Skip to main content

On Lie algebras of vectorfields, Lie algebras of differential operators and (nonlinear) filtering

  • Conference paper
  • First Online:
Geometry Symposium Utrecht 1980

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 894))

  • 418 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. Benes, to appear in Stochastics, 1980.

    Google Scholar 

  2. N. Bourbaki, Groupes et Algèbres de Lie, Ch. 1: Algèbres de Lie, Hermann, 1960.

    Google Scholar 

  3. R. W. Brockett, Remarks on Finite Dimensional Nonlinear Estimation, In: C. Lobry (ed), Analyse des Systèmes (Bordeaux 1978), 47–56, Astérisque 7576, Soc. Math. de France, 1980.

    Google Scholar 

  4. R. W. Brockett, Classification and Equivalence in Estimation Theory, Proc. 1979 IEEE CDC (Ft Lauderdale, Dec. 1979).

    Google Scholar 

  5. R. W. Brockett, J. M. C. Clark, The Geometry of the Conditional Density Equation, Proc. Int. Conf. on Analysis and Opt. of Stoch, Systems, Oxford 1978.

    Google Scholar 

  6. R. W. Brocket, Lectures on Lie Algebras in Systems and Filtering, In: M. Hazewinkel, J. C. Willems (eds), Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel Publ. Cy., to appear 1981.

    Google Scholar 

  7. J. M. C. Clark, An Introduction to Stochastic Differential Equations on Manifolds, In: D. Q. Mayne, R. W. Brockett (eds), Geometric Methods in System Theory, Reidel, 1973, 131–149.

    Google Scholar 

  8. M. H. A. Davis, S. I. Marcus, An Introduction to Nonlinear Filtering, In: M. Hazewinkel, J. C. Willems (eds), Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel Publ. Cy, to appear, 1981.

    Google Scholar 

  9. M. Demazure. Classification des Algèbres de Lie Filtrés, Séminaire Bourbaki 1966/1967, Exp. 326, Benjamin, 1967.

    Google Scholar 

  10. L. Gillman, M. Jerison, Rings of Continuous Functions, V. Nostrand, 1960.

    Google Scholar 

  11. C. Godbillon, Cohomologie d'Algèbres de Lie de Champ de Vecteurs Formels, Séminaire Bourbaki 1972/1973, Exposé 421, Springer LNM 383, 1974.

    Google Scholar 

  12. P. de la Harpe, H. Omori, About Interactions Between Banach-Lie Groups and Finite Dimensional Manifolds, J. Math. Kyoto Univ. 12, 3 (1972), 543–570.

    MathSciNet  MATH  Google Scholar 

  13. M. Hazewinkel, On Deformations, Approximations and Nonlinear Filtering, to appear, Systems and Control Letters, 1, 1 (1981).

    Google Scholar 

    Google Scholar 

  14. M. Hazewinkel, S. I. Marcus, unpublished.

    Google Scholar 

  15. M. Hazewinkel, S. I. Marcus, Some Results and Speculations on the Role of Lie Algebras in Filtering. In: M. Hazewinkel, J. C. Willems (eds), Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel Publ. Cy, to appear 1981.

    Google Scholar 

  16. M. Hazewinkel, S. Marcus, On Lie Algebras and Finite Dimensional Filtering, submitted to Stochastics.

    Google Scholar 

  17. M. Hazewinkel, C.-H. Liu, S. I. Marcus, Some Examples of Lie Algebraic Structure in Nonlinear Estimation, In: Proc. JACC (San Francisco 1980), TP7-C.

    Google Scholar 

  18. M. Hazewinkel, S. I. Marcus, H. J. Sussmann, Nonexistence of Exact Finite Dimensional Filters for the Cubic Sensor Problem. In preparation.

    Google Scholar 

  19. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Acad. Press, 1978.

    Google Scholar 

  20. A. Joseph, Commuting Polynomials in Quantum Canonical Operators and Relizations of Lie Algebras, J. Math. Physics 13 (1972), 351–357.

    Article  MATH  Google Scholar 

  21. A. J. Krener, On the Equivalence of Control System and the Linearization of Nonlinear Systems, SIAM J. Control 11 (1973), 670–676.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. S. Krishnaprasad, S. I. Marcus, Some Nonlinear Filtering Problems Arising in Recursive Identification. In: M. Hazewinkel, J. C. Willems (eds), Stochastic systems: The Mathematics of Filtering and Identification and Applications, Reidel Publ. Cy, to appear 1981.

    Google Scholar 

  23. C.-H. Liu, S. I. Marcus, The Lie Algebraic Structure of a Class of Finite Dimensional Nonlinear Filters. In: “Filterdag Rotterdam 1980”, M. Hazewinkel (ed), Report 8011, Econometric Institute, Erasmus Univ., Rotterdam, 1980.

    Google Scholar 

  24. S. I. Marcus, S. K. Mitter, D. Ocone, Finite Dimensional Nonlinear Estimation for a Class of Systems in Continuous and Discrete Time, Proc. Int. Conf. on Analysis and Optimization of Stochastic Systems, Oxford 1978.

    Google Scholar 

  25. S. K. Mitter, On the Analogy Between the Mathematical Problems of Nonlinear Filtering and Quantum Physics, Richerche di Automatica, to appear.

    Google Scholar 

  26. S. K. Mitter, Filtering Theory and Quantum Fields. In: C. Lobry (ed), Analyse des Systèmes (Bordeaux 1978), 199–206, Astérisque 7576, Soc. Math. de France, 1980.

    Google Scholar 

  27. S. K. Mitter, Lectures on Filtering and Quantum Theory. In: M. Hazewinkel, J. C. Willems (eds), Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel Publ. Cy, to appear 1981.

    Google Scholar 

  28. V. Pittie, Characteristic Classes of Foliations, Pitman, 1976.

    Google Scholar 

  29. I. Singer, S. Sternberg, On the Infinite Groups of Lie and Cartan, J. d'Analyse Math. 15 (1965), 1–114.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. J. Sussmann, Existence and Uniqueness of Minimal Realizations of Nonlinear Systems, Math. Syst. Theory 10 (1977), 263–284.

    Article  MathSciNet  Google Scholar 

  31. H. J. Sussmann, Rigorous Results on the Cubic Sensor Problem. In: M. Hazewinkel, J. C. Willems (eds), Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel Publ. Cy, to appear 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

E. Looijenga D. Siersma F. Takens

Additional information

Dedicated to my teacher and friend Nico Kuiper on the occasion of his 60th birthday with gratitude for the attitude to mathematics that he taught me by example and instruction.

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Hazewinskel, M. (1981). On Lie algebras of vectorfields, Lie algebras of differential operators and (nonlinear) filtering. In: Looijenga, E., Siersma, D., Takens, F. (eds) Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics, vol 894. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096225

Download citation

  • DOI: https://doi.org/10.1007/BFb0096225

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11167-2

  • Online ISBN: 978-3-540-38641-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics