Relaxation results for hybrid inclusions
type: Item
instance of: scholarly article
title: Relaxation results for hybrid inclusions
zbMATH Open document ID: 1169.37017
author: Chaohong Cai
author: Rafal Goebel
author: Andrew Teel
published in: Set-Valued Analysis
publication date: 2009-03-16T00:00:00Z
review text: The goal of the paper is to extend the Filippov-Ważevski relaxation theorem, established for differential inclusions, to the setting of hybrid inclusions. Hybrid inclusions can be written as \[ {\mathcal H}:\quad \left\{ \begin{matrix} \dot x \in F(x)&\quad x \in C \\ x^{+} \in G(x) &\quad x \in D\end{matrix} \right. \tag{1} \] That is, the state \(x\) evolution in time is described by the flow map \(F\) when \(x\) is in the flow set \(C\), and \(x\) can also jump according to the jump map \(G\) from the jump set \(D\). A relaxed hybrid inclusion \[ {\mathcal H}^{\text{con}}:\quad \left\{ \begin{matrix} \dot x \in \text{con}\; F(x)&\quad x \in C \\ x^{+} \in G(x) &\quad x \in D\end{matrix} \right. \tag{2} \] is obtained from the original \(\mathcal H\) by convexification of the flow map \(F\). The paper establishes conditions when solutions to \(\mathcal H\) are dense in the set of solutions to \({\mathcal H}^{\text{con}}\), with respect to the distance between their graphs. As an application of this relaxation result to stability theory, it is shown that uniform global asymptotic stability of a compact set for Eq.~1 implies that property for Eq.~2. The relaxation theorem leads to results on continuous dependence of the solution set to hybrid inclusions on initial conditions. The case of constrained differential inclusions and the case of systems with logic variables are considered also.
Mathematics Subject Classification ID: 37N35
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 39A99
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 5529267
zbMATH Keywords: relaxation
zbMATH Keywords: hybrid systems
zbMATH Keywords: differential inclusions
zbMATH Keywords: difference inclusions
zbMATH Keywords: constraints
zbMATH Keywords: continuous dependence on initial conditions
MaRDI profile type: Publication
full work available at URL: s11228-007-0067-3
OpenAlex ID: W2048991385
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DOI: 10.1007/S11228-007-0067-3
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description: scientific article; zbMATH DE number 5529267