Time Optimal Return of a Dynamic Object

TL;DR

This paper presents an explicit solution for the time-optimal return of a particle to the origin using bounded force, with analysis of control strategies and motion regimes relevant to control theory and flight mechanics.

Contribution

It provides a novel explicit integration of the controlled motion equations and analyzes optimal control depending on problem parameters.

Findings

Explicit solutions for optimal return time and control are derived.

Qualitative behavior of solutions is characterized.

Comparison with other motion regimes is provided.

Abstract

We solve the problem concerning a time optimal return of a particle with a prescribed velocity to the origin by applying a magnitude-bounded force. The equations of controlled motion are derived and explicitly integrated, and the optimal open-loop control and optimal time are analyzed depending on the parameters of the problem. The qualitative behavior of the solution is established, and the solution is compared with other regimes of motion. The results are of interest for control theory and its applications to controlled flight mechanics.