Abstract : In a Hilbert space $\mathcal H$, we study the convergence properties when $t \to + \infty$ of the trajectories of the second-order differential equation
\begin{equation*}
\mbox{(IGS)}_{\gamma} \quad \quad \ddot{x}(t) + \gamma(t) \dot{x}(t) + \nabla \Phi (x(t))=0,
\end{equation*}
where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, and $\gamma(t)$ is a time-dependent positive viscous damping coefficient. This study aims to offer a unifying vision on the subject, and to complement the article by Attouch and Cabot (J. Diff. Equations, 2017). We obtain convergence
rates for the values $\Phi(x(t))-\inf_{\mathcal H} \Phi$ and the velocities under general conditions involving only
$\gamma (\cdot)$ and its derivatives. In particular, in the case $\gamma(t) = \frac{\alpha}{t}$, which is directly connected to the Nesterov accelerated gradient method, our approach allows to cover all the positive values of $\alpha$, including the subcritical case $\alpha <3$. Our approach is based on the introduction of a new class of Lyapunov functions.
