Abstract
Using the second Lyapunov–Krasovskii method, sufficient conditions are obtained for the asymptotic stochastic global stability, for the global stability, and mean-square stability of trivial solutions to systems of diffusion stochastic functional–differential equations with Markov switchings, and the theory is illustrated by two model problems.
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References
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes [in Russian], Vol. 1, Fizmatgiz, Moscow (1994).
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes [in Russian], Vol. 2, Fizmatgiz, Moscow (1994).
E. B. Dynkin, Markov Processes, Academic Press, NY (1965).
I. Ya. Katz, The Method of Lyapunov Functions in Problems of the Stability and Stabilization of Random-Structure Systems [in Russian], UGAPS, Ekaterinburg (1998).
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Naukova Dumka, Kyiv (1968).
A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations [in Russian], Naukova Dumka, Kyiv (1987).
R. Z. Khasminskii, Stability of Systems of Differential Equations Under Random Disturbances of their Parameters [in Russian], Nauka, Moscow (1969).
V. I. Tikhonov and M. A. Mironov, Markov Processes [in Russian], Nauka, Moscow (1981).
V. B. Kolmanovskii and V. R. Nosov, Stability and Periodic Conditions of Controlled Hereditary Systems [in Russian], Nauka, Moscow (1981).
N. Danford and J. T. Schwartz, Linear Operators. General Theory (Wiley Classics Library), Pt. 1, Wiley (1988).
I. I. Gikhman and A. V. Skorokhod, Controlled Random Processes [in Russian], Naukova Dumka, Kyiv (1977).
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications [in Russian], Naukova Dumka, Kyiv (1982).
V. K. Yasinsky and E. V. Yasinsky, Problems on the Stability and Stabilization of Dynamic Systems with Finite Aftereffect [in Ukrainian], TViMS, Kyiv (2005).
J. L. Doob, Stochastic Processes, John Wiley, New York (1953).
V. K. Yasinsky, E. V. Yasinsky, and I. V. Yurchenko, Stabilization in Dynamic Systems of Random Structure [in Ukrainian], Zoloti Lytavry, Chernivtsi (2011).
H. J. Kushner, Stochastic Stability and Control, Academic Press (1967).
A. M. Lyapunov, The General Stability Problem [in Russian], Khark. Mat. Obshch., Kharkov (1892).
K. L. Chung, Lectures from Markov Processes to Brownian, Springer, New York (1982).
V. S. Korolyuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, London–Singapore–Hong Kong (2005).
V. Mizel and V. Trutzer, “Stochastic hereditary equations: existence and asymptotic stability,” J. Integral Equations, 7, 1–72 (1984).
V. M. Shurenkov, “On the theory of Markov renewal,” Theor. Probab. Appl., 19, No. 2, 247–265 (1984).
V. Yasinsky and V. Bereza, “The asymptotic uniform stability of solutions of neutral stochastic differential–functional equations with Poisson switchings,” Theory of Stochastic Processes, 29(25), No. 3–4, 211–217 (2003).
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Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2013, pp. 77–91.
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Yasinsky, V.K. Stability of stochastic dynamic random-structure systems with aftereffect and Markov switchings. Cybern Syst Anal 49, 706–719 (2013). https://doi.org/10.1007/s10559-013-9558-4
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DOI: https://doi.org/10.1007/s10559-013-9558-4