A Mathematical Theory of Dynamic Systems Built on Differential Calculus in Semi-Normed Spaces

The paper contains several results belonging to the theory of systems with infinite memory, using the differential calculus in locally convex spaces. These results are the following: the general constitutive functional can have, as a first approximation, an integral representation; the constitutive functional could be expressed by a double integral, so obtaining a better approximation; the speed of the present state modification is in a linear dependence on the history of the speed by which the inputs were changed, the whole time elapsed till the present moment; the state of the system in a next moment is obtained also by means of some formulae using the derivative of the constitutive functional; the problem of optimal control of the system evolution, formulated for this general functional representation, leads to the equations of the Calculus of Variations.