© 2016 Author(s).The controlled motion which is represented by the linear ODE system with constant coefficients is considered. The admissible control is a piecewise constant function that blanks selected frequency components of the solution of linear equations at the moment T. As "the expenditure" functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the Expenditure. To solve this problem the method is proposed which leads to explicit formulas. All results of research are formulated as the theorem. These results can be applied not only in mechanical controlled systems, but also in any problem that can be described by the system of ordinary differential equations with control.