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

Abstract

The main topic of this paper is the controllability/reachabilityproblems of the maximal invariant sets of non-linear discrete-time multiplevaluediterative dynamical systems. We prove that the controllability/reachabilityproblems of the maximal full-invariant sets of classical control dynamicalsystems are equivalent to those of the maximal quasi-invariant sets of disturbedcontrol dynamical systems, when modeled by the iterative dynamics ofmultiple-valued self-maps. Also, we prove that the afore-mentioned maximalfull-invariant sets and maximal quasi-invariant sets are countably infinite stepcontrollable under some appropriate conditions. We take an abstract set theoreticalapproach, so that our main theorems remain valid regardless of thetopological structure of the space or the analytical structure of the dynamics.