Control theory problems and the Rashevskii-Chow theorem on a Cartan group (Q6631338)

The canonical Cartan group \(\mathbb{K}\) is a certain 5-dimensional Lie group, diffeomorphic to \(\mathbb{R}^5\), whose group structure depends on three positive constants. It is equipped with a rank-2 distribution defined by two left-invariant vector fields on \(\mathbb{K}\). This distribution is bracket-generating, ensuring that any two points in \(\mathbb{K}\) can be connected by a horizontal curve, as guaranteed by the Rashevskii-Chow Theorem. In this context, horizontal curves correspond to the trajectories of a ball rolling on a plane without slipping.\N\NThe authors address the problem of finding the minimum number \(k\) such that, locally, any two points in \(\mathbb{K}\) can be connected by a horizontal curve composed of \(k\) segments, each segment being an integral curve of \(X\) or \(Y\) (referred to as a horizontal \(k\)-broken line). They prove that the minimum \(k\) for this connection is \(4\). Furthermore, they establish that the minimum \(k\) for which a closed horizontal \(k\)-broken line exists is \(6\).