Abstract
We study control-affine systems with n − 1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank (equivalently, they may be considered as 1-dimensional systems with n − 1 inputs entering nonlinearly). We provide a complete classification of such generic systems and their one-parameter families. We show that a generic family for n > 2 is equivalent (with respect to feedback or orbital feedback transformations) to one of nine canonical forms which differ from those for n = 2 by quadratic terms only. We also describe all generic bifurcations of 1-parameter families of systems of the above form.
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This work was partially supported by Ministry of Research and Higher Education grant N201 039 32/2703 and by the European Community Marie Curie Fellowship through the CTS contract HMPT-CT-2001-00278.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Rupniewski, M.W., Respondek, W. A classification of generic families of control-affine systems and their bifurcations. Math. Control Signals Syst. 21, 303–336 (2010). https://doi.org/10.1007/s00498-010-0047-2
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DOI: https://doi.org/10.1007/s00498-010-0047-2