A note on the uniform perturbation index

Abstract

For a given differential-algebraic equation (DAE) the perturbation index gives a measure for the sensitivity of a solution w.r.t. small perturbations. If we consider, however, classes of DAEs (e.g. all DAEs that arise as semi-discretizations of a given partial DAE by the method of lines) then the error bound in the definition of the perturbation index may become arbitrarily large even if the perturbation index does not exceed 1. We illustrate this fact by 2 examples and define as alternative the uniform perturbartion index that gives simultaneously error bounds for all DAEs of a fiven class. We prove that in one example each individual DAE has perturbation index 1 but the uniform perturbation index is 2. Another example illustrates that the class of all finite difference semi-discretizations may even have no uniform perturbation index if the given partial DAE has perturbation index 2.