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the aim of this work is to study the possibility of the disposal of the space debris in the dense layers of the Earth's atmosphere. We use information about the fragments of the space debris location that stored in NORAD databases. There two types of the orbits of the space debris are explored: the high elliptical orbit and the circular orbits. The algorithm of the operation of the utilization of the fragments of the space debris in the dense layers of the Earth's atmosphere using the special spacecraft debris collector is composed. The spacecraft debris collector must be equipped with electro propulsion engines with low-thrust. The mathematical model of the plane motion of the spacecraft debris collector, in the form of the system of osculating elements, is composed. Problem of the descent of the fragments of the space debris reduced to the problem of the reducing of the height of the perigee of elliptical orbit. Obtained the locally - optimal control program of the orientation of the thrust vector. Using the L.S. Pontryagin's maximum principle, we solved the boundary value problem of the optimal control (decrease) of the perigee of the elliptical orbit. The results of the parametric study of the problem in the space of the initial orbits and the acceleration traction of the electro propulsion engines is shown. Numerical simulation of the space debris reduction from the low circular orbits is performed. Determined the number of revolutions passed a fragment of the space debris around the Earth during braking due to aerodynamic drag. Calculated the estimates of the gains of the characteristic velocity to perform the operation. [ABSTRACT FROM AUTHOR]