Systematic construction of Lyapunov functions for nonlinear systems in critical cases

In this paper the local stability and stabilizability of fairly general classes of smooth nonlinear systems are analyzed using normal form methods. A brief comparison is made between normal form and center manifold methods. A new procedure for constructing a Lyapunov function for a locally asymptotically stable nonlinear system is presented. These ideas are also used for finding a smooth locally stabilizing state feedback for a nonlinear control system.