On the stability of multiple time-scale systems

Abstract

The stability of time-invariant multiparameter singular perturbation problems is considered and the implications of two time-scale stability results for multiple time-scale systems are clarified. An example shows that the asymptotic stability of a multiparameter singular perturbation problem under the ‘bounded mutual ratios’ assumption for arbitrary bounds on the ratios of the small parameters does not imply asymptotic stability under the multiple time scales assumption for any ordering of the smallness of the parameters. However, this conclusion does apply when only two small parameters are present and the fast variables are scalar-valued. A multiparameter singularly perturbed system may be asymptotically stable for all sufficiently small (and positive) values of the perturbation parameters, even though the boundary layer system does not satisfy the D-stability criterion. These examples are discussed in the light of the ‘strong D-stability’ condition which must be imposed to obtain results that are robus...