Lyapunov Functions Obtained from First Order Approximations

We study the construction of Lyapunov functions based on first order approximations. We first consider the (transverse) local exponential stability of an invariant manifold and largely rephrase [3]. We show how to construct a Lyapunov function with this framework that characterizes this local stability property. We then consider global stability of an equilibrium point, and show that the first order approximation along solutions of the system allows to construct a global Lyapunov function. This result can be regarded as a new inverse Lyapunov theorem arising from Riemannian metric.