Set Invariance for Delay Difference Equations

Abstract : This paper deals with set invariance for time delay systems. The first goal of the paper is to review the known necessary or su?cient conditions for the existence of invariant sets with respect to dynamical systems described by discrete-time delay difference equations (dDDEs). Secondly, we address the construction of invariant sets in the original state space (also called D-invariant sets) by exploiting the forward mappings. As novelties, the present paper contains a su?cient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and su?cient condition for the existence of Dinvariant sets in relation to time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance.Inshort, tis shown that the convex hull of two D-invariant sets is not D-invariant but the convex hull of a non-convex D-invariant set is D-invariant.