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. 2018 Feb 16;8:3188. doi: 10.1038/s41598-018-21398-7

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© The Author(s) 2018

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Reachability, Controllability-to-0 and Complete Controllability problems. (a) The reachability (or controllability-from-0) problem is difficult along the stable eigendirections of A (red curves in the leftmost panel) and easy along the unstable ones (blue). This is reflected in the surfaces of $ℰ_{r} (t_{f}) = x_{f}^{T} W_{r}^{- 1} (t_{f}) x_{f}$ shown in the 3 rightmost panels. In particular, the reachability problem requires limited control energy when A is antistable (rightmost panel). (b) The controllability-to-0 problem is difficult along the unstable eigendirections of A (red) and easy along the stable ones (blue). The input energy surfaces, $ℰ_{c} (t_{f}) = x_{o}^{T} W_{c}^{- 1} (t_{f}) x_{o}$ , reflect these properties. The case of A stable requires the least control energy. (c) The problem studied in this paper, complete controllability, is a mixture of the two cases, collecting the worst-case situations of both. When the real part of the eigenvalues of A is squeezed towards the imaginary axis as in the right panels of Fig. 2(a), the input energy reduces accordingly.