Dynamic Lyapunov functions: Properties and applications

Lyapunov functions are a fundamental tool to investigate the stability properties of equilibrium points of linear and nonlinear systems. The existence of Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. Nevertheless the actual computation (of the analytic expression) of the function may be difficult. Herein we introduce the concept of Dynamic Lyapunov function together with results relating the stability properties of an equilibrium point to the existence of a Dynamic Lyapunov function. A positive definite function is combined with additional dynamics that render Dynamic Lyapunov functions easier to construct than Lyapunov functions. In fact the construction of the former does not require the knowledge of the solution of the underlying ordinary differential equation or of any partial differential equation or inequality. Moreover applications of Dynamic Lyapunov functions to the analysis and design of control systems are presented.