### Abstract

Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum of squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

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

Article number | 6194280 |

Pages (from-to) | 2281-2293 |

Number of pages | 13 |

Journal | IEEE Transactions on Automatic Control |

Volume | 57 |

Issue number | 9 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Computational complexity
- linear matrix inequalities (LMIs)
- Lyapunov functions
- nonlinear systems
- ordinary differential equations
- stability
- sum-of-squares

### ASJC Scopus subject areas

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

### Cite this

*IEEE Transactions on Automatic Control*,

*57*(9), 2281-2293. [6194280]. https://doi.org/10.1109/TAC.2012.2190163

**A converse sum of squares Lyapunov result with a degree bound.** / Peet, Matthew; Papachristodoulou, Antonis.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 57, no. 9, 6194280, pp. 2281-2293. https://doi.org/10.1109/TAC.2012.2190163

}

TY - JOUR

T1 - A converse sum of squares Lyapunov result with a degree bound

AU - Peet, Matthew

AU - Papachristodoulou, Antonis

PY - 2012

Y1 - 2012

N2 - Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum of squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

AB - Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum of squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

KW - Computational complexity

KW - linear matrix inequalities (LMIs)

KW - Lyapunov functions

KW - nonlinear systems

KW - ordinary differential equations

KW - stability

KW - sum-of-squares

UR - http://www.scopus.com/inward/record.url?scp=84865697207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865697207&partnerID=8YFLogxK

U2 - 10.1109/TAC.2012.2190163

DO - 10.1109/TAC.2012.2190163

M3 - Article

VL - 57

SP - 2281

EP - 2293

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 9

M1 - 6194280

ER -