Abstract:
I will talk about a new convenient method for describing plane convex
compact sets and their polars, which generalizes the classical
trigonometric functions $\cos$ and $\sin$. Properties of this pair of
functions in the case of the unit circle are inherited by two pairs of
functions $\cos_\Omega$, $\sin_\Omega$ and $\cos_{\Omega^\circ},
\sin_{\Omega^\circ} $ constructed for the set $\Omega$ and its polar
$\Omega^\circ$. This method has proven to be very useful for explicitly
describing solutions of optimal control problems with two-dimensional
control. With its help, in 2018, I was able to explicitly find geodesics
in a series of subfinsler problems for the cases of Heisenberg, Grushin,
Martin, Engel, and Cartan. In 2019, together with Yu. L. Sachkov and
A. A. Ardentov we explicitly solved more than 10 classical problems. For
example, in the talk I will show Finsler geodesics on the Lobachevsky
plane.
