Flow-invariant rectangular sets and componentwise asymptotic stability of interval matrix systems

Given an interval matrix system with discrete- or continuous-time dynamics, the flow-invariance of an arbitrarily time-dependent rectangular set with respect to this system is introduced as a concept of geometric nature. The case of rectangular sets with exponential time-dependence is separately explored. If there exist flow-invariant rectangular sets approaching the state space origin for an infinite time horizon, then the interval matrix system exhibits two special types of asymptotic stability, which we have called componentwise asymptotic stability and componentwise exponential asymptotic stability. An interval matrix system is shown to be componentwise exponential asymptotically stable if and only if it is componentwise asymptotically stable. A necessary and sufficient condition for the componentwise asymptotic stability of an interval matrix system is derived in a matrix form. Brief comments focus on three classes of interval matrix systems for which the componentwise asymptotic stability is equivalent to the standard asymptotic stability (defined in terms of any consistent finite-dimensional norm).