# American Institute of Mathematical Sciences

February  2016, 10(1): 63-78. doi: 10.3934/amc.2016.10.63

## 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, Germany, Germany

Received  December 2014 Revised  July 2015 Published  March 2016

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.
Citation: Uwe Helmke, Jens Jordan, Julia Lieb. Probability estimates for reachability of linear systems defined over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 63-78. doi: 10.3934/amc.2016.10.63
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