A theorem of the alternative for SOS Lyapunov functions

In this paper duality theory is used to derive a theorem of the alternative for the existence of a sums of squares (SOS) Lyapunov function for a nonlinear system described by a polynomial vector field. We show that moments of occupation measures of unstable trajectories are dual feasible solutions providing a natural interpretation of the elements in the dual space. We show that moments corresponding to equilibria, orbits, and unbounded solutions indeed provide a certificate of infeasibility of the SOS Lyapunov problem. Additionally, equilibrium points may be recovered from special dual solutions.