Finitely additive measures in constructions of extension for abstract attainability problems

This lecture notes are intended for the students taking courses in mathematical control theory. They are concerned with the attainability problem with constraints. The exposition is oriented to the linear control problems with the impulse constraints and the possible discontinuity in the coefficients under control actions. In addition, the weakening of constraints is assumed. The ordinary attainable set is replaced with the attraction set. To construct this attraction set we realize the extension procedure; the generalized elements are defined as finitely additive measures with the property of the weak absolute continuity with respect to the restriction of the Lebesgue measure. The auxiliary constructions of general topology and measure theory are used.