Mechanical systems and rendez-vous controllability

Motivated by the study of the controlled Kepler problem, we analyze the controllability properties of some classes of mechanical systems. Consider a control system with drift. Our aim is to determine if, given a pair of initial states at a fixed initial time, assuming that the controller acts only on one of the two corresponding trajectories, it is possible for the controlled trajectory to reach the uncontrolled one in finite time. We prove that this is the case for the controlled Kepler problem, taking into account both elliptic and non-elliptic orbits. We extend the result to other classes of mechanical controlled systems.