Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

Jiang Liu, Naijun Zhan, Hengjun Zhao

Abstract: The notion of Lyapunov function plays a key role in the 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. Furthermore, we present a method for automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is relatively complete in the sense that it is able to discover all polynomial RLFs with a given predefined template for any PDS. Therefore it can also generate all polynomial RLFs for the PDS by enumerating all polynomial templates.