A first approximation of the quasipotential in problems of the stability of systems with random nondegenerate perturbations
type: Item
instance of: scholarly article
title: A first approximation of the quasipotential in problems of the stability of systems with random nondegenerate perturbations
zbMATH Open document ID: 0880.34059
DOI: 10.1016/0021-8928(95)00006-B
author: G. N. Mil'shtejn
author: Lev Ryashko
published in: Journal of Applied Mathematics and Mechanics
publication date: 1995-01-01T00:00:00Z
review text: A Lyapunov function of a special form -- the quasi-potential -- is used here to consider the stability of \[ dx= f(x)dt+ \varepsilon\sigma(x)dw(t), \] a system with small non-degenerate random perturbations. Specifically, the authors investigate the quasipotential in a neighbourhood of the stationary point and give an estimate of the error in finding the first approximation of the quasipotential. Consideration is also given to the stability problem when a closed curve (or orbit), rather than a point, is taken as the limiting variant set of the deterministic system \(dx= f(x)dt\). The existence and uniqueness of the solution to the Lyapunov matrix equation with periodic coefficients that is obtained are examined. The plane orbit (two-dimensional) case is also examined, and some illustrative examples are given.
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 93E99
zbMATH DE Number: 1032256
zbMATH Keywords: Lyapunov function
zbMATH Keywords: stability
zbMATH Keywords: small non-degenerate random perturbations
zbMATH Keywords: Lyapunov matrix equation
zbMATH Keywords: periodic coefficients
reviewed by: Andrew Dale
MaRDI profile type: Publication
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Wikidata QID: Q127642277
description: scientific article; zbMATH DE number 1032256