Convex trigonometry in problems with two-dimensional control

The talk will discuss a new convenient method of describing flat convex compact sets and their polar sets, which generalizes classical trigonometric functions $cos$ and $sin$. The properties of this pair of functions in the case of unit circle are inherited by two pairs of functions $cos_\Omega$, $sin_\Omega$, and $\cos_{\Omega^\circ}$, $\sin_{\Omega^\circ}$ for the set $\Omega$ and its polar set $\Omega^\circ$. This method turned out to be very useful in optimal control problems with two-dimensional control. In 2018, using this method the explicit geodesic flows in a series of sub-Finsler problems for the cases of Heisenberg, Grushin, Martinet, Engel, and Cartan were found. In 2019, jointly with Yu.L. Sachkov and A.A. Ardentov, more than 10 classical problems were explicitly solved. For instance, I will talk on Finsler geodesic flows on the Lobachevsky plane.