The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems

Byungik Kahng; Miguel Mendes

doi:10.3934/proc.2013.2013.393

Article Contents

2013, Volume 2013: 393-406. Doi: 10.3934/proc.2013.2013.393 open access

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The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems

Byungik Kahng^1, and
Miguel Mendes^2,

1.
Department of Mathematics and Information Sciences, University of North Texas at Dallas, Dallas, TX 75241
2.
Departamento de Engenharia Civil, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias 4200 - 465 Porto

Received: August 31, 2012

Revised: February 28, 2013

Published: October 2013

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Abstract

The main topic of this paper is the controllability/reachability problems of the maximal invariant sets of non-linear discrete-time multiple-valued iterative dynamical systems. We prove that the controllability/reachability problems of the maximal full-invariant sets of classical control dynamical systems are equivalent to those of the maximal quasi-invariant sets of disturbed control dynamical systems, when modeled by the iterative dynamics of multiple-valued self-maps. Also, we prove that the afore-mentioned maximal full-invariant sets and maximal quasi-invariant sets are countably infinite step controllable under some appropriate conditions. We take an abstract set theoretical approach, so that our main theorems remain valid regardless of the topological structure of the space or the analytical structure of the dynamics.

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Mathematics Subject Classification: Primary: 93C05, 93C25, 93C55; Secondary: 37E99.

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