Alexey Podobryaev
Publications:
Podobryaev A.
Symmetric Extremal Trajectories in LeftInvariant Optimal Control Problems
2019, Vol. 15, no. 4, pp. 569575
Abstract
We consider leftinvariant optimal control problems on connected Lie groups. We describe
the symmetries of the exponential map that are induced by the symmetries of the vertical part
of the Hamiltonian system of the Pontryagin maximum principle. These symmetries play a key
role in investigation of optimality of extremal trajectories. For connected Lie groups such that
the generic coadjoint orbit has codimension not more than 1 and a connected stabilizer we
introduce a general construction for such symmetries of the exponential map.

Podobryaev A.
Antipodal Points and Diameter of a Sphere
2018, Vol. 14, no. 4, pp. 579581
Abstract
We give an example of a Riemannian manifold homeomorphic to a sphere such that its
diameter cannot be realized as a distance between antipodal points. We consider a Berger sphere,
i.e., a threedimensional sphere with Riemannian metric that is compressed along the fibers of
the Hopf fibration. We give a condition for a Berger sphere to have the desired property. We
use our previous results on a cut locus of Berger spheres obtained by the method from geometric
control theory.
