### 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.

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.