  We study time-minimum optimal control for a class of quantum two-dimensional
dissipative systems whose dynamics are governed by the Lindblad equation and
where control inputs acts only in the Hamiltonian. The dynamics of the control
system are analyzed as a bi-linear control system on the Bloch ball after a
decoupling of such dynamics into intra- and inter-unitary orbits. The
(singular) control problem consists of finding a trajectory of the state
variables solving a radial equation in the minimum amount of time, starting at
the completely mixed state and ending at the state with the maximum achievable
purity.
  The boundary value problem determined by the time-minimum singular optimal
control problem is studied numerically. If controls are unbounded, simulations
show that multiple local minimal solutions might exist. To find the unique
globally minimal solution, we must repeat the algorithm for various initial
conditions and find the best solution out of all of the candidates. If controls
are bounded, optimal controls are given by bang-bang controls using the
Pontryagin minimum principle. Using a switching map we construct optimal
solutions consisting of singular arcs. If controls are bounded, the analysis of
our model also implies classical analysis done previously for this problem.
