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.
In REF , Ahmadi and Parrilo show that in the continuous time case, existence of the Lyapunov function of Mason et al. further implies existence of a Lyapunov function that can be found with sum of squares techniques.
Sum of squares certificates for stability of planar, homogeneous, and switched systems
IEEE Transactions On Automatic Control, Vol. 62, No. 10, October 2017
Computer Science Mathematics