Set invariance for delay difference equations

Journal article (2015)

Authors

M. T. Laraba CentraleSupelec
Sorin Olaru CentraleSupelec
S. I. Niculescu CentraleSupelec
Franco Blanchini Università degli Studi di Udine
G. Giordano Università degli Studi di Udine
D. Casagrande Università degli Studi di Udine
Stefano Miani Università degli Studi di Udine
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Time-delay systems Set invariance Discrete-time delay difference equations
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Published Date

01-07-2015

Language

English

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Abstract

This paper deals with set invariance for time delay systems. The first goal of the paper is to review the known necessary or sufficient 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 sufficient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and sufficient condition for the existence of 22-invariant sets in relation to time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance. In short, it is 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,.