Converse Lyapunov functions for exponentially stable periodic orbits
type: Item
instance of: scholarly article
title: Converse Lyapunov functions for exponentially stable periodic orbits
zbMATH Open document ID: 0827.34042
DOI: 10.1016/0167-6911(94)90078-7
author: V. Pereyra
published in: Systems \& Control Letters
publication date: 1994-09-19T00:00:00Z
review text: The authors consider an autonomous system in \(\mathbb{R}^n\), (S) \(\dot x= f(x)\), with a periodic orbit \(\eta\). First they provide a special set of local coordinates \((\rho, \theta)\) in a neighbourhood of \(\eta\), which highlight the local transversal behaviour of the flow. Then they associate to (S) a linear \(\theta\)-periodic system (P) in the \(\rho\) coordinates. They prove that \(\eta\) is exponentially stable for (S) if and only if the origin is asymptotically stable for (P). Under the hypothesis that \(\eta\) is exponentially stable for (S), they use the Lyapunov equation associated to (P) to produce a Lyapunov function \(V(x)\) that proves the exponential stability of \(\eta\). Finally, they provide an example in which this result is used to stabilize a periodic orbit of an affine control system.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 93D30
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 640408
zbMATH Keywords: periodic orbit
zbMATH Keywords: Lyapunov function
zbMATH Keywords: exponential stability
zbMATH Keywords: affine control system
MaRDI profile type: Publication
full work available at URL: 0167-6911(94)90078-7
OpenAlex ID: W2174809691
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description: scientific article; zbMATH DE number 640408