Abstract
We obtain sufficient conditions for the stability of a dynamical system in a semi-Markov medium under the conditions of diffusion approximation by using asymptotic properties of the compensation operator for a semi-Markov process and properties of the Lyapunov function for an averaged system.
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T. Hersh and G. Griego, “Random evolution—theory and applications,” Univ. New Mexico. Techn. Rep., 180, 15–38 (1969).
M. Pinsky, “Random evolution,” Springer Lect. Notes Math., 451, 89–100 (1975).
G. L. Blankenship and G. C. Papanicolaou, “Stability and control of stochastic systems with wide band noise disturbances,” SIAM J. Appl. Math., 34, 437–476 (1978).
V. S. Korolyuk, “Stability of stochastic systems in the scheme of diffusion approximation,” Ukr. Mat. Zh., 50, No. 1, 36–47 (1998).
A. V. Swishchuk, “Stability of semi-Markov evolutionary stochastic systems in averaging and diffusion approximation schemes,” in: Asymptotic Analysis of Random Evolutions, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1994), pp. 255–269.
M. N. Sviridenko, “Martingale characterization of limit distributions in the space of functions without discontinuities of the second kind,” Mat. Zam., 43, No. 5, 398–402 (1998).
V. S. Korolyuk and A. V. Swishchuk, Evolution of Systems in Random Media, CRC Press (1995).
V. S. Korolyuk and A. V. Swishchuk, Semi-Markov Random Evolution [in Russian], Naukova Dumka, Kiev (1992).
V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Kluwer, Dordrecht (1999).
V. S. Korolyuk and Ya. M. Chabanyuk, “Stability of a dynamical system with semi-Markov switchings under conditions of stability of the averaged system,” Ukr. Mat. Zh., 54, No. 2, 195–204 (2002).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1290–1296, September, 2007.
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Chabanyuk, Y.M. Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation. Ukr Math J 59, 1441–1452 (2007). https://doi.org/10.1007/s11253-007-0097-x
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DOI: https://doi.org/10.1007/s11253-007-0097-x