Majorant matrices in the qualitative analysis of interval dynamical systems

The paper explores the role played by the majorant matrices in the qualitative analysis of interval systems with discrete- or continuous-time dynamics. We prove that, besides the classical usage in stability testing, the majorant matrices provide valuable information for the study of the exponentially decreasing sets, invariant with respect to the trajectories of the interval systems. The invariant sets are characterized by arbitrary shapes, defined in terms of Hölder vector p-norms, 1 ≤ p ≤ ∞. We formulate results for two types of interval systems: (i) described by interval matrices of general form and (ii) described by some classes of interval matrices. For the systems of type (i), the results represent necessary and sufficient conditions, when the shape of the invariant sets is defined by the norms p =1, ∞, and sufficient conditions, when the shape is defined by the norms 1 <; p <; ∞. For the systems of type (ii), the results represent necessary and sufficient conditions for all norms 1 ≤ p ≤ ∞. The qualitative analysis tools developed by our work are easy to apply in practice, since the construction of the majorant matrix is a straightforward task for any interval system.