Maximal Invariant Sets of Multiple Valued Iterative Dynamics in Disturbed Control Systems

Invariant set theory is an important tool in the control theory. It has rich history that goes back over a century, yet it is still an active research topic both in pure mathematics and theoretical engineering. It is easy to reduce a traditional discrete-time control dynamical system to an iterative dynamics of one endomorphism in the phase space. It is not easy to do the same in the presence of the disturbance. The purpose of this paper is to show that we can overcome this difficulty using the dynamics of multiple valued maps. First, we show that the dynamics of disturbed control systems can be modeled by the multiple valued iterative dynamics. Second, we define and study the invariant sets, the maximal invariant sets, and the positively maximal invariant sets of the multiple valued iterative dynamical systems. Finally, as an application, we study the reachability problem of the maximal positively invariant sets of the multiple valued iterative dynamical systems. Keywords— Multiple valued dynamics, Control dynamical system with disturbance, Maximal invariant set, Reachability, Controlability.