Perturbations

And according to the Collins Dictionary of Mathematics:

perturbation n , 1. (of an equation or of an optimization problem) a change (usually slight) in the values of some of the underlying parameters, made to obtain the desired solution or to study the stability of a given solution. 2. (Mechanics) a small displacement in the orbit of a particle.

I quote this “definition” to make it abundantly clear that the perturbations under consideration in this book are assumed to be small.

The concept “small perturbation” is used extensively in mathematics and other fields to investigate the behavior of an object in the neighborhood of a given point in the assumed space. The picture (from WIKIPEDIA) shows a neighborhood of a point p\in V that consists of all the perturbations in p whose size is not greater than a given value (radius of the disc shown).

Typically, the size of the perturbations of interest is not stipulated a priori. So, certain perturbations that might eventually be considered, can turn out to be quite large, sometimes very large. Often, the term “small” indicates that small perturbations are considered first. Larger perturbations are considered only if the investigation of all smaller perturbations failed to achieve the objective of the analysis.

Needless to say, in many applications the numeric value of the “size” of the perturbations depends on the unit used to specify it (e.g. mm, cm, m, km, etc).

In some applications, it is essential to keep the size of the perturbations small because the analysis is valid only in a small neighborhood around the nominal value of the parameter of interest. For instance, this could be the case in situations where the stability conditions under consideration are approximations that are valid only in a small neighborhood of the nominal value of the parameter of interest.

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