Methods for Optimal Control and Control Theory

Article # 24_2018

12 p.

PDF

Anton Popov, Yuri Sachkov
Two-side bound of a root of an equation containing complete elliptic integrals

We prove uniqueness of root of an equation arising in a problem of geometric control theory. The problem consists of description of singularity of the sub-Riemannian sphere on the Engel group near abnormal length minimizer.
During the proof, several new inequalities for complete elliptic integrals were obtained. For example, we proved that the function K(k)E(k) is increasing at the segment [0, 1); this fact was not noticed before in literature.
The method of investigation developed and the results obtained can be useful both for the study of elliptic integrals and for solving problems were such integrals arise (e.g. in problems of sub-Riemannian geometry). (In Russian).

Key words: asymptotics, complete elliptic integral, sub-Riemannian geometry.

article citation

DOI

Article # 28_2018

42 p.

PDF

Alexey Mashtakov
On the step-2 nilpotent (n, n(n + 1)/2) sub-Riemannian structures

We consider the Sub-Riemannian (SR) problem on the step-2 free nilpotent Lie groups Gn. This problem is classical in SR geometry and in some sense the simplest open problem nowadays. Although the problem satisfies to a wide group of symmetries, the cut locus is known only in the cases of small dimensions n = 2, 3. In the general case there exists a conjecture by Rizzi–Serres claiming that the cut locus consists on stable points of the specific symmetry. In this paper, we derive the geodesic equations via PMP and study the symmetries of the corresponding Hamiltonian system. Then, using method of reduction over the symmetries, we propose an idea to prove the conjecture for the general n ≥ 2. We study the cases n = 2, 3, 4 in details and show pictures (for n = 2, 3) of the SR wave front with indicated the cut locus on it. (In Russian).

Key words: Sub-Riemannian geometry, geodesic, shortest, cut set, Carnot group.

article citation

DOI