Geodesic
Extremal trajectories in a~nilpotent sub-Riemannian problem on the Engel group
Sbornik. Mathematics, Tome 202 (2011) no. 11, pp. 1593-1615

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On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis. Bibliography: 25 titles.
Keywords: optimal control, sub-Riemannian geometry, geometric methods, Engel group. 
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     author = {A. A. Ardentov and Yu. L. Sachkov},
     title = {Extremal trajectories in a~nilpotent {sub-Riemannian} problem on the {Engel} group},
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A. A. Ardentov; Yu. L. Sachkov. Extremal trajectories in a~nilpotent sub-Riemannian problem on the Engel group. Sbornik. Mathematics, Tome 202 (2011) no. 11, pp. 1593-1615. http://geodesic.mathdoc.fr/item/SM_2011_202_11_a1/