Geodesic
A Sub-Finsler Problem on the Cartan Group
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control and differential equations, Tome 304 (2019), pp. 49-67

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We study a sub-Finsler geometric problem on the free nilpotent group of rank $2$ and step $3$. Such a group is also called the Cartan group and has a natural structure of Carnot group, which we metrize by considering the $\ell _\infty $ norm on its first layer. We adopt the point of view of time-optimal control theory. We characterize extremal curves via the Pontryagin maximum principle. We describe abnormal and singular arcs and construct the bang–bang flow. 
@article{TM_2019_304_a3,
     author = {A. A. Ardentov and E. Le Donne and Yu. L. Sachkov},
     title = {A {Sub-Finsler} {Problem} on the {Cartan} {Group}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {49--67},
     publisher = {mathdoc},
     volume = {304},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2019_304_a3/}
}
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A. A. Ardentov; E. Le Donne; Yu. L. Sachkov. A Sub-Finsler Problem on the Cartan Group. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control and differential equations, Tome 304 (2019), pp. 49-67. http://geodesic.mathdoc.fr/item/TM_2019_304_a3/