Abstract
Let \({\dot{q}=X_{0}+\sum_{j=1}^{k}u_{j}X_{j}}\) be a control affine system on a manifold M, let C be a convex compact subset of \({\mathbb{R}^{k}}\), dim C > 0, let q 0 be a fixed point of M, and let U be a neighbourhood of q 0. We consider three reachable sets from q 0 for our system which are generated by square integrable controls with values in C, riC—the relative interior of C, and rbC—the relative boundary of C, respectively, with contraints on a state variable q of the form \({q\in U}\). Among other things, we investigate the relation between closures, interiors and boundaries of the three reachable sets. We also show how methods of the sub-Lorentzian geometry can serve as an auxiliary tool in the study of control affine systems.
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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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Grochowski, M. Some properties of reachable sets for control affine systems. Anal.Math.Phys. 1, 3–13 (2011). https://doi.org/10.1007/s13324-010-0001-y
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DOI: https://doi.org/10.1007/s13324-010-0001-y