Probability estimates for reachability of linear systems defined over finite fields

Uwe Helmke; Jens  Jordan; Julia  Lieb

doi:10.3934/amc.2016.10.63

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2016, Volume 10, Issue 1: 63-78. Doi: 10.3934/amc.2016.10.63

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Probability estimates for reachability of linear systems defined over finite fields

1.
Institut für Mathematik; Lehrstuhl für Mathematik II, Universität Würzburg, Am Hubland, 97074 Würzburg,
2.
Institute of Mathematics, University of Würzburg, 97074 Würzburg

Received: December 2014

Revised: July 2015

Published: February 2016

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Abstract

This paper deals with the probability that random linear systems defined over a finite field are reachable. Explicit formulas are derived for the probabilities that a linear input-state system is reachable, that the reachability matrix has a prescribed rank, as well as for the number of cyclic vectors of a cyclic matrix. We also estimate the probability that the parallel connection of finitely many single-input systems is reachable. These results may be viewed as a first step to calculate the probability that a network of linear systems is reachable.

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Mathematics Subject Classification: Primary: 93B05, 93C05, 11T06; Secondary: 93B25, 93C55.

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