Stability of Multiparameter Singular Perturbation Problems with Parameter Bounds, 11. Time-Invariant Systems with Arbitrary Perturbations.

Abstract

The asymptotic stability of a general linear time-invariant multiparameter singular perturbation problem is studied with no a priori assumptions on the relative magnitudes of the small parameters. Thus the results apply, for example, to multiple time scale problems as well as to multiparameter singular perturbation problems possessing only two time scales. An explicit upper bound is obtained for a weighted norm of the vector of singular perturbation parameters such that asymptotic stability of the system is ensured if this bound is respected. In a companion paper [1], the time-varying case is studied using Liapunov stability theory, assuming that the small parameters have bounded mutual ratios (two time scales). The present paper makes use of fixed-point methods, in addition to Liapunov techniques, to arrive at the desired upper bound.