Invariance properties of interval dynamical systems

Abstract

This article analyses the flow (positive) invariance of different types of time-dependent sets with respect to the dynamics of interval systems (state-space models described by interval matrices) in both discrete and continuous time. The investigated sets have general shapes, defined in terms of Ho 00 lder vector p-norms, 1 p 1. The time dependence of the sets is separately considered arbitrary and exponentially decreasing. We develop procedures that explore set invariance by using (i) a single test matrix, introduced as a majorant of the interval matrix and (ii) all the distinct vertices of the matrix polytope corresponding to the interval matrix. In case (i) we provide sufficient conditions for all p, 1 p 1, whereas the necessity is guaranteed only for p 2 f1, 1g; in case (ii) we formulate necessary and sufficient conditions valid for all p, 1 p 1. Our analysis shows that the existence of invariant sets (with arbitrary or exponential time-dependence, defined by a p-norm) approaching the state-space origin for infinite time horizon implies the asymptotic stability. Some examples are presented to provide intuitive support for the theoretical concepts and to illustrate the applicability of the main results.