A first cubic upper bound on the local reachability index for some positive 2-D systems

Abstract

The calculation of the smallest number of steps needed to deterministically reach all local states of an nth-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