On the Stability of Stochastic Parametrically Forced Equations with Rank One Forcing

TL;DR

This paper simplifies the stability analysis of stochastic parametrically forced linear systems, especially for rank one forcing, by converting complex spectral sums into residue sums, making computations more efficient.

Contribution

It introduces a new method to analyze stability using residues of the power spectral density for systems with rank one forcing, extending previous spectral approaches.

Findings

Simplified formulas for stability analysis of stochastic systems.

Efficient residue-based computation for rank one forcing.

Extension of spectral methods to broader classes of stochastic systems.

Abstract

We derive simplified formulas for analyzing the stability of stochastic parametrically forced linear systems. This extends the results in [T. Blass and L.A. Romero, SIAM J. Control Optim. 51(2):1099--1127, 2013] where, assuming the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this leads to an enormous simplification.