### Abstract

We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result is extended to show that such sos-based certificates of stability are guaranteed to exist for all stable switched linear systems. For this class of systems, we further show that if the derivative inequality of the Lyapunov function has an sos certificate, then the Lyapunov function itself is automatically a sum of squares. These converse results establish cases where semidefinite programming is guaranteed to succeed in finding proofs of Lyapunov inequalities. Finally, we demonstrate some merits of replacing the sos requirement on a polynomial Lyapunov function with an sos requirement on its top homogeneous component. In particular, we show that this is a weaker algebraic requirement in addition to being cheaper to impose computationally.

Original language | English (US) |
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Article number | 7803555 |

Pages (from-to) | 5269-5274 |

Number of pages | 6 |

Journal | IEEE Transactions on Automatic Control |

Volume | 62 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2017 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Asymptotic stability
- polynomial vector fields
- semidefinite programming
- sum of squares lyapunov functions

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## Cite this

*IEEE Transactions on Automatic Control*,

*62*(10), 5269-5274. [7803555]. https://doi.org/10.1109/TAC.2016.2647253