Time Optimal Return of a Dynamic Object

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.