Vol 210, No 8 (2019)

It is proved that two Chevalley groups with indecomposable root systems of rank $>1$ over commutative rings (which contain in addition $1/2$ for the types $\mathbf A_2$, $\mathbf B_l$, $\mathbf C_l$, $\mathbf F_4$, and $\mathbf G_2$, and $1/3$ for the type $\mathbf G_2$) are isomorphic or elementarily equivalent if and only if the corresponding root systems coincide, the weight lattices of the representation of the Lie algebra coincide, and the rings are isomorphic or elementarily equivalent, respectively. The isomorphisms of adjoint (elementary) Chevalley groups over the rings of the above types are also described. Bibliography: 25 titles.

A new convenient method for describing flat convex compact sets and their polar sets is proposed. It generalizes the classical trigonometric functions $\sin$ and $\cos$. It is apparent that this method can be very useful for an explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Finsler problems with two-dimensional control lying in an arbitrary convex set $\Omega$ is investigated. Namely, problems on the Heisenberg, Engel, and Cartan groups and also Grushin's and Martinet's cases are considered. Particular attention is paid to the case when $\Omega$ is a convex polygon.Bibliography: 13 titles.