Dmitriy Stepanov

We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a threedimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

The simplest almost Riemannian problem with a Martinet point is studied. Extremal and optimal trajectories are studied by analytic, symbolic, and numeric techniques.
Analytically the following results were obtained: existence of optimal trajectories was proved, absence of abnormal trajectories was shawn, a Hamiltonian system for normal extremals was derived, its symmetry and Maxwell points was described. Symbolically all polynomial integrals of the Hamiltonian system of degree not greater than 54 were described. Numerically the optimal synthesis was constructed.