A semi-algebraic approach for the computation of Lyapunov functions

In this paper we deal with the problem of computing Lyapunov functions for stability verification of differential systems. We concern on symbolic methods and start the discussion with a classical quantifier elimination model for computing Lyapunov functions in a given polynomial form, especially in quadratic forms. Then we propose a new semi-algebraic method by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first using real solution classification to construct a semi-algebraic system and then solving this semi-algebraic system. Our semi-algebraic approach is more efficient in practice, especially for low-order systems. This efficiency will be evaluated empirically.