The controlled motion of a mechanical system is represented by the linear ODE system with constant coefficients. The admissible control is a piecewise polynomial 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 that consists of analytical and numerical parts is proposed. All results of research are formulated as the theorem.