Converse results on existence of sum of squares Lyapunov functions

Amir Ali Ahmadi, Pablo A. Parrilo

Research output: Chapter in Book/Report/Conference proceedingConference contribution

22 Scopus citations

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 languageEnglish (US)
Title of host publication2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011
Pages6516-6521
Number of pages6
DOIs
StatePublished - Dec 1 2011
Externally publishedYes
Event2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 - Orlando, FL, United States
Duration: Dec 12 2011Dec 15 2011

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0191-2216

Other

Other2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011
CountryUnited States
CityOrlando, FL
Period12/12/1112/15/11

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

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  • Cite this

    Ahmadi, A. A., & Parrilo, P. A. (2011). Converse results on existence of sum of squares Lyapunov functions. In 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