Characterization of stochastic processes which stabilize linear companion form systems.
type: Item
instance of: scholarly article
title: Characterization of stochastic processes which stabilize linear companion form systems.
zbMATH Open document ID: 1045.60059
DOI: 10.1016/S0304-4149(00)00012-0
author: John Kao
author: Volker Wihstutz
published in: Stochastic Processes and their Applications
publication date: 2004-09-07T00:00:00Z
review text: Consider the stochastic differential equation \[ dx^\varepsilon =Ax^\varepsilon d t +Ux^\varepsilon \circ dF^\varepsilon _t, \tag{1} \] (in Stratonovich sense), \(\varepsilon >0\), where \(A\) is a \(d\times d\) companion matrix to a linear differential equation of order \(d\), \[ U= \begin{pmatrix} 0 & \hdots & 0 &0\\ \\ u_1 & \hdots & u_{d-1} & u_d \end{pmatrix}, \] \(F^\varepsilon _t:= F_{t/\varepsilon }\) and the semimartingale \(F_t\) is defined by the equation \[ dF_t= f_0(\xi _t)d t +\sum _{k=1}^r f_k(\xi _t)\circ dW^k_t, \] where \(f_0,\dots ,f_r\) are real-valued \(C^\infty \)-functions on a connected smooth Riemann manifold \(M\), and \(\xi _t\) is a stationary ergodic process on \(M\). Stability of the trivial solution to the system (1) is investigated. The class of processes \(F\) is characterized for which the system (1) is stabilized and the Lyapunov spectrum is studied.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 93E15
zbMATH DE Number: 2098340
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: stabilization by noise
zbMATH Keywords: stochastic linear systems
MaRDI profile type: Publication
cites work: Q4210478
cites work: Q3715980
cites work: Wide Sense Stationary Solutions of Linear Systems with Additive Noise
cites work: Q3734804
cites work: Vibrational control of nonlinear systems: Vibrational stabilizability
cites work: Q3721531
cites work: Q3883249
cites work: A classification of the second order degenerate elliptic operators and its probabilistic characterization
cites work: Supplements and corrections to the paper: A classification of the second order degenerate elliptic operators and its probabilistic characterization
cites work: Stabilization of companion form systems by mean zero noise
cites work: Q4840149
cites work: Q3341616
cites work: Principle of vibrational control: Theory and applications
cites work: THE STRUCTURE OF EIGENFUNCTIONS OF ONE-DIMENSIONAL UNORDERED STRUCTURES
cites work: Q4048325
cites work: Stationary solutions for linear systems with additive noise
cites work: Q5646734
cites work: Q3886594
cites work: Lyapunov Exponent and Rotation Number of Two-Dimensional Linear Stochastic Systems with Small Diffusion
cites work: Lyapunov exponents of real–noise–driven nilpotent systems and harmonic oscillators
description: scientific article; zbMATH DE number 2098340