Discovering polynomial Lyapunov functions for continuous dynamical systems

In this paper we analyze locally asymptotic stability of polynomial dynamical systems by discovering local Lyapunov functions beyond quadratic forms. We first derive an algebraizable sufficient condition for the existence of a polynomial Lyapunov function. Then we apply a real root classification based method step by step to under-approximate this derived condition as a semi-algebraic system such that the semi-algebraic system only involves the coefficients of the pre-assumed polynomial. Afterward, we compute a sample point in the corresponding semi-algebraic set for the coefficients resulting in a local Lyapunov function. Moreover, we test our approach on some examples using a prototype implementation and compare it with the generic quantifier elimination based method and the sum of squares based method. These computation and comparison results show the applicability and efficiency of our approach.