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On controlled invariance of regular distributions

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Abstract

This paper considers the problem of controlled invariance of involutive regular distribution, both for smooth and real analytic cases. After a review of some existing work, a precise formulation of the problem of local and global controlled invariance of involutive regular distributions for both affine control systems and affine distributions is introduced. A complete characterization for local controlled invariance of involutive regular distributions for affine control systems is presented. A geometric interpretation for this characterization is provided. A result on local controlled invariance for real analytic affine distribution is given. Then, we investigate conditions that allow passages from local controlled invariance to global controlled invariance, for both smooth and real analytic affine distributions. We clarify existing results in the literature. Finally, for manifolds with a symmetry Lie group action, the problem of global controlled invariance is considered.

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Acknowledgements

This work was performed while the author was visiting the Department of Mathematics and Statistics, Queen’s University. The author would like to thank Professor Andrew Lewis for some enlightening talks, for example on the use of Cartan’s Theorem B. The author would also like to thank the anonymous reviewers for their careful reading of the manuscript and a lot of helpful and constructive suggestions which help improve the presentation of the paper a lot. This work was supported by CSC and the NSFC Grant 61703211.

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  1. School of Automation, CICAEET, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Qianqian Xia

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Xia, Q. On controlled invariance of regular distributions. Math. Control Signals Syst. 33, 79–107 (2021). https://doi.org/10.1007/s00498-020-00275-7

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