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2013
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# Yuri Sachkov

Program Systems Institute, Russian Academy of Sciences

## Publications:

 Sachkov Y. L. Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems 2020, vol. 25, no. 1, pp.  33-39 Abstract We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented. Keywords: optimal control, sub-Finsler geometry, Lie groups, Pontryagin maximum principle Citation: Sachkov Y. L.,  Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 33-39 DOI:10.1134/S1560354720010050
 Ardentov A. A., Le Donne E., Sachkov Y. L. Sub-Finsler Geodesics on the Cartan Group 2019, vol. 24, no. 1, pp.  36-60 Abstract This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that any normal extremal is either bang-bang, or singular, or mixed. Consequently, we study mixed extremals. In particular, we prove that every two points can be connected by a piecewise smooth minimizer, and we give a uniform bound on the number of such pieces. Keywords: Sub-Finsler geometry, time-optimal control, geometric control, Cartan group Citation: Ardentov A. A., Le Donne E., Sachkov Y. L.,  Sub-Finsler Geodesics on the Cartan Group, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 36-60 DOI:10.1134/S1560354719010027
 Ardentov A. A., Sachkov Y. L. Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group 2017, vol. 22, no. 8, pp.  909–936 Abstract We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian 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 one-parameter 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 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional 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. Keywords: sub-Riemannian geometry, optimal control, Engel group, Maxwell strata, cut locus, mobile robot, Euler's elasticae Citation: Ardentov A. A., Sachkov Y. L.,  Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 909–936 DOI:10.1134/S1560354717080020
 Mashtakov A. P., Ardentov A. A., Sachkov Y. L. Relation Between Euler’s Elasticae and Sub-Riemannian Geodesics on $SE(2)$ 2016, vol. 21, no. 7-8, pp.  832-839 Abstract In this note we describe a relation between Euler’s elasticae and sub-Riemannian 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. Keywords: elastica, sub-Riemannian geodesic, group of rototranslations Citation: Mashtakov A. P., Ardentov A. A., Sachkov Y. L.,  Relation Between Euler’s Elasticae and Sub-Riemannian Geodesics on $SE(2)$, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 832-839 DOI:10.1134/S1560354716070066