Asymptotic for the perturbed heavy ball system with vanishing damping term

We investigate the long time behavior of solutions to the differential equation $\ddot{x}(t)+\frac{c}{\left( t+1\right) ^{\alpha}}\dot{x}(t)+\nabla \Phi\left( x(t)\right) =g(t),~t\geq0, $where$c$isnonnegativeconstant,$\alpha\in\lbrack0,1[,$ $\mathrm{\Phi }$ is a $C^1$ convex function on a Hilbert space $\mathcal{H}$ and $g\in L^1(0,+\mathrm{\infty };\mathcal{H})\mathrm{.}$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\to +\mathrm{\infty }$ to a minimizer of the function $\mathrm{\Phi }$ if one exists.