Series expansions for analytic systems linear in control
type: Item
instance of: scholarly article
title: Series expansions for analytic systems linear in control
zbMATH Open document ID: 1001.93028
DOI: 10.1016/S0005-1098(02)00042-0
published in: Automatica
publication date: 2002-09-05T00:00:00Z
review text: For the solution \(x(t,\varepsilon)\) of the initial value problem \(x(0,\varepsilon)=0\) of the system \[ \dot x=f(x)+\varepsilon g(t)\quad (x\in\mathbb R^n,\;t\in\mathbb R_+,\;\varepsilon\in\mathbb R_+), \] where \(f(x)\) is analytic in a neighborhood of the origin, \(f(0)=0\), \(g(t)\) is piecewise continuous and uniformly bounded, the paper presents a series expansion \[ x(t,\varepsilon)=\sum_{k=1}^{+\infty}x_k(t)\varepsilon ^k \tag{1} \] that can be computed via explicit recursive expressions. Conditions sufficient for the uniform convergence of (1) over finite- and infinite-time intervals are obtained. For the second-order polynomial \(f(x)\), a simplified series and convergence analysis are presented. Some open problems are discussed.
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 34A45
zbMATH DE Number: 1797126
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: perturbation theory
zbMATH Keywords: Volterra series
zbMATH Keywords: series expansion
author: Francesco Bullo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1797126