Yuri Sachkov
Program Systems Institute, Russian Academy of Sciences
Publications:
Ardentov A. A., Sachkov Y. L.
Maxwell Strata and Cut Locus in the SubRiemannian Problem on the Engel Group
2017, vol. 22, no. 8, pp. 909–936
Abstract
We consider the nilpotent leftinvariant subRiemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 subRiemannian structure on a 4manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest subRiemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a oneparameter group of dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification with 6 threedimensional strata, 12 twodimensional strata, and 2 onedimensional strata. Threedimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Twodimensional strata of the cut locus consist of conjugate points. Finally, onedimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of subRiemannian geodesics to the 2dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.

Mashtakov A. P., Ardentov A. A., Sachkov Y. L.
Relation Between Euler’s Elasticae and SubRiemannian Geodesics on $SE(2)$
2016, vol. 21, no. 78, pp. 832839
Abstract
In this note we describe a relation between Euler’s elasticae and subRiemannian geodesics on $SE(2)$. Analyzing the Hamiltonian system of the Pontryagin maximum principle, we show that these two curves coincide only in the case when they are segments of a straight line.
