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

  • Esteban Bailo
  • Josep Gelonch
  • Sergio Romero-VivóEmail author
Original Paper
  • 39 Downloads

Abstract

The calculation of the smallest number of steps needed to deterministically reach all local states of an \(n\hbox {th}\)-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 \(n+3\left( \left\lfloor n/ 2\right\rfloor \right) ^2\) 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 \(n\ge 6\) 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 \(n_1>n_2\ge 2\) and \(n-n_1-n_2\ge 1\). The second one has two main cycles, a cycle where only the remaining vertices \(n_1+1, \ldots , n\) appear and a cycle containing only the vertices \(n_1+1, \ldots , n_1+n_2-1\). Moreover, the last vertices are connected through the 2-arc \((n_1+n_2-1, n)\). Furthermore, if \(n\ge 12\) 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 \(\frac{n^3+9n^2+45n+108}{27}\), which shows that some local states can be deterministically reached much further than initially proposed in the literature.

Keywords

Positive two dimensional (2-D) systems Fornasini–Marchesini models Hurwitz products Influence digraph Local reachability index Composition table 

Mathematics Subject Classification

05C99 15A48 93B03 93C55 

Notes

Acknowledgements

We are gratefully thankful to the reviewers for their valuable remarks. This work has been partially supported by the European Union [FEDER funds] and Ministerio de Ciencia e Innovación through Grants MTM-2013-43678-P and DPI2016-78831-C2-1-R.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department de MatemàticaUniversitat de LleidaLleidaSpain
  2. 2.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain
  3. 3.Centro de Investigación Biomédica en Red de Diabetes y Enfermedades Metabólicas Asociadas (CIBERDEM)Instituto de Salud Carlos IIIMadridSpain

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