A. Mashtakov, A. Podobryaev, Geodesic Flow of the Sub-Riemannian Structure of Engel Type with Strictly Abnormal Extremals

Authors: A. Mashtakov, A. Podobryaev
Title: Geodesic Flow of the Sub-Riemannian Structure of Engel Type with Strictly Abnormal Extremals
Abstract:

We consider a left-invariant sub-Riemannian problem of Engel type on the central extension of the special linear group. Interest in this problem comes from the fact that it has strictly  abnormal trajectories, and the normal geodesic flow is Liouville integrable. An extremal  trajectory is called strictly abnormal if it is not present among normal geodesics. It is known that the most complicated singularities of the sub-Riemannian metric arise near abnormal  trajectories. The presence of a strictly abnormal trajectory in combination with the integrability  of the normal geodesic flow makes the problem under consideration a model example for  studying the singularities of the sub-Riemannian metric. We apply to the problem the invariant formulation of Pontryagin maximum principle (PMP), in which the vertical subsystem  (for adjoint variables) of the Hamiltonian system of PMP is independent of the state variables.  We show that the vertical subsystem is reduced to the equation of a skewed pendulum. The  first integrals of the system are found and an explicit solution is obtained in a special case. In  the general case, we carry out a qualitative analysis of the phase flow of the Hamiltonian  system. 

Citation: A. Mashtakov, A. Podobryaev. Geodesic Flow of the Sub-Riemannian Structure of Engel Type with Strictly Abnormal Extremals // 16th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference). pp. 1-3, IEEE (2022) DOI: 10.1109/STAB54858.2022.9807528.
Conference: 16th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference)
Place:

Moscow, ICS RAS

Year: 2022
Pages: 1-3