Let phi : H --> R be a C-1 function on a real Hilbert space H, and let gamma > 0 be a positive damping parameter. For any (singular) repulsive potential V : H \{0} --> R+, i.e., such that lim(z-->0) V (z) = +infinity, and any control function epsilon : R+ --> R+ \ {0} which tends to zero as t --> +infinity, we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators (HBFCsing2) [GRAPHICS] This system is the singular version of the regular (HBFCreg2) system studied in [A. Cabot and M.-O. Czarnecki, SIAM J. Control Optim., 41 (2002), pp. 1254-1280], where the potential V is defined on the whole space H. The purpose of this paper is to obtain whenever possible the same existence and convergence results in the singular case as in the regular case considered by A. Cabot and M.-O. Czarnecki. This study is mainly motivated by a better convergence behavior of (HBFCsing2) with the same sharp condition on the control e exhibited in [H. Attouch and M.-O. Czarnecki, J. Differential Equations, 179 (2002), pp. 278-310] and by A. Cabot and M.-O. Czarnecki. Precisely, when H = R, and if epsilon is a "slow" control, i.e., integral(0)(+infinity) epsilon(t)dt = +infinity, then the trajectories x and y converge to extremal points of the set S = {lambda is an element of R, delphi(lambda) = 0} of the equilibria of phi. The awkward case in A. Cabot and M.-O. Czarnecki, where the trajectories may have the same limit, disappears. Of importance, from a physical point of view, we thus can consider actual, for example electromagnetic, repulsive potentials.