Higher derivatives of the end-point map of a control-linear system via adapted coordinates

Michał Jóźwikowski; Bartłomiej Sikorski

doi:10.3934/mcrf.2025018

Article Contents

2026, Volume 16: 279-314. Doi: 10.3934/mcrf.2025018

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Higher derivatives of the end-point map of a control-linear system via adapted coordinates

Michał Jóźwikowski^{1, 2, ,} and
Bartłomiej Sikorski^{1, 3,}

1.
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
2.
Department of Mathematics, University of Fribourg, Switzerland
3.
Faculty of Physics, University of Warsaw, Poland

^*Corresponding author: Michał Jóźwikowski
^*Corresponding author: Michał Jóźwikowski

Received: August 2023

Revised: January 2025

Early access: March 2025

Published: March 2026

Part of this research was conducted during the employment of MJ at the University of Fribourg, financed by the ERC Starting Grant Geometry of Metric Groups, grant agreement 713998 GeoMeG. BS was supported by the Ministry of Science and Higher Education of the Republic of Poland project "Szkoła Orłów", project number 500-D110-06-0465160.

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Abstract

We study the end-point map of a control-linear system in a neighborhood of an arbitrarily chosen trajectory. In particular, we want to calculate the $ k $-th order derivative of this map in a given direction. A priori it is a solution of a quite complicated ODE depending on all derivatives of order less than or equal to $ k $. We prove that there exists a special coordinate system adapted to the geometry of the problem, which changes the system of ODEs describing all derivatives of the end-point map up to order $ k $ to equations of a control-affine (non-autonomous control-linear) system, with the direction of derivation playing the role of the new control. As an application, we study controllability criteria for this system, obtaining first and second-order necessary optimality conditions of sub-Riemannian geodesics. In particular, for the case of an abnormal minimizer, we can interpret Goh conditions as non-controllability conditions of this control-affine system for $ k = 2 $. We make the conjecture that for higher $ k $'s its non-controllability corresponds to recently obtained higher-order analogs of the Goh conditions [Boarotto, Monti, Palmurella, 2020], [Boarotto, Monti, Socionovo, 2022]. We also use our results to obtain an estimate on the residue of the Taylor expansion of the end-point map.

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Mathematics Subject Classification: Primary: 93C10, 93B11, 93C73; Secondary: 53C17.

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