Abstract: While traditional asymptotic control methods are limited in convergence rate and precision, finite-time control theory provides a superior alternative. However, its practical application is constrained by the dependence of the settling time on the system's initial conditions. To overcome this limitation, fixed-time stability theory was developed, which establishes a deterministic upper bound for the settling time independent of initial states. Further advancing this concept, predefined-time (or prescribed-time) stability theory allows this upper bound to be directly assigned as a tunable parameter by the designer. This theoretical evolution has opened new avenues for achieving fast and accurate control of complex nonlinear systems. This paper systematically reviews this important theoretical progression. It clarifies the core definitions of various finite-time stability concepts and offers a detailed comparative analysis of the corresponding Lyapunov-based stability criteria, highlighting their differences in settling time estimation. Finally, the characteristics of these criteria are summarized, and future research directions are outlined to provide a clear theoretical framework and guidance for the field.

Key words: Finite-time stability, Fixed-time stability, Predefined-time stability, Prescribed-time stability, Convergence time, Lyapunov stability