Subspaces of Maximal Singularity for Homogeneous Control Systems
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Subspaces of Maximal Singularity for Homogeneous Control Systems G. M. Sklyar1 · S. Yu. Ignatovich2 Accepted: 28 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Non-local properties of non-linear systems, linear with respect to control function, are studied. We consider a class of such systems that are homogeneous at the origin. At a point different from the origin the system may be non-homogeneous, so its homogeneous approximation is of interest. However, at such points, the system may become less singular than at the origin. We describe a set of points where the system is as singular as at the origin. It is shown that such a set forms a linear subspace. The main tool used in the paper is a free algebra approach proposed in our previous papers. Keywords Non-linear control system · Homogeneous approximation · Series of iterated integrals · Core Lie subalgebra · Set of maximal singularity Mathematics Subject Classification (2010) 93B11 · 93B25
1 Introduction and Motivation The homogeneous approximation problem was a center of attention of experts in the nonlinear control theory and the differential geometry during several decades. We mention some papers of the 1980s and 1990s [1, 4–6, 8, 24, 28], which had a big impact on the field. Obtained results were summarized in the survey of H. Hermes [9] and, later, in the paper of A. Bella¨ıche [3], which was considered to be a major reference in the homogeneous approximation problem. The latter paper presented an explanation of “approximating properties”
G. M. Sklyar
[email protected] S. Yu. Ignatovich [email protected] 1
Institute of Mathematics, University of Szczecin, Wielkopolska str. 15, Szczecin 70-451, Poland
2
V. N. Karazin Kharkiv National University, Svobody sqr. 4, Kharkiv 61022, Ukraine
G. M. Sklyar and S. Yu. Ignatovich
and proposed a method of finding a homogeneous approximation for control systems of the form: x˙ =
m
ui Xi (x),
x ∈ U (0) ⊂ Rn , u1 , . . . , um ∈ R,
(1)
i=1
where X1 (x), . . . , Xm (x) are real analytic vector fields in a neighborhood of the origin. However, this method involves finding a homogeneous approximation only after passing to special “privileged” coordinates. A construction of privileged coordinates (with applications to the controllability and time optimality) was proposed by G. Stefani [24–26]. Therefore, such a construction of a homogeneous approximation uses coordinates. This obstructs answering simpler questions, e.g., for two given systems to determine whether they have the same homogeneous approximation (without finding it). Thus, the topic was not exhausted by publishing [3]. A coordinate-free description of the homogeneous approximation was obtained by A. A. Agrachev and A. Marigo [2]. Another perspective tool was proposed by the authors of the present paper. We suggested developing the free algebraic technique, which was first applied to non-linear control systems by M. Fliess [7]; some useful discussions can be found in [13, 15]. Namely, M. Fliess propo
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