  The notion of Lyapunov function plays a key role in design and verification
of dynamical systems, as well as hybrid and cyber-physical systems. In this
paper, to analyze the asymptotic stability of a dynamical system, we generalize
standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by
considering higher order Lie derivatives of certain functions along the
system's vector field. Furthermore, we present a complete method to
automatically discovering polynomial RLFs for polynomial dynamical systems
(PDSs). Our method is complete in the sense that it is able to discover all
polynomial RLFs by enumerating all polynomial templates for any PDS.
