. (1997) On controllability of distributed systems, Proc. Natl. Acad. Sci. USA 94, 4828–4835.CrossRefGoogle Scholar
  9. Marchuk, G.I. (1995) Adjoint Equations and Analysis of Complex Systems, Kluwer, Dordrecht.Google Scholar
  10. Marchuk, G.I., Agoshkov, V.I., and Shutyaev, V.P. (1996) Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, CRC Press Inc., New York.Google Scholar
  11. Marchuk, G.I. and Penenko, V.V. (1978) Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment, in G.I. Marchuk (ed.), Modelling and Optimization of Complex Systems: Proc. of the IFIP-TC7 Working conf., Springer, New York, pp. 240–252.Google Scholar
  12. Marchuk, G.I. and Shutyaev, V.P. (1994) Iteration methods for solving a data assimilation problem, Russ. J. Numer. Anal. Math. Modelling 9, 265–279.CrossRefGoogle Scholar
  13. Marchuk, G., Shutyaev, V., and Zalesny V. (2001) Approaches to the solution of data assimilation problems, in J.L. Menaldi, E. Rofman, and A. Sulem (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, pp.489–497.Google Scholar
  14. Marchuk, G.I. and Zalesny, V.B. (1993) A numerical technique for geophysical data assimilation problem using Pontryagin’s principle and splitting-up method, Russ. J. Numer. Anal. Math. Modelling 8, 311–326.CrossRefGoogle Scholar
  15. Navon, I.M. (1986) A review of variational and optimization methods in meteorology, in Y.K. Sasaki (ed.), Variational Methods in Geosciences, Elsevier, New York, pp. 29–34.Google Scholar
  16. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mischenko, E.F. (1962) The Mathematical Theory of Optimal Processes, John Wiley, New York.Google Scholar
  17. Sasaki, Y.K. (1970) Some basic formalisms in numerical variational analysis, Monthly Weather Review 98, 857–883.Google Scholar
  18. Shutyaev, V.P. (1995) Some properties of the control operator in the problem of data assimilation and iterative algorithms, Russ. J. Numer. Anal. Math. Modelling 10, 357–371.CrossRefGoogle Scholar
  19. Shutyaev, V.P. (2001) Control Operators and Iterative Algorithms for Variational Data Assimilation Problems, Nauka, Moscow.Google Scholar
  20. Tikhonov, A.N. (1963) On the solution of ill-posed problems and the regularization method, Dokl. Akad. Nauk SSSR 151, 501–504.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • V. Shutyaev
    • 1
  1. 1.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

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