  In a Hilbert space setting $\mathcal H$, we study the fast convergence
properties as $t \to + \infty$ of the trajectories of the second-order
differential equation $ \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi
(x(t)) = g(t)$, where $\nabla\Phi$ is the gradient of a convex continuously
differentiable function $\Phi: \mathcal H \to \mathbb R$, $\alpha$ is a
positive parameter, and $g: [t_0, + \infty[ \rightarrow \mathcal H$ is a
"small" perturbation term. In this damped inertial system, the viscous damping
coefficient $\frac{\alpha}{t}$ vanishes asymptotically, but not too rapidly.
  For $\alpha \geq 3$, and $\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty$,
just assuming that the solution set is non empty, we show that any trajectory
of the above system satisfies the fast convergence property $\Phi(x(t))-
\min_{\mathcal H}\Phi \leq \frac{C}{t^2}$. For $\alpha > 3$, we show that any
trajectory converges weakly to a minimizer of $\Phi$, and we show the strong
convergence property in various practical situations. This complements the
results obtained by Su-Boyd- Cand\`es, and Attouch-Peypouquet-Redont, in the
unperturbed case $g=0$. The parallel study of the time discretized version of
this system provides new insight on the effect of errors, or perturbations on
Nesterov's type algorithms. We obtain fast convergence of the values, and
convergence of the trajectories for a perturbed version of the variant of FISTA
recently considered by Chambolle-Dossal, and Su-Boyd-Cand\`es.
