### Abstract

Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a homogeneous polynomial vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. The latter result is extended to develop a converse sos Lyapunov theorem for robust stability of switched linear systems.

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

Title of host publication | 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 |

Pages | 6516-6521 |

Number of pages | 6 |

DOIs | |

State | Published - Dec 1 2011 |

Externally published | Yes |

Event | 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 - Orlando, FL, United States Duration: Dec 12 2011 → Dec 15 2011 |

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
---|---|

ISSN (Print) | 0191-2216 |

### Other

Other | 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 |
---|---|

Country | United States |

City | Orlando, FL |

Period | 12/12/11 → 12/15/11 |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

## Fingerprint Dive into the research topics of 'Converse results on existence of sum of squares Lyapunov functions'. Together they form a unique fingerprint.

## Cite this

*2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011*(pp. 6516-6521). [6161493] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2011.6161493