Asymptotic stability of singularly perturbed systems that have marginally stable boundary-layer systems (Q1179528)

A class of linear singularly perturbed systems with singular perturbation parameter \(\mu>0\) is considered. To assure asymptotic stability of the full-order system for \(\mu>0\) sufficiently small, it is customary to require that both the reduced-order system \((\mu=0)\) and the boundary- layer system are asymptotically stable. Here the author relaxes the requirement on the boundary-layer system to stability (i.e., stability, but not necessarily asymptotic stability) and show that, subject to one additional condition, the full-order system is asymptotically stable for sufficiently small \(\mu\). The result is illustrated by an application in which the author considers the stability robustness of a feedback- controlled mechanical system with respect to an unmodeled flexibility.