On the Estimation of Small Perturbations in Ordinary Differential Equations

Abstract

The essential subject of this work is concerned with those problems represented by a system of ordinary differential equations involving one or several small perturbing functions to be determined in order to obtain either a solution given in advance (control problems) or a solution that approximates a set of measurements that may be affected by random errors. The traditional solution of such problems consists of the parameter identification of a model of the perturbations by means of statistical methods. The inconvenience of these procedures is that some peculiarities in the residuals that are small but physically significant can be smoothed out either by insufficiency of the model or by the intrinsic nature of the statistical methods. we present here a direct method based only on the assumptions that the solutions of the differential equations can be expressed piecewise in short intervals by a Taylor convergent expansion and that the unknown perturbations can be approximated by elementary functions.