Minimal-time trajectories of a linear control system on a homogeneous space of the 2D Lie group

Through the Pontryagin maximum principle, we solve a minimal-time problem for a linear control system on a cylinder, considered as a homogeneous space of the solvable Lie group of dimension two. The main result explicitly shows the existence of an optimal trajectory connecting every couple of arbitrary states on the manifold. It also gives a way to calculate the corresponding minimal time. Finally, the system admits points with two distinct minimal-time trajectories connecting them.