Some properties and applications of minimum time control

This thesis contributes to the optimal time study of control-affine systems. These problems arise naturally from physics, and contains, for instance, mechanical systems. We tackle the study of their singularities, while minimizing the final time, meaning the time on which the aim is reached. We give a precise study of the extremal flow, for mechanical systems, for starter, and then, in general. \linebreak This leads to the knowledge of the flow regularity: it is smooth on a stratification around the singular set. We then apply those results to mechanical systems, and orbit transfer problems, with two and three bodies, giving an upper bound to the number of singularities occurring during a transfer. \linebreak We then change our viewpoint to study the optimality of such extremal in general, and give an optimality criteria than can be easily checked \linebreak numerically. In the last chapter we study the singularities of the controlled Kepler problem through another path: we prove a non-integrability theorem - in the Liouville sens - for the Hamiltonian system given by the minimum time orbit transfer (or rendez-vous) problem in the Kepler configuration.