Abstract:
Methods of nilpotent approximation are actively developed for the last years. For example the famous Gromov theorem on approximation of metrics on Carnot-Carathéodory spaces by left-invariant subriemannian metrics on nilpotent Lie groups is widely known. General optimal control problems (and also many Hamiltonian systems with discontinuous right hand side) may be approximated by nilpotent-convex problems of optimal control. These problems can be studied precisely because of the rich internal geometry. For example the old question about the structure of the set of singularity points of optimal controls has been solved recently. There are well known artificial examples by Filippov and Silin with an optimal control having discontinuities on a positive measure set. This phenomenon never happen in nilpotent-convex problems: optimal control can have a countable number of switchings at most (this situation is general due to Kupka–Zelikin–Borisov theorem).
As one of examples I will show a series of problems with multidimensional control in a ball. Optimal control here is moving along an irrational winding of a Clifford torus lying in the boundary of the ball and the optimal trajectory is a generalised logarithmic spiral spanned by the winding. Moreover the moving process along the whole winding takes finite time and at the final moment the optimal trajectory hits the origin. The proof of irrationality of the winding is being reduced to the investigation of linear independence over $\mathbb Q$ of roots of a series of special polynomials and is based on Galois theory.