On $K-P$ sub-Riemannian problems and their cut locus.

Domenico D'Alessandro, Benjamin Sheller

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 \cite{Bredon} 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 {\bf 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 $SU(2)$.