### 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 language | English (US) |
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Title of host publication | 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 |

Pages | 1074-1081 |

Number of pages | 8 |

Volume | 2 |

ISBN (Print) | 9781605600864 |

State | Published - 2007 |

Externally published | Yes |

Event | 45th Annual Allerton Conference on Communication, Control, and Computing 2007 - Monticello, United States Duration: Sep 26 2007 → Sep 28 2007 |

### Other

Other | 45th Annual Allerton Conference on Communication, Control, and Computing 2007 |
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Country | United States |

City | Monticello |

Period | 9/26/07 → 9/28/07 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science Applications
- Computer Networks and Communications

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

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 -