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Stability in impulsive systems with markov perturbations in averaging scheme. 3. Weak convergence of solutions of impulsive systems

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Abstract

For a stochastic dynamic system with a small parameter, the uniform boundedness of the p-th moment of the solution (p > 1), the weak convergence of the solution of the system to the solution of Ito stochastic differential equation, and the weak convergence of normalized deviations are proved. The stability of linear systems with a small parameter and Markov perturbations is analyzed.

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References

  1. P. Billingsley, Convergence of Probability Measures, Wiley (1968).

  2. I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications [in Russian], Naukova Dumka, Kyiv (1982).

    Google Scholar 

  3. I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Vol. 3, Nauka, Moscow (1975).

    Google Scholar 

  4. Yu. L. Daletskii and M. G. Krein, Stability of Solutions to Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  5. N. Dunford and J. T. Schwartz, Linear Operators: General Theory, Pure and Appl. Math., Vol. 7, Interscience, New York (1958).

    Google Scholar 

  6. J. L. Doob, Stochastic Processes, Wiley & Sons, New York (1953).

    MATH  Google Scholar 

  7. E. B. Dynkin, Markov Processes [in Russian], Nauka, Moscow (1963).

    Google Scholar 

  8. J. Jacod and A. N. Shiryaev, Limit Theorems for Random Processes [in Russian], Vol. 1, Nauka, Moscow (1994).

    Google Scholar 

  9. J. Jacod and A. N. Shiryaev, Limit Theorems for Random Processes [in Russian], Vol. 2, Nauka, Moscow (1996).

    Google Scholar 

  10. V. B. Kolmanovskii and V. R. Nosov, Stability and Periodic Modes of Controllable Hereditary Systems [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  11. V. S. Korolyuk, Ye. F. Tsarkov, and V. K. Yasinsky, Probability, Statistics, and Random Processes, Vol. 3, Random Processes. Computer Modeling [in Ukrainian], Zoloti Lytavry, Chernivtsi (2009).

  12. A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulsive Loads [in Russian], Vyshcha Shkola, Kyiv (1967).

    Google Scholar 

  13. M. L. Sverdan and Ye. F. Tsarkov, Stability of Impulsive Systems [in Russian], Izd. RTU, Riga (1994).

    Google Scholar 

  14. A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations [in Russian], Naukova Dumka, Kyiv (1978).

    Google Scholar 

  15. R. Z. Khas’minskii, Stability of Systems of Differential Equations with Randomly Perturbed Parameters [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  16. Ye. F. Tsarkov, V. K. Yasinsky, and I. V. Malyk, “Stability in impulsive systems with Markov perturbations in averaging scheme. 1. Averaging principle for impulsive Markov systems,” Cybern. Syst. Analysis, 46, No. 6, 975–985 (2010).

    Article  Google Scholar 

  17. Ye. F. Tsarkov, V. K. Yasinsky, and I. V. Malyk, “Stability in impulsive systems with Markov perturbations in averaging scheme. 2. Averaging principle for impulsive Markov systems and stability analysis based on averaged equations,” Cybern. Syst. Analysis, 47, No. 1, 44–54 (2011).

    Article  Google Scholar 

  18. Ye. F. Tsarkov, Random Perturbations of Functional Differential Equations [in Russian], Zinatne, Riga (1989).

    Google Scholar 

  19. G. Blankenship and G. C. Papanicolaou, “Stability and control of stochastic system with wide–band noise disturbance. 1,” SIAM J. Appl. Mat., 34, 437–476 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. S. Korolyuk, “Averaging and stability of dynamical system with rapid Markov switchings,” Univ. of Umea, Umea, S 90167, Febr. (1991).

  21. V. S. Korolyuk and N. Limnios, “Diffusion approximation of integral functional in double merging and averaging scheme,” Theory Probab. and Math. Statist., 60, 87–94 (2000).

    MathSciNet  Google Scholar 

  22. Ye. Tsarkov, “Averaging in dynamical system with Markov jumps,” Inst. Dynamical Syst., Univ. of Bremen, No. 282, April, Bremen (1993).

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Authors and Affiliations

  1. Riga Technical University, Riga, Latvia

    Ye. F. Tsarkov

  2. Yu. Fed’kovych National University, Chernivtsi, Ukraine

    V. K. Yasynsky & I. V. Malyk

Authors
  1. Ye. F. Tsarkov
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  2. V. K. Yasynsky
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  3. I. V. Malyk
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Corresponding author

Correspondence to Ye. F. Tsarkov.

Additional information

Continued from Cybernetics and Systems Analysis, No. 6 (2010), No. 1 (2011).

Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 127–145, May–June 2011.

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Tsarkov, Y.F., Yasynsky, V.K. & Malyk, I.V. Stability in impulsive systems with markov perturbations in averaging scheme. 3. Weak convergence of solutions of impulsive systems. Cybern Syst Anal 47, 442–458 (2011). https://doi.org/10.1007/s10559-011-9326-2

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  • DOI: https://doi.org/10.1007/s10559-011-9326-2

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