Skip to main content
Springer Nature Link
Log in

On minimal splitting subspaces and markovian representations

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

Given a Hilbert spaceH, letH 1 andH 2 be two arbitrary subspaces. The problem of finding all minimal splitting subspaces ofH with respect toH 1 andH 2 is solved. This result is applied to the stochastic realization problem. Each minimal stochastic realization of a given vector processy defines a family of state spaces. It is shown that these families are precisely those families of minimal splitting subspaces (with respect to the past and the future ofy) which satisfy a certain growth condition.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Akaike, Markovian representation of stochastic processes by canonical variables,SIAM J. Control, 13 162–173 (1975).

    Google Scholar 

  2. B. D. O. Anderson, The inverse problem of stationary covariance generation,J. Statistical Physics, 1 133–147 (1969).

    Google Scholar 

  3. P. Faurre, Realizations markoviennes de processus stationnaires,Research Report no. 13,March 1973,IRIA (LABORIA), Le Chesnay, France.

  4. A. Lindquist and G. Picci, On the stochastic realization problem,SIAM J. Control, (to be published).

  5. A Lindquist and G. Picci, On the structure of minimal splitting subspaces in stochastic realization theory,Proc. 1977 Conf. Decision and Control, New Orleans, 42–48 1977.

  6. A Lindquist and G. Picci, A state-space theory for stationary stochastic processes,Proc. 21st Midwest Symposium on Circuits and Systems, Ames, Iowa, August 1978.

  7. H. P. McKean, Brownian motion with a several dimensional time,Theory Prob. Appl. (USSR), VIII 335–354 (1963).

    Google Scholar 

  8. J. Neveu,Processus Aléatoires Gaussiens, Presses de l'Université de Montréal, 1968.

  9. G. Picci, Stochastic realization of Gaussian processes,Proc. IEEE 64 112–122 (1976).

    Google Scholar 

  10. Yu. A. Rozanov, On two selected topics connected with stochastic systems theory,Applied Mathematics & Optimization, 3 73–80 (1976).

    Google Scholar 

  11. Yu. A. Rozanov,Stationary Random Processes, Holden-Day, San Francisco, 1967.

    Google Scholar 

  12. G. Ruckebusch, Representations Markoviennes de processus Gaussiens Stationnaires,Ph.D Thesis, Univ. of Paris VI, 1975.

  13. G. Ruckebusch, Représentations Markoviennes de Processus Gaussiens Stationnaires,C. R. Acad. Sc. Paris, Series A, 282 649–651 (1976).

    Google Scholar 

  14. G. Ruckebusch, Représentations Markoviennes de Processus Gaussiens Stationnaires et Applications Statistiques,Journées de Statistique des Processus Stochastiques, Grenoble, June 1977, Springer Verlag Lecture Notes in Mathematics 636.

  15. G. Ruckebusch, A state space approach to the stochastic realization problem,Proc. 1978 Intern. Symp. Circuits and Systems, New York.

  16. J. C. Willems, Dissipative dynamical systems, Part II: Linear systems with quadratic supply rates,Archive for Rational Mechanics and Analysis, 45, 352–393 (1972).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, 40506, Lexington, Kentucky, USA

    Anders Lindquist

  2. Laboratorio per Ricerche di Dinamica dei Sistemi e di Elettronica Biomedica, Consiglio Nazionale delle Ricerche, 35100, Padova, Italy

    Giorgio Picci

  3. Centre de Mathematiques Appliquées, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Guy Ruckebusch

Authors
  1. Anders Lindquist
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Picci
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guy Ruckebusch
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research sponsored by the Air Force Office of Scientific Research under Grant No. AFOSR-78-3519.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lindquist, A., Picci, G. & Ruckebusch, G. On minimal splitting subspaces and markovian representations. Math. Systems Theory 12, 271–279 (1978). https://doi.org/10.1007/BF01776578

Download citation

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01776578

Keywords