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

Abstract

The problem of finding minimizing geodesics for a manifold <tex>$M$</tex> with a sub-Riemannian structure is equivalent to the time optimal control of a driftless system on <tex>$M$</tex> with a bound on the control. We consider here a class of sub-Riemannian problems on the classical Lie groups <tex>$G$</tex> where the dynamical equations are of the form <tex>$\\dot{x}=\\sum\\nolimits_{j}X_{j}(x)u_{j}$</tex> and the <tex>$X_{j}=X_{j}(x)$</tex> are right invariant vector fields on <tex>$G$</tex> and <tex>$u_{j}:=u_{j}(t)$</tex> the controls. The vector fields <tex>$X_{j}$</tex> are assumed to belong to the P part of a Cartan K-P decomposition. These types of problems admit a group of symmetries <tex>$K$</tex> which act on <tex>$G$</tex> by conjugation. Under the assumption that the minimal isotropy group in <tex>$K$</tex> is discrete, we prove that we can reduce the problem to a Riemannian problem on the regular part of the associated quotient space <tex>$G/K$</tex>. On this part we define the corresponding quotient metric. For the special cases of the K-P decomposition of <tex>$SU(n)$</tex> of type AIII we prove that the assumption on the minimal isotropy group is verified. As an example of application of the techniques discussed we find the cut locus of a K-P optimal control problem on <tex>$SU(2)$</tex>.