  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.
