Localization of Compact Invariant Sets of Discrete-Time Systems with Disturbances

A method for constructing localizing sets for discrete-time systems with disturbances is proposed and the properties of localizing sets are established. Conditions for the existence of maximal invariant compact sets for a discrete-time system with disturbances are obtained and a method for finding such sets is suggested. It is proved that any positively invariant compact set of system that is contained in a set also contained in the set. A necessary and sufficient condition for negative invariance with the additional compactness condition is stated. For systems with disturbances, there are analogues of properties of undisturbed discrete-time systems associated with shifts of localizing sets along trajectories of the system. For a compact set, the corresponding set is the maximal positively invariant compact set of system among the positively invariant compact sets contained in G.