Mathematical modeling of dynamical systems by generalized functions

The distributions or generalized functions are linear and continuous functionals defined by the class of functions which become null outside of a compact set and have derivatives of any order. The calculus with distributions was used to the modeling of linear systems. Generalized functions are also useful in the study of non-linear systems. In this paper, it is proved that the distributions with compact support represent a first approximation in the mathematical modeling of a system with infinite fading memory. The demonstration of this statement is the main part of the paper. The mathematical tool used is the differential calculus in the locally-convex topological space of the histories of inputs in system. The last part refers to the ε-distribution, R. Vallée's recent concept, and enumerates some applications.