Issue |
ESAIM: COCV
Volume 31, 2025
|
|
---|---|---|
Article Number | 7 | |
Number of page(s) | 26 | |
DOI | https://doi.org/10.1051/cocv/2024079 | |
Published online | 24 January 2025 |
Conjugate points of dynamic pairs and control systems
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
* Corresponding author: b.jakubczyk@impan.pl
Received:
27
October
2023
Accepted:
27
October
2024
We study the geometry of dynamic pairs (X, V) on a manifold M, where X is a vector field and V is a distribution on M, both satisfying a regularity condition. Special cases are pairs defined by systems of second order ODEs, geodesic sprays in Riemannian, Finslerian and Lagranian geometries, semi-Hamiltonian systems and control-affine systems. Analogs of conjugate points from the calculus of variations are defined for the pair (X, V). The main results give estimates for the position of conjugate points in terms of a curvature operator, analogously to the Cartan–Hadamard and Bonet–Myers theorems. Contrary to classical cases, no metric is given a priori, the distribution V may be nonintegrable and the curvature operator is defined in terms of (X, V).
Mathematics Subject Classification: 93C10 / 34C14 / 53A55
Key words: Dynamic pairs / control systems / geodesic flows / conjugate points / curvature / Jacobi fields
© The authors. Published by EDP Sciences, SMAI 2025
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.