Generalized Lyapunov function theorems and its applications in switched systems

Generalized Lyapunov theorems of nonlinear systems are developed wherein all regularity assumptions on traditional Lyapunov function are removed. In particular, stability theorems of nonlinear systems are presented by replacing “VV along the system trajectories is non-increasing” with “VV along the system trajectories may increase its value during some proper time intervals”. Furthermore, stability theorems of discrete-time and continuous-time switched system with unstable subsystems are derived, using the generalized Lyapunov theorems. Two numerical examples are included to illustrate the effectiveness of the method.