Optimal Time Control of Uniform Linear Motion: Problem Formulation and Switching Time Characterization Via Pontryagin’s Maximum Principle.

Abstract:

Optimal control theory seeks to determine effective controls for dynamical systems, driven by the pressing need for efficient use of energy, time, and resources. This paper examines the optimal time-control problem for the uniform linear motion of a train between two fixed points. The problem is rigorously formulated using differential equations and the Newtonian principles of motion. The Pontryagin Maximum Principle (PMP) is applied to derive the necessary conditions for optimality and to determine the optimal control that minimises the travel time of the body. Both analytical and geometrical approaches are employed to reconstruct optimal trajectories and characterise the switching curve that governs transitions between control values. The study demonstrates that the optimal control is uniquely determined by the initial state of the system relative to the switching curve, providing a practical and computationally tractable feedback control conditions.