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On the Partial Stability in Probability of Nonlinear Stochastic Systems

  • Stochastic Systems
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Abstract

A general class of the nonlinear time-varying systems of Itô stochastic differential equations is considered. Two problems on the partial stability in probability are studied as follows: 1) the stability with respect to a given part of the variables of the trivial equilibrium; 2) the stability with respect to a given part of the variables of the partial equilibrium. The stochastic Lyapunov functions-based conditions of the partial stability in probability are established. In addition to the main Lyapunov function, an auxiliary (generally speaking, vector-valued) function is introduced for correcting the domain in which the main Lyapunov function is constructed. A comparison with the well-known results on the partial stability of the systems of stochastic differential equations is given. An example that well illustrates the peculiarities of the suggested approach is described. Also a possible unified approach to analyze the partial stability of the time-invariant and time-varying systems of stochastic differential equations is discussed.

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Author information

Authors and Affiliations

  1. Ural Federal University, Yekaterinburg, Russia

    V. I. Vorotnikov

  2. Gubkin Russian State University of Oil and Gas (National Research University), Moscow, Russia

    Yu. G. Martyshenko

Authors
  1. V. I. Vorotnikov
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  2. Yu. G. Martyshenko
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Corresponding authors

Correspondence to V. I. Vorotnikov or Yu. G. Martyshenko.

Additional information

Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 5, pp. 86–98.

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Vorotnikov, V.I., Martyshenko, Y.G. On the Partial Stability in Probability of Nonlinear Stochastic Systems. Autom Remote Control 80, 856–866 (2019). https://doi.org/10.1134/S0005117919050059

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  • DOI: https://doi.org/10.1134/S0005117919050059

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