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 $c$ is nonnegative constant, $\alpha\in\lbrack0,1[,$ $\Phi$ is a $C^{ 1}$ convex function on a Hilbert space $\mathcal{ H}$ and $g\in L^{ 1} (0,+\infty;\mathcal{ H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\rightarrow+\infty$ to a minimizer of the function $\Phi$ if one exists.
Categories: math.OC