Minimal and maximal Lyapunov exponents of bilinear control systems
type: Item
instance of: scholarly article
title: Minimal and maximal Lyapunov exponents of bilinear control systems
zbMATH Open document ID: 0769.34037
author: Fritz Colonius
author: Wolfgang Kliemann
published in: Journal of Differential Equations
publication date: 1993-05-16T00:00:00Z
review text: This paper deals with control systems of the form (1) \(\dot x=A_ 0(u_ 0)x+ \sum_{i=1}^ m u_ i A_ i(u_ 0)x\), \(x\in\mathbb{R}^ d\), where \(u_ 0: \mathbb{R}\to\Omega\) represents a locally integrable constrained control (\(\Omega\) is a compact subset of \(\mathbb{R}^ l\)) and \(u=(u_ 1,\dots,u_ m):\mathbb{R} \to\mathbb{R}^ m\) is a locally integrable control (not subject to constraints). The stabilization (or destabilization) properties of (1) are related to the minimal and maximal Lyapunov exponents \(\sup_ u \sup_{x_ 0} \lambda(x_ 0,u)\), \(\inf_ u \inf_{x_ 0} \lambda(x_ 0,u)\), where \(\lambda(x_ 0,u)=\limsup {{\log| x(t,x_ 0,u)|} \over t}\) and \(x(t,x_ 0,u)\) denotes the solution of (1) corresponding to the control \(u\) and the initial value \(x_ 0\). The paper is devoted to a careful investigation of maximal and minimal Lyapunov exponents and the related uniform notions. The authors give several characterizations of these notions, as well as of the set of initial values for which these extremal exponents are realized. The analysis is carried out by considering an optimization problem for the system induced by (1) on the projective space \(\mathbb{P}\) of \(\mathbb{R}^ d\). Some applications are included (high gain stabilization, stabilization of linear systems by time-varying feedback, the linear oscillator problem).
reviewed by: Andrea Bacciotti
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 166900
zbMATH Keywords: high gain stabilization
zbMATH Keywords: control systems
zbMATH Keywords: stabilization
zbMATH Keywords: minimal and maximal Lyapunov exponents
zbMATH Keywords: stabilization of linear systems by time- varying feedback
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1012
OpenAlex ID: W1991570978
DOI: 10.1006/JDEQ.1993.1012
description: scientific article; zbMATH DE number 166900