ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Iterative Method for Stochastic Stability of Linear Differential Equation with Periodic Coefficients Analysis

Author(s):

A. A. GUBKIN

Russia, 620083, Yekaterinburg, str. Lenina st., 51,
Urals state university,
Department of mathematics and mechanics

andreygubkin@mail.ru

L. B. RYASHKO

Russia, 620083, Yekaterinburg, str. Lenina st., 51,
Urals state university,
Department of mathematics and mechanics

lev.ryashko@usu.ru

Abstract:

The paper is devoted to the stability analysis for stochastic differential equations with periodic coefficients under parametric random disturbances. Such systems are basic mathematical models for many real oscillatory processes. Multiplicative kind of disturbances makes their analysis more difficult.
In the systems with random disturbances one can study different kinds of stability. The kind considered here is the stability in mean squares. In the basis of the method developed in this paper lies the spectral criterion, allowing to reduce the question on stability to evaluating of spectral radius of some positive operator. For evaluating the spectral radius a simple iterative method is suggested.
The main theoretical result in the paper is the proof of iterative method convergence, obtained with the help of theory of positive operators. Quite simple sufficient conditions of convergence allow to use the method for wide class of systems analysis.
The suggested method in the difficult task of stability domains construction is more effective in comparison to traditionally used method of second moments.
The work was supported by RFFI grant 04-01-96098ural.

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