Sum of squares certificates for stability of planar, homogeneous, and switched systems

TL;DR

This paper proves that for certain polynomial and switched systems, the existence of polynomial Lyapunov functions guarantees sum of squares certificates, enabling efficient stability verification via semidefinite programming.

Contribution

It establishes new theoretical links between polynomial Lyapunov functions and sum of squares certificates for stability, including for switched systems.

Findings

Sum of squares certificates exist for stable homogeneous and planar polynomial systems.

If the derivative inequality has an sos certificate, the Lyapunov function is also sos.

Replacing sos on the Lyapunov function with sos on its top component is weaker and computationally cheaper.

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.