We discuss here the stabilization problem for an ordinary differential equation (ODE) dynamical model. To make such a control, one can form a Kolesnikov’s subset attracting the phase trajectories to its neighborhood in the phase space via defining the appropriate feedback signal. Kolesnikov’s target attractor algorithm provides the exponential convergence, but at the same time it demands the permanent power supply pumping the energy to the system even if the control goal is achieved. To decrease the power cost of Kolesnikov’s control, we re-formulate the feedback in the form of Caputo’s fractional derivative. In this case the solution to the ODE together with the feedback control signal could be found with the Rida-Arafa method based on the generalized Mittag-Leffler function. We prove that for the certain constraints over the initial condition and the target stabilization level, the integer-dimensional Kolesnikov algorithm can be replaced with the fractional target attractor feedback to provide the minimal power cost.