Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper presents a proof that existence of a polyno-mial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear or-dinary differential equations on bounded sets. The main result implies that if there exists an n-times continuously differentiate Lyapunov function which proves exponen-tial stability on a bounded subset of ℝ<sup>n</sup> , then there exists a polynomial Lyapunov function which proves exponen-tial stability on the same region. Such a continuous Lya-punov function will exist if , for example , the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. Our investigation is motivated by the use of polynomial optimization algo-rithms to construct polynomial Lyapunov functions for systems of nonlinear ordinary differential equations.

Original languageEnglish (US)
Title of host publication45th Annual Allerton Conference on Communication, Control, and Computing 2007
PublisherUniversity of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering
Pages1074-1081
Number of pages8
Volume2
ISBN (Print)9781605600864
StatePublished - 2007
Externally publishedYes
Event45th Annual Allerton Conference on Communication, Control, and Computing 2007 - Monticello, United States
Duration: Sep 26 2007Sep 28 2007

Other

Other45th Annual Allerton Conference on Communication, Control, and Computing 2007
CountryUnited States
CityMonticello
Period9/26/079/28/07

Fingerprint

Lyapunov functions
Nonlinear systems
Polynomials
Differential equations
Asymptotic stability
Set theory
Ordinary differential equations

ASJC Scopus subject areas

  • Computer Science Applications
  • Computer Networks and Communications

Cite this

Peet, M. (2007). Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions. In 45th Annual Allerton Conference on Communication, Control, and Computing 2007 (Vol. 2, pp. 1074-1081). University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering.

Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions. / Peet, Matthew.

45th Annual Allerton Conference on Communication, Control, and Computing 2007. Vol. 2 University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering, 2007. p. 1074-1081.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Peet, M 2007, Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions. in 45th Annual Allerton Conference on Communication, Control, and Computing 2007. vol. 2, University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering, pp. 1074-1081, 45th Annual Allerton Conference on Communication, Control, and Computing 2007, Monticello, United States, 9/26/07.
Peet M. Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions. In 45th Annual Allerton Conference on Communication, Control, and Computing 2007. Vol. 2. University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering. 2007. p. 1074-1081
Peet, Matthew. / Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions. 45th Annual Allerton Conference on Communication, Control, and Computing 2007. Vol. 2 University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering, 2007. pp. 1074-1081
@inproceedings{e5277ff88e064e01ba970314ff368cb9,
title = "Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions",
abstract = "This paper presents a proof that existence of a polyno-mial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear or-dinary differential equations on bounded sets. The main result implies that if there exists an n-times continuously differentiate Lyapunov function which proves exponen-tial stability on a bounded subset of ℝn , then there exists a polynomial Lyapunov function which proves exponen-tial stability on the same region. Such a continuous Lya-punov function will exist if , for example , the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. Our investigation is motivated by the use of polynomial optimization algo-rithms to construct polynomial Lyapunov functions for systems of nonlinear ordinary differential equations.",
author = "Matthew Peet",
year = "2007",
language = "English (US)",
isbn = "9781605600864",
volume = "2",
pages = "1074--1081",
booktitle = "45th Annual Allerton Conference on Communication, Control, and Computing 2007",
publisher = "University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering",

}

TY - GEN

T1 - Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions

AU - Peet, Matthew

PY - 2007

Y1 - 2007

N2 - This paper presents a proof that existence of a polyno-mial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear or-dinary differential equations on bounded sets. The main result implies that if there exists an n-times continuously differentiate Lyapunov function which proves exponen-tial stability on a bounded subset of ℝn , then there exists a polynomial Lyapunov function which proves exponen-tial stability on the same region. Such a continuous Lya-punov function will exist if , for example , the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. Our investigation is motivated by the use of polynomial optimization algo-rithms to construct polynomial Lyapunov functions for systems of nonlinear ordinary differential equations.

AB - This paper presents a proof that existence of a polyno-mial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear or-dinary differential equations on bounded sets. The main result implies that if there exists an n-times continuously differentiate Lyapunov function which proves exponen-tial stability on a bounded subset of ℝn , then there exists a polynomial Lyapunov function which proves exponen-tial stability on the same region. Such a continuous Lya-punov function will exist if , for example , the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. Our investigation is motivated by the use of polynomial optimization algo-rithms to construct polynomial Lyapunov functions for systems of nonlinear ordinary differential equations.

UR - http://www.scopus.com/inward/record.url?scp=84940643413&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940643413&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781605600864

VL - 2

SP - 1074

EP - 1081

BT - 45th Annual Allerton Conference on Communication, Control, and Computing 2007

PB - University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering

ER -