1. A first cubic upper bound on the local reachability index for some positive 2-D systems.
- Author
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Bailo, Esteban, Gelonch, Josep, and Romero-Vivó, Sergio
- Abstract
The calculation of the smallest number of steps needed to deterministically reach all local states of an n th -order positive 2-D system, which is called local reachability index ( I LR ) of that system, was recently tackled by means of the use of a suitable composition table. The greatest index I LR obtained in the previous literature was n + 3 n / 2 2 for some appropriated values of n. Taking as a basis both a combinatorial approach of such systems and the construction of suitable geometric sets in the plane, an upper bound on I LR depending on the dimension n for a new family of systems is characterized. The 2-D influence digraph of this family of order n ≥ 6 consists of two subdigraphs corresponding to a unique source s. The first one is a cycle involving the first n 1 vertices and is connected to the another subdigraph through the 1-arc (2 , n 1 + n 2) , being the natural numbers n 1 and n 2 such that n 1 > n 2 ≥ 2 and n - n 1 - n 2 ≥ 1 . The second one has two main cycles, a cycle where only the remaining vertices n 1 + 1 , ... , n appear and a cycle containing only the vertices n 1 + 1 , ... , n 1 + n 2 - 1 . Moreover, the last vertices are connected through the 2-arc (n 1 + n 2 - 1 , n) . Furthermore, if n ≥ 12 and is a multiple of 3, for appropriate n 1 and n 2 , the I LR of that family is at least cubic, exactly, it must be n 3 + 9 n 2 + 45 n + 108 27 , which shows that some local states can be deterministically reached much further than initially proposed in the literature. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
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