Sum of squares certificates for stability of planar, homogeneous, and switched systems
This provides theoretical guarantees for the success of sum-of-squares optimization in verifying stability for important classes of dynamical systems, addressing a known gap between existence of Lyapunov functions and their sos representation.
The paper proves that for certain classes of systems (homogeneous polynomial, planar polynomial, and switched linear), the existence of a polynomial Lyapunov function implies the existence of one that is a sum of squares (sos) with sos derivative, guaranteeing that semidefinite programming can find stability proofs. It also shows that requiring only the top homogeneous component to be sos is computationally cheaper and algebraically weaker.
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.