Some exact results on stability and growth of linear parameter excited stochastic systems

  • W. Kliemann
Part II: Research Reports
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 16)


We investigate the stability and growth of the linear parameter-excited stochastic system xt=At xt, which At is a stationary diffusion process. The interplay between stochastic systems and associated deterministic control systems allows us to derive results on the ergodicity of (xt/|xt|, At) and so on the growth of xt. Using an effective computation procedure the 2-dimensional case is solved completely.


Invariant Measure Stochastic System Nonlinear Stochastic System Switching Surface Return Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Arnold, L. Wihstutz, V. On the Stability and Growth of Real Noise Parameter-excited Linear Systems in: Kölzow, Kallianpur (eds) Proceedings of the Conference on Measure Theory and Stochastic Analysis, Oberwolfach July 1977 Springer Lecture Notes in Mathematics, to appear 1979Google Scholar
  2. [2]
    Benderski, M.M. Pastur, L.A. On the Asymptotic Behaviour of the Solutions of a Second Order Equation with Random Coefficients (russ.) Teor. Funkcii Funkcional. Anal. i Prilozen. Vyp 22 (1975) p. 3–14Google Scholar
  3. [3]
    Bhattacharya, R.N. Criteria for Recurrence and Existence of Invariant Measures for Multidimensional Diffusions Ann. Probability 6 (1978) p. 541–553Google Scholar
  4. [4]
    Blankenship, G. Stability of Linear Differential Equations with Random Coefficients IEEE Trans. AC 22 (1977) p. 834–838Google Scholar
  5. [5]
    Blankenship, G. Papanicolaou, G.C. Stability and Control of Stochastic Systems with Wide-Band Noise Disturbances I SIAM J. Appl. Math. 34 (1978) p. 437–476Google Scholar
  6. [6]
    Brockett, R.W. Parametrically Stochastic Linear Differential Equations Mathematical Programming Study 5 (1976) p. 8–21Google Scholar
  7. [7]
    Chasminskii, R.Z. Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems Theory Prob. Appl. 12 (1967) p. 167–172Google Scholar
  8. [8]
    Chasminskii, R.Z. Stability of Systems of Differential Equations with Random Perturbation of their Parameters (russ.) Nauka, Moscow (1969)Google Scholar
  9. [9]
    Dauer, J.P. Approximate Controllability of Nonlinear Systems with Restrained Controls J. Math. Ana. Appl. 46 (1974) p. 126–131Google Scholar
  10. [10]
    Hermann, R. Krener, A.J. Nonlinear Controllability and Observability IEEE Trans. AC 22 (1977) p. 728–740Google Scholar
  11. [11]
    Infante, E.F. On the Stability of some Linear Nonautonomous Random Systems ASME J. Appl. Math. 36 (1968) p. 7–12Google Scholar
  12. [12]
    Johnson, R.A. Ergodic Theory and Linear Differential Equations J. Diff. Equations 28 (1978) p. 23–34Google Scholar
  13. [13]
    Kliemann, W. Stationäre nichtlineare stochastische Systeme Ph. D. Thesis, Bremen (1979)Google Scholar
  14. [14]
    Kolmogoroff, A.N. Zur Umkehrbarkeit der statistischen Naturgesetze Math. Ann. 113 (1937) p. 766–772Google Scholar
  15. [15]
    Kunita, H. Supports of Diffusion Processes and Controllability Problems Proc. of Intern. Symp. SDE Kyoto (1976) p. 163–185Google Scholar
  16. [16]
    Lobry, C. Controllability of Nonlinear Systems on Compact Manifolds SIAM J. Control 12 (1974) p. 1–4Google Scholar
  17. [17]
    Lukes, D.L. Global Controllability of Nonlinear Systems SIAM J. Control 10 (1972) p. 112–126 11 (1973) p. 186Google Scholar
  18. [18]
    Maruyama, G. Tanaka, H. Ergodic Property of N-Dimensional Recurrent Markov Processes Mem. Fac. Sci. Kyushu Univ. Ser. A 13 (1959) p. 157–172Google Scholar
  19. [19]
    Rümelin, W. Stability and Growth of the Solution of ÿ+fty=0 where ft is a Positive Stationary Markov Process Transactions of the Eighth Prague Conference Volume B (1978) p. 149–161Google Scholar
  20. [20]
    Stroock, D.W. Varadhan, S.R.S. On the Support of Diffusion Processes with Applications to the Strong Maximum Principle Proc. Sixth Berkeley Symp. Mat. Stat. Prob. Vol 3 (1972) p. 333–359Google Scholar
  21. [21]
    Wihstutz, V. Ueber Stabilität und Wachstum von Lösungen linearer Differentialgleichungen mit stationären zufälligen Parametern Ph. D. Thesis Bremen (1975)Google Scholar
  22. [22]
    Yosida, K. Functional Analysis Springer, Berlin-Heidelberg-New York 4th Ed. (1974)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • W. Kliemann
    • 1
  1. 1.Forschungsgschwerpunkt "Dynamische Systeme"Univerität BremenGermany

Personalised recommendations