Geodesic
Construction of Lyapunov Functions for Second-Order Linear Stochastic Stationary Systems
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 149 (2018), pp. 118-128.

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In this paper, we present a construction of Lyapunov functions for second-order linear stochastic systems with constant coefficients. Based on this construction, we state necessary and sufficient conditions of the mean-square exponential stability of two-dimensional linear stationary systems. We obtain analytical expressions for the bifurcation value of the intensity of white noise acting on the parameters of the system. As an example, we consider equations of elastic vibrations whose coefficients are perturbed by white noise.
Keywords: stability, Lyapunov function, linear stochastic differential system, Gaussian white noise process. 
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M. M. Shumafov; V. B. Tlyachev. Construction of Lyapunov Functions for Second-Order Linear Stochastic Stationary Systems. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 149 (2018), pp. 118-128. http://geodesic.mathdoc.fr/item/INTO_2018_149_a13/

[1] Barbashin E. A., Funktsii Lyapunova, Nauka, M., 1970 | MR

[2] Lyapunov A. M., “Obschaya zadacha ob ustoichivosti dvizheniya”, Sobranie sochinenii, v. 2, ed. Lyapunov A. M., Izd-vo AN SSSR, M.-L., 1956, 7–263 | MR

[3] Donalson D. D., Lyapunov's direct method in the analysis of nonlinear control systems, McGraw-Hill, New York, 1965

[4] Kats I. Ya., Krasovskii N. N., “On the stability of systems with random parameters”, J. Appl. Math. Mech., 24:5 (1960), 1225–1246 | DOI | MR | Zbl

[5] Khasminskii R. Z., Stochastic stability of differential equations, Springer-Verlag, Heidelberg–Dordrecht–London–New York, 2012 | MR | Zbl

[6] Kushner H. J., Stochastic stability and control, Academic Press, New York–London, 1967 | MR | Zbl

[7] Liao X., Wang L. Q., Yu P., Stability of dynamical systems, Elsevier, Amsterdam, 2007 | MR | Zbl

[8] Oksendal B., Stochastic differential equations. An introduction with applications, Springer-Verlag, Berlin–Heidelberg, New York, 2010 | MR