Asymptotic exponential cones of Metzler matrices and their use in the solution of an algebraic problem
type: Item
instance of: scholarly article
title: Asymptotic exponential cones of Metzler matrices and their use in the solution of an algebraic problem
zbMATH Open document ID: 1207.15038
author: Paolo Santesso
author: Maria Elena Valcher
published in: Linear Algebra and its Applications
publication date: 2010-02-09T00:00:00Z
review text: Several algebraic problems arising in the context of positive switched systems are still unsolved. In particular, in the investigating reachability property for continuous-time switched systems, the problem has originated under what conditions one can find nonnegative integers \(\tau_1, \tau_2, \dots, \tau_m\) such that a strictly positive vector \({\mathbf v}\) belongs to the boundary of the cone generated by the columns of the matrix \(e^{A_{i_{1}} \tau_1}, e^{A_{i_{2}} \tau_2}, \dots, e^{A_{i_{m}} \tau_m}\), where \(A_{i_{1}}, A_{i_{2}}, \dots, A_{i_{m}}\) are \(m\) Metzler matrices (real square matrices with nonnegative off-diagonal entries). In this paper, the analysis, which was started by the authors in [Linear Algebra Appl. 425, No.~2--3, 283--302 (2007; Zbl 1123.93033)], is developed further by widening and deepening the knowledge about asymptotic exponential cones of a family of Metzler matrices and these results are used to provide necessary and/or sufficient conditions for the solvability of the given algebraic problem. The problem solution is also investigated in the specific case when dealing with at most two exponential matrices.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 15A16
zbMATH DE Number: 5667902
zbMATH Keywords: continuous-time positive system
zbMATH Keywords: Metzler matrix
zbMATH Keywords: exponential matrix
zbMATH Keywords: zero pattern
zbMATH Keywords: directed graph
zbMATH Keywords: communicating classes
zbMATH Keywords: Frobenius normal form
zbMATH Keywords: asymptotic exponential law
zbMATH Keywords: simplicial cone
zbMATH Keywords: boundary of a cone
MaRDI profile type: Publication
full work available at URL: j.laa.2009.10.012
OpenAlex ID: W2067329304
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DOI: 10.1016/J.LAA.2009.10.012
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description: scientific article; zbMATH DE number 5667902