### Abstract

This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of BBR^{n}, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the vector field is at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, polynomials can be used to approximate a differentiable function, using the Sobolev norm W^{1}, ∞ to any desired accuracy. This approximation result is combined with the second-order Taylor series expansion to show that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. The investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.

Original language | English (US) |
---|---|

Pages (from-to) | 979-987 |

Number of pages | 9 |

Journal | IEEE Transactions on Automatic Control |

Volume | 54 |

Issue number | 5 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

### Fingerprint

### Keywords

- Exponential stability
- Lyapunov methods
- Polynomial approximation
- Polynomials
- Sum of squares

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Control and Systems Engineering
- Computer Science Applications

### Cite this

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

Research output: Contribution to journal › Article

}

TY - JOUR

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

AU - Peet, Matthew

PY - 2009

Y1 - 2009

N2 - This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of BBRn, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the vector field is at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, polynomials can be used to approximate a differentiable function, using the Sobolev norm W1, ∞ to any desired accuracy. This approximation result is combined with the second-order Taylor series expansion to show that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. The investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.

AB - This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of BBRn, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the vector field is at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, polynomials can be used to approximate a differentiable function, using the Sobolev norm W1, ∞ to any desired accuracy. This approximation result is combined with the second-order Taylor series expansion to show that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. The investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.

KW - Exponential stability

KW - Lyapunov methods

KW - Polynomial approximation

KW - Polynomials

KW - Sum of squares

UR - http://www.scopus.com/inward/record.url?scp=67349084187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349084187&partnerID=8YFLogxK

U2 - 10.1109/TAC.2009.2017116

DO - 10.1109/TAC.2009.2017116

M3 - Article

VL - 54

SP - 979

EP - 987

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 5

ER -