Òptimes

Òptimes is a term used in time-dependent optimization to denote the set of time points at which a system attains an optimum value of a defined objective function. The concept appears in fields such as dynamic programming, optimal control, and scheduling, where the state evolves over time and the objective depends on both state and time. The word is derived from “optimum” with a diacritic used by some authors to mark a specialized class of optimal moments; its use is not standardized across the literature.

In practice, òptimes describe moments along a trajectory when the instantaneous or accumulated performance meets a local or global optimum according to the chosen criteria, such as minimum completion time, minimum energy, or lowest cost. Identifying òptimes typically involves solving time-dependent boundary-value problems, applying the Pontryagin Maximum Principle, or performing dynamic-programming recursions. Depending on the model, the set of òptimes can be discrete (switching instants) or form a continuous interval within the horizon, and may depend on initial conditions, constraints, and parameter values.

Applications of the concept include real-time scheduling, robotic path planning, energy-management systems, and manufacturing optimization, where determining the critical moments for action yields efficiency gains. Òptimes relate to other concepts like switching times, event times, and critical points in optimal control, serving as a useful heuristic in complex, time-sensitive decision problems. See also: optimal control, dynamic programming, time-dependent optimization.