Skip to main content
Log in

Asymptotic behavior of solutions of pulse systems with small parameter and markov switchings. I. Uniform boundedness of solutions

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We consider pulse systems with Markov switchings. We study the problems of uniform boundedness of solutions of these systems and the stability of the systems with respect to the limit equation.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. M. Samoilenko and N. A. Perestyuk,Differential Equations with Pulse Influence [in Russian], Vyshcha Shkola, Kiev 1987.

    Google Scholar 

  2. V. V. Anisimov,Stochastic Processes with Discrete Component [in Russian], Naukova Dumka, Kiev 1988.

    MATH  Google Scholar 

  3. A. M. Samoilenko and A. N. Stanzhitskii, “On fluctuations in the averaging scheme for differential equations with random pulse influence,”Ukr. Mat. Zh,41, No. 5, 631–641 (1989).

    Article  MathSciNet  Google Scholar 

  4. A. V. Skorokhod,Asymptotic Methods in the Theory of Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1987).

    MATH  Google Scholar 

  5. R. Z. Khas’minskii,Stability of Systems of Differential Equations under Random Perturbations of Their Parameters [in Russian], Nauka, Moscow 1969.

    Google Scholar 

  6. M. L. Sverdan and E. F. Tsar’kov,Stability of Stochastic Pulse Systems [in Russian], Riga Technical University, Riga 1994.

    Google Scholar 

  7. E. F. Tsar’kov,Random Perturbations of Differential-Functional Equations [in Russian], Zinatne, Riga 1989.

    MATH  Google Scholar 

  8. G. Blankenship and G. C. Papanicolaou, “Stability and control of stochastic systems with wide-band noise disturbances,”SI AM JAppL Math.,34, 437–476 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  9. V. S. Korolyuk,Averaging and Stability of Dynamical Systems with Rapid Markov Switchings, No. 90167, Umea University, Umea(1991).

  10. Ye. Tsarkov,Averaging in Dynamical Systems with Markov Jumps, No. 282, Institute of Dynamical Systems, Bremen University, Bremen (1993).

  11. V. V. Anisimov, “Averaging principle for switching processes,”Teor. Ver. Mat. Statist., No. 46, 3–13 (1992).

    Google Scholar 

  12. V. V. Anisimov, “Asymptotic methods in the theory of switching processes,” in:Proceedings of the 2nd Ukrainian-HungarianConference (Mukachevo, Ukraine) [in Russian], TBiMC, Kiev (1995), pp. 1–28.

    Google Scholar 

  13. V. V. Anisimov, “Switching processes: averaging principle, diffusion approximation and application,”Acta AppL Math.,40, 95–191 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  14. I.I. Gikhman and A. V. Skorokhod,Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev 1968.

    Google Scholar 

  15. E. B. Dynkin,Markov Processes [in Russian], Fizmatgiz, Moscow 1963.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sverdan, M.L., Tsar’kov, E.F. & Yasyns’kyi, V.K. Asymptotic behavior of solutions of pulse systems with small parameter and markov switchings. I. Uniform boundedness of solutions. Ukr Math J 48, 1561–1573 (1996). https://doi.org/10.1007/BF02377824

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02377824

Keywords

Navigation