From Asymptotic to Designer-Assigned-Time Control: A Review of Stability Notions, Design Mechanisms, and Controller Architectures

Abstract: Many control tasks require a target to be reached not only eventually but on time. Finite-, fixed-, predefined-, and prescribed-time control address this need, yet the labels are used loosely: a settling time that grows with the initial condition, a bound that holds for all initial conditions, a deadline chosen by the designer, and a limit attained only at the terminal instant often share one name. This review aims to make such claims comparable. It separates three questions that are often conflated: what temporal property is promised, which feature of the Lyapunov analysis produces it, and which controller architecture carries it into the closed loop. A recurring question is whether a guarantee proven for an idealized loop survives once observers, adaptation, disturbances, actuator limits, and digital implementation are included. The examined studies are therefore audited one by one, recording what each claims, which variable is actually certified, and how the result is validated. The audit shows that estimation and approximation errors often reduce an exact guarantee to a practical one, and that experimental evidence comes mostly from fast electromechanical systems. A scalar benchmark with two complementary tunings shows how much of the apparent difference between methods stems from conservative bounds, initial-condition dependence, and numerical tolerance, and how saturation and sampling can delay or remove a deadline. The review closes with open problems, among them deciding which deadlines a given plant can meet, combining certificates across interconnected subsystems, and preserving a guarantee through implementation.