Stabilization of linear systems by rotation
type: Item
instance of: scholarly article
title: Stabilization of linear systems by rotation
zbMATH Open document ID: 1124.34031
author: Tobias Damm
author: Hans Crauel
author: Achim Ilchmann
published in: Journal of Differential Equations
publication date: 2007-03-27T00:00:00Z
full work available at URL: dbt mods 00005390
review text: The problem addressed in this paper can be seen as a stabilization problem. Given a linear system \(\dot x= Ax+ u\), \(x\in\mathbb{R}^n\), the authors seek a stabilizing law of the form \(u = k(t)\Sigma x\), where \(\Sigma\) is skew-symmetric and \(k(t)\) is real valued, under the standing assumption that the trace of \(A\) is negative. The first result states the existence of a \(k^*\geq 0\) such that the system is stabilizable with \(|k|\geq k^*\) and \(\Sigma= \Sigma_A= U\Sigma_n U^T\). Here, \(U\) is an orthogonal matrix whose columns are eigenvectors of \(A+ A^T\), and the elements of \(\Sigma_n\) are \[ \sigma= \begin{cases} 0 &\text{if }i= j,\\ -1 &\text{if }i< j,\\ 1 &\text{if }i> j.\end{cases} \] The authors give an example which shows that the conclusion is not true if \(k\) is replaced by an arbitrary \(k(t)\geq k^*\). However, they prove that stabilization is achieved if either \(k(t)\) is nondecreasing and \(k\geq k^*_1\) or \(k(t)= kp(t)\) where \(p(t)\) is periodic and \(k\geq k^*_2\) (\(k^*_1\) and \(k^*_2\) are in general greater than \(k^*\)). The latter statement is compared with the classical principle of vibrational control. In fact the principle of vibrational control is improved under some respects by the present result. It is also proved that a stabilizing gain \(k(t)\) can be obtained in a adaptive and robust way, as a solution of \(\dot k= \| x\|^p\). Numerical examples are given. The proofs require an accurate analysis of properties of eigenvalues of skew-symmetric matrices and perturbation matrix theory.
reviewed by: Andrea Bacciotti
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 93D99
zbMATH DE Number: 5137563
zbMATH Keywords: linear systems
zbMATH Keywords: stability
zbMATH Keywords: stabilization
MaRDI profile type: Publication
OpenAlex ID: W2133765701
Wikidata QID: Q122927523
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DOI: 10.1016/J.JDE.2006.12.005
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description: scientific article; zbMATH DE number 5137563