Sum of squares certificates for stability of planar, homogeneous, and switched systems
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 sosbased 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1801.00070
 Bibcode:
 2018arXiv180100070A
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Systems and Control;
 Mathematics  Algebraic Geometry;
 Mathematics  Dynamical Systems
 EPrint:
 12 pages. The arxiv version includes some more details than the published version