Converse results on existence of sum of squares Lyapunov functions
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Parrilo_Converse results.pdf
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288.55 KB
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Adobe PDF
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Author(s)
Ahmadi, Amir Ali
Parrilo, Pablo A.
Date Issued
March 2012
Journal
50th IEEE Conference on Decision and Control and European Control Conference 2011 (CDC-ECC)
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Ahmadi, Amir Ali, and Pablo A. Parrilo. “Converse Results on Existence of Sum of Squares Lyapunov Functions.” 50th IEEE Conference on Decision and Control and European Control Conference 2011 (CDC-ECC). 6516–6521.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1109/CDC.2011.6161493
