Exponential Quality of Time-Varying Dynamical Systems: Stability and Tracking

Exponential properties of dynamical systems express their high-quality behaviour. Such properties are attractive from both the stability and tracking point of views. Krasovskii proved, by applying the Lyapunov methodology for nonlinear systems, the necessary and sufficient conditions for the exponential stability via existence of a system Lyapunov function obeying the corresponding conditions. However, the problem of the determination of an algorithm for an exact construction of the Lyapunov function has been left unsolved. The problem of the conditions that are both necessary and sufficient for a single algorithmic exact construction of a Lyapunov function for a given nonlinear system has recently been solved in the framework of asymptotic stability. Various criteria provide a new methodology for the application of the Lyapunov method to nonlinear systems. Any of them provides the final reply to the question whether x = 0 is asymptotically stable. The procedure ends after a single application of a criterion.