Event description
Abstract: In 1970, N. Tanaka introduced a method for constructing a canonical frame for distributions with constant Tanaka symbol. In 2009, B. Doubrov and I, building on earlier works of A. Agrachev, R. Gamkrelidze, and myself, employed a symplectification procedure to obtain a canonical frame for distributions independent of their Tanaka symbol. However, this required an additional assumption—the maximality of class of the distribution.
In a recent joint work with N. Day, we proved that all bracket generating rank-2 distributions with 5-dimensional cube are of maximal class at a generic point. This result allows one to assign a canonical frame at a generic point to every rank-2 distribution that is not of Goursat type. On the optimal control side, this result implies that for bracket-generating rank-2 distributions with 5-dimensional cube, there exist plenty of abnormal extremal trajectories starting from a generic point.
Further, in the rank-2 case, I will give an interpretation of the symplectification procedure in terms of a classical construction known as Cartan prolongation and discuss the question of the minimal number of iterative Cartan prolongations needed for the Tanaka symbols to become unified or finitely unified.
In contrast with the rank-2 situation, we found examples of rank-3 distributions with 6-dimensional square that are not of maximal class. In particular, I will present a (3, 8) distribution of non-maximal class whose symmetry algebra has dimension 29 and contains a semidirect sum of the exceptional Lie algebra 𝔤₂ with a copy of its adjoint module.