  The problems of optimizing the value of an arbitrary observable of the
two-level system at both a fixed time and the shortest possible time is
theoretically explored. Complete identification and classification along with
comprehensive analysis of globally optimal control policies and traps (i.e.
policies which are locally but not globally optimal) is presented. The central
question addressed is whether the control landscape remains trap-free if
control constraints of the inequality type are imposed. The answer is
astonishingly controversial, namely, although formally it is always negative,
in practice it is positive provided that the control time is fixed and chosen
long enough.
