Generalized pseudo-product structures and finite type distributions via abnormal extremals

TL;DR

This paper extends Tanaka's classical results to pseudo-product structures with non-degree-1 distributions, establishing finiteness of symmetries for controllable distributions via abnormal extremals.

Contribution

It generalizes the notion of universal prolongation and finiteness criteria, solving an open problem about symmetry finiteness for certain distributions.

Findings

Distributions controllable by abnormal extremals have finite-dimensional symmetries.

Generalized prolongation criterion applies to non-degree-1 distributions.

Settles an open problem from 2013 on symmetry finiteness.

Abstract

We generalize the classical Tanaka result on the finiteness of symmetry algebra for non-degenerate pseudo-product structures to the case when the completely-integrable distributions defining the pseudo-product structure are no longer concentrated in the degree $-1$. In order to do this, we modify the notion of universal prolongation of graded nilpotent Lie algebras and generalize the original finiteness criterion of Tanaka. Using this result, we demonstrate that in real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries. This result settles Problem V in the affirmative from the 2013 list of open problems by Andrei Agrachev. Additionally, we discuss applications to symmetries and natural equivalence problems for systems of ODEs of mixed order.