A spectral operator-theoretic framework for global stability

Alexandre Mauroy, Igor Mezić

Research output: Contribution in Book/Catalog/Report/Conference proceedingConference contribution

Abstract

The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

Original languageEnglish
Title of host publication2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages5234-5239
Number of pages6
ISBN (Print)9781467357173
DOIs
Publication statusPublished - 2013
Externally publishedYes
Event52nd IEEE Conference on Decision and Control, CDC 2013 - Florence, Italy
Duration: 10 Dec 201313 Dec 2013

Conference

Conference52nd IEEE Conference on Decision and Control, CDC 2013
CountryItaly
CityFlorence
Period10/12/1313/12/13

Fingerprint

Global Stability
Stability Analysis
Global Analysis
Spectral Properties
Operator
Stability Theory
Lyapunov Function
Linear Operator
Eigenfunctions
Convergence of numerical methods
Lyapunov functions
Nonlinear Systems
Fixed point
System stability
Numerical Methods
Eigenvalues and eigenfunctions
Nonlinear systems
Metric
Numerical methods
Framework

Cite this

Mauroy, A., & Mezić, I. (2013). A spectral operator-theoretic framework for global stability. In 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013 (pp. 5234-5239). [6760712] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2013.6760712
Mauroy, Alexandre ; Mezić, Igor. / A spectral operator-theoretic framework for global stability. 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013. Institute of Electrical and Electronics Engineers Inc., 2013. pp. 5234-5239
@inproceedings{55c414b21f624d549e8f92c3db785b50,
title = "A spectral operator-theoretic framework for global stability",
abstract = "The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.",
author = "Alexandre Mauroy and Igor Mezić",
year = "2013",
doi = "10.1109/CDC.2013.6760712",
language = "English",
isbn = "9781467357173",
pages = "5234--5239",
booktitle = "2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013",
publisher = "Institute of Electrical and Electronics Engineers Inc.",

}

Mauroy, A & Mezić, I 2013, A spectral operator-theoretic framework for global stability. in 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013., 6760712, Institute of Electrical and Electronics Engineers Inc., pp. 5234-5239, 52nd IEEE Conference on Decision and Control, CDC 2013, Florence, Italy, 10/12/13. https://doi.org/10.1109/CDC.2013.6760712

A spectral operator-theoretic framework for global stability. / Mauroy, Alexandre; Mezić, Igor.

2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013. Institute of Electrical and Electronics Engineers Inc., 2013. p. 5234-5239 6760712.

Research output: Contribution in Book/Catalog/Report/Conference proceedingConference contribution

TY - GEN

T1 - A spectral operator-theoretic framework for global stability

AU - Mauroy, Alexandre

AU - Mezić, Igor

PY - 2013

Y1 - 2013

N2 - The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

AB - The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

UR - http://www.scopus.com/inward/record.url?scp=84902341187&partnerID=8YFLogxK

U2 - 10.1109/CDC.2013.6760712

DO - 10.1109/CDC.2013.6760712

M3 - Conference contribution

SN - 9781467357173

SP - 5234

EP - 5239

BT - 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013

PB - Institute of Electrical and Electronics Engineers Inc.

ER -

Mauroy A, Mezić I. A spectral operator-theoretic framework for global stability. In 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013. Institute of Electrical and Electronics Engineers Inc. 2013. p. 5234-5239. 6760712 https://doi.org/10.1109/CDC.2013.6760712