Abstract
Feedback control is one of the key approaches in the controlled dynamical systems, allowing to stabilize complex constructs on some desirable trajectories (or attractors). It is well known that the disparity between the homotopy type of the configuration space of the system and the attractor precludes existence of a continuous vector field realizing such a stabilization, thus forcing the discontinuity of the control function at some points of the configuration space. How complicated should be this discontinuity set? In this note we present some simple lower bounds on the ranks of homology groups of such sets of discontinuities (the cuts), and discuss some examples.
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Acknowledgements
This project was started and developed under the influence of Dan Koditschek, telling the stories about robots, and interpreting them as mathematical queries. I am most indebted to him.
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The research is supported by the Multidisciplinary University Research Initiatives (MURI) Program SLICE.
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Baryshnikov, Y. Topological perplexity of feedback stabilization. J Appl. and Comput. Topology 7, 75–87 (2023). https://doi.org/10.1007/s41468-022-00098-2
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DOI: https://doi.org/10.1007/s41468-022-00098-2