  In a real Hilbert space $\mathcal H$, we study the fast convergence
properties as $t \to + \infty$ of the trajectories of the second-order
evolution equation $$ \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi
(x(t)) = 0, $$ where $\nabla \Phi$ is the gradient of a convex continuously
differentiable function $\Phi : \mathcal H \rightarrow \mathbb R$, and $\alpha$
is a positive parameter. In this inertial system, the viscous damping
coefficient $\frac{\alpha}{t}$ vanishes asymptotically in a moderate way. For
$\alpha > 3$, we show that any trajectory converges weakly to a minimizer of
$\Phi$, just assuming that the set of minimizers is nonempty. The strong
convergence is established in various practical situations. These results
complement the $\mathcal O(t^{-2})$ rate of convergence for the values obtained
by Su, Boyd and Cand\`es. Time discretization of this system, and some of its
variants, provides new fast converging algorithms, expanding the field of rapid
methods for structured convex minimization introduced by Nesterov, and further
developed by Beck and Teboulle. This study also complements recent advances due
to Chambolle and Dossal.
