On the Partial Stability in Probability of Nonlinear 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.

About the authors

V. I. Vorotnikov

Ural Federal University

Author for correspondence.
Email: vorotnikov-vi@rambler.ru
Russian Federation, Yekaterinburg

Yu. G. Martyshenko

Gubkin Russian State University of Oil and Gas (National Research University)

Author for correspondence.
Email: j-mart@mail.ru
Russian Federation, Moscow

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