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 repulsive potential V:H-->R+ and any control function epsilon:R+-->R+ which tends to zero as t-->+infinity, we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators [GRAPHICS] We first provide general existence results and show that delphi(x (t))-->0 and delphi(y(t))-->0 when t-->+infinity, assuming either that the trajectory (x,y) is bounded, or that the potential V is bounded and that phi satisfies the following limit condition: (LIM) For every sequence (z(n)) subset of H such that lim(n-->+infinity) \z(n)\=+infinity, there exists a subsequence (z(phi(n))) such that [GRAPHICS] If epsilon(t) does not tend to zero too rapidly as t-->+infinity, then the term epsilon(t)delV(x-y) asymptotically repulses the trajectories one from the other. Precisely, when H=R, and if 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 (when S = empty set), or they have the same limit. In particular, when S is reduced to an interval for example, if is convex-this allows us to obtain a global description of the set S. We provide numerical experiments which make our convergence results more precise.