On stability-preserving mappings of dynamical systems. I: Theory
type: Item
instance of: scholarly article
title: On stability-preserving mappings of dynamical systems. I: Theory
zbMATH Open document ID: 0855.34053
author: Anthony N. Michel
author: Kevin M. Passino
author: Kaining Wang
published in: Automation and Remote Control
publication date: 1997-02-16T00:00:00Z
review text: In this paper, a general dynamical system \(S\) is defined as a family of maps \(p : \mathbb{R}^+ \times A \times \mathbb{R}^+ \to X\), where \(X\) is a metric space, \(A \subseteq X\), \(p(t,a,t_0)\) is defined for all \(t \geq t_0\) and \(p(t_0, a, t_0) = a\) \((\mathbb{R}^+\) can be replaced by \(\mathbb{N}\), so that continuous time and discrete time systems are treated simultaneously). For a set \(M \subseteq X\) which is invariant with respect to all \(p \in S\), several notions of stability are given. These include generalizations of classical notions like uniform stability, asymptotic stability, exponential stability, boundedness of solution and so on. Given two dynamical systems \(S_1\) and \(S_2\) on metric spaces \(X_1\) and \(X_2\) respectively, assume that there exists \(V : X_1 \times \mathbb{R}^+ \to X_2\) which maps \(S_1\) on \(S_2\). Assume that \(M_1 \subseteq X_1\), \(M_2 \subseteq X_2\) are invariant with respect to \(S_1\), \(S_2\) respectively, and assume further that \(V\) maps \(M_1\) on \(M_2\). The authors prove a sufficient condition for \(V\) being stability preserving, which to say that \(S_1\) possesses one of the aforementioned stability properties if and only if \(S_2\) possesses the same property.
reviewed by: Andrea Bacciotti
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 93C30
Mathematics Subject Classification ID: 93A10
zbMATH DE Number: 919491
zbMATH Keywords: general dynamical system
zbMATH Keywords: stability preserving
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 919491