Positive invariance of multiple valued iterative dynamical systems in disturbed control models

It is well known that a classical non-linear discrete-time control dynamical system can be reduced to an iterative dynamical system of one endomorphism in the phase space. In this paper, we study the similar reduction in the presence of disturbance, via multiple valued iterative dynamics. This reduction raises some intriguing problems in the invariant set theory of the disturbed control dynamical systems. The multiple valuedness of our system necessitates us to consider two distinct types of predecessor operations, strong and weak, where each type generates its own invariant set theory. A natural question that follows is, how the strong and the weak invariant set theories of the multiple valued iterative dynamical systems differ. The purpose of this paper is to answer this question and discuss its implications in modeling the disturbed control dynamical systems.