This article is cited in 10 papers

Stochastic Systems

On the partial stability in probability of nonlinear stochastic systems

V. I. Vorotnikov^a, Yu. G. Martyshenko<sup>b

a</sup> Ural Federal University, Yekaterinburg, Russia
^b Gubkin Russian State University of Oil and Gas (National Research University), Moscow, Russia

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.

Keywords: systems of Itô, stochastic differential equations, partial stability in probability, the method of Lyapunov functions.

Presented by the member of Editorial Board: M. M. Khrustalev

Received: 20.04.2018
Revised: 10.09.2018
Accepted: 08.11.2018

DOI: 10.1134/S0005231019050052

 English version: Automation and Remote Control, 2019, 80:5, 856–866

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