New results in multiparameter singular perturbations

Abstract

The author will summarize new results on stability, control, and invariant manifold theory for multiparameter singular perturbation problems. The feature distinguishing this work from previous studies is that no constraint is placed on the relative magnitudes of the singular perturbation parameters. Thus, the multiple time scale case and the two time scale setting (bounded ratios) are treated us special cases. Specialized results employing the concept of 'strong D-stability' recently introduced by the author are given for time-invariant systems. Stability of systems with several small and large parasitic parameters is discussed. Controllability of multiparameter singularly perturbed systems has been studied. For the nonlinear time-invariant case, the research program has included a study of the existence of a locally invariant and locally attracting manifold of the 'center manifold type' for systems with several small parameters.