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In a real Hilbert space $\mathcal H$, we study the fast convergenceproperties as $t \to + \infty$ of the trajectories of the second-orderevolution equation $$ \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi(x(t)) = 0, $$ where $\nabla \Phi$ is the gradient of a convex continuouslydifferentiable function $\Phi : \mathcal H \rightarrow \mathbb R$, and $\alpha$is a positive parameter. In this inertial system, the viscous dampingcoefficient $\frac{\alpha}{t}$ 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 strongconvergence is established in various practical situations. These resultscomplement the $\mathcal O(t^{-2})$ rate of convergence for the values obtainedby Su, Boyd and Cand\`es. Time discretization of this system, and some of itsvariants, provides new fast converging algorithms, expanding the field of rapidmethods for structured convex minimization introduced by Nesterov, and furtherdeveloped by Beck and Teboulle. This study also complements recent advances dueto Chambolle and Dossal.

Fast Convergence of an Inertial Gradient-like System with Vanishing Viscosity