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Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems

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Abstract

The Bogoliubov–Mitropolsky small parameter method is used to study the behavior of stochastic differential systems in the analysis of the corresponding properties of solutions of averaged systems.

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References

  1. I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Naukova Dumka, Kyiv (1968).

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. D. A. Kalnyne and V. N. Tsarkova, “Random perturbations of the parameters of linear iterations,” Problemy Sluchainogo Poiska [in Russian], Issue 11, Zinatne, Riga (1988), pp. 150–166.

  4. N. N. Bogoliubov and Yu. A. Mitropolsky, Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kyiv (1971).

    Google Scholar 

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

    Google Scholar 

  6. A. M. Samoilenko and A. N. Stanzhitskii, “Fluctuations in the averaging scheme for differential equations with random impulsive loads,” Ukr. Mat. Zh., No. 5, 631–641 (1989).

    Google Scholar 

  7. R. Z. Khas’minskii, “Random processes defined by differential equations with small parameter,” Teor. Veroyatn. i yeyo Primenen., 2, No. 2, 240–259 (1966).

    Google Scholar 

  8. R. Z. Khas’minskii, “Limit theorem for solutions of a differential equation with random right-hand side,” Teor. Veroyatn. i yeyo Primenen., 2, No. 3, 444–462 (1966).

    Google Scholar 

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

  10. 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  MATH  MathSciNet  Google Scholar 

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

  12. 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 

  13. V. S. Korolyuk and N. Limnios, “Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space,” Theory Probab. and Math. Statist., 70, 71–82 (2005).

    Article  MathSciNet  Google Scholar 

  14. Ye. Tsarkov, “Asymptotic methods for stability analysis of Markov impulse dynamical systems,” Nonlinear Dynamic and Systems Theory, 1, No. 2, 103–115 (2002).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  16. 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).

  17. Ye. F. Tsarkov and M. L. Sverdan, Stability of Stochastic Impulsive Systems [in Russian], Izd. Rizhsk. Tekhn. Univ., Riga (1994).

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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. Yasinsky & I. V. Malyk

Authors
  1. Ye. F. Tsarkov
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  2. V. K. Yasinsky
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  3. I. V. Malyk
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Correspondence to Ye. F. Tsarkov.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 128–139, November–December 2010.

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Tsarkov, Y.F., Yasinsky, V.K. & Malyk, I.V. Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems. Cybern Syst Anal 46, 975–985 (2010). https://doi.org/10.1007/s10559-010-9279-x

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  • DOI: https://doi.org/10.1007/s10559-010-9279-x

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