  The problem of finding minimizing geodesics for a manifold M with a
sub-Riemannian structure is equivalent to the time optimal control of a
driftless system on M with a bound on the control. We consider here a class of
sub-Riemannian problems on the classical Lie groups G where the dynamical
equations are of the form \dot x=\sum_j X_j(x) u_j and the X_j=X_j(x) are right
invariant vector fields on G and u_j:=u_j(t) the controls. The vector fields
X_j are assumed to belong to the P part of a Cartan K-P decomposition. These
types of problems admit a group of symmetries K which act on G by conjugation.
Under the assumption that the minimal isotropy group in K
  is discrete, we prove that we can reduce the problem to a Riemannian problem
on the regular part of the associated quotient space G/K. On this part we
define the corresponding quotient metric. For the special cases of the K-P
decomposition of SU(n) of type AIII we prove that the assumption on the minimal
isotropy group is verified. Moreover, under the assumption that the quotient
space G/K with the given metric has negative curvature we prove that the cut
locus has to belong to the singular part of G. As an example of applications of
these techniques we characterize the cut locus for a problem on SU(2) of
interest in the control of quantum systems.
