On global Lyapunov functions of nonlinear autonomous systems

In this paper we study a method for constructing Lyapunov functions defined on the whole region of attraction of an exponentially stable equilibrium point of nonlinear autonomous systems. The method involves solving a partial differential equation. We prove that a solution exists in the region of attraction. When the eigenvalues of the linearization of the system are non-resonant, a power series solution can be found, and its region of convergence can be used to determine the region of attraction.