In this paper, we present a Lyapunov function for a system (may contain double integrator) with saturating controller. As usual this Lyapunov function is composed of a quadratic term and an integral term. However, instead of using a positive definite quadratic term, a positive semidefinite quadratic term is used. By using this Lyapunov function we show that for a double integrator system it can be globally stabilized by a saturating linear controller, however, for a triple-integrator system the saturating linear controller does not exist, which agrees with the known result.
