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In a Hilbert space setting $\mathcal H$, we study the fast convergenceproperties as $t \to + \infty$ of the trajectories of the second-orderdifferential 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 continuouslydifferentiable function $\Phi: \mathcal H \to \mathbb R$, $\alpha$ is apositive parameter, and $g: [t_0, + \infty[ \rightarrow \mathcal H$ is a"small" perturbation term. In this damped inertial system, the viscous dampingcoefficient $\frac{\alpha}{t}$ 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 trajectoryof 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 anytrajectory converges weakly to a minimizer of $\Phi$, and we show the strongconvergence property in various practical situations. This complements theresults obtained by Su-Boyd- Cand\`es, and Attouch-Peypouquet-Redont, in theunperturbed case $g=0$. The parallel study of the time discretized version ofthis system provides new insight on the effect of errors, or perturbations onNesterov's type algorithms. We obtain fast convergence of the values, andconvergence of the trajectories for a perturbed version of the variant of FISTArecently considered by Chambolle-Dossal, and Su-Boyd-Cand\`es.

Fast inertial dynamics and FISTA algorithms in convex optimization. Perturbation aspects