Asymptotic for the perturbed heavy ball system with vanishing damping term

We investigate the long time behavior of solutions to the differential equation ddotx(t)+fraccleft( t+1right) αdotx(t)+nabla Phileft( x(t)right) =g(t),~t≥0, where c is nonnegative constant, α∈lbrack0,1[, Phi is a C1 convex function on a Hilbert space H and g∈ L1 (0,+∞;H). We obtain sufficient conditions on the source term g(t) ensuring the weak or the strong convergence of any trajectory x(t) as trightarrow+∞ to a minimizer of the function Phi if one exists.