On K-P sub-Riemannian Problems and their Cut Locus

Abstract

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.