Stability and Set-Invariance Testing for Interval Systems

Abstract

Many works dealing with the stability analysis of interval systems developed criteria
based on matrices that majorize (in a certain sense) the interval matrices describing the system
dynamics. Besides this already classical employment, we prove that the majorant matrices also
contain valuable information for the study of the exponentially decreasing sets, invariant with
respect to the trajectories of the interval systems. The interval systems are considered with both
discrete- and continuous-time dynamics. The invariant sets are characterized by arbitrary shapes,
defined in terms of Holder vector p-norms, 1 <= p <= infinite . Our results cover two types of interval
systems, namely described by interval matrices of general form and by some particular classes of
interval matrices. For the general case, we formulate necessary and sufficient conditions, when
the shape of the invariant sets is defined by the norms p = 1, infinite, and sufficient conditions, when the
shape is defined by the norms 1< p < infinite. For the particular cases, we provide necessary and
sufficient conditions for all norms 1 <= p <= infinite.