The optimal controlled motion of a mechanical system, that is determined by the linear system ODE with constant coefficients and piecewise constant control components, is considered. The number of control switching points and the heights of control steps are considered as preset. The optimized functional is equal to the total area of all steps of all control components ("Expenditure criteria"). In the absence of control, the solution of the system is equal to the sum of the components (frequency components) corresponding to different eigenvalues of the matrix of the system. Admissible controls are those that turn to zero (at a non predetermined time moment) the previously chosen frequency components of the solution. An algorithm for the finding of control switching points, based on the necessary minimum conditions for Expenditure criteria, is proposed.