Generalized pseudo-product structures and finite type distributions via abnormal extremals
Abstract
We generalize the classical Tanaka result on the finiteness of the Lie algebra of infinitesimal symmetries 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.
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BibTeX
@misc{doubrov2026generalized,
title = {{Generalized pseudo-product structures and finite type distributions via abnormal extremals}},
author = {Boris Doubrov and Igor Zelenko},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.12307},
eprint = {2605.12307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.12307},
}
RIS
TY - UNPB
TI - Generalized pseudo-product structures and finite type distributions via abnormal extremals
AU - Boris Doubrov
AU - Igor Zelenko
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2605.12307
UR - https://arxiv.org/abs/2605.12307
AB - We generalize the classical Tanaka result on the finiteness of the Lie algebra of infinitesimal symmetries 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.
ER -