Control theory problems and the Rashevskii–Chow theorem on a Cartan group

We consider the problem of controlling the nonlinear $5$-dimensional systems that are induced by horizontal vector fields $X$ and $Y$ which together with their commutators generate some Cartan algebra depending linearly on two piecewise constant controls. We also study the properties of solutions to the systems on interpreting a solution as a horizontal $k$-broken line $L_k$ on the canonical Cartan group $\Bbb K$, where the segments of $L_k$ are segments of integral curves of the vector fields of the form $aX+bY$ with $a,b=\mathrm{const}$. As regards $\Bbb K$, we prove that $4$ is the minimal number $N_{\Bbb K}$ such that every two points $u,v\in\Bbb K$ can be joined by some $L_k$ with $k\leq N_{\Bbb K}$. Thus, we obtain the best version of the Rashevskii–Chow theorem on the Cartan group. We also show that the minimal number of segments of a closed horizontal broken line on $\Bbb K$ equals 6.