Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient

In a Hilbert space H , we study the convergence properties when $t→+∞$ of the trajectories of the second-order differential equation equation*ẍ(t) + γ(t) ẋ(t) + ∇ Φ (x(t)) = 0, \;\;\;\;\;\;\;\; (IGS)γequation* where ∇Φ is the gradient of a convex continuously differentiable function Φ: H→ R , and $γ(t)$ 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 Φ(x(t))-infH Φ and the velocities under general conditions involving only $γ(·)$ and its derivatives. In particular, in the case γ(t) = α/t , which is directly connected to the Nesterov accelerated gradient method, our approach allows us to cover all the positive values of $α$ , including the subcritical case $α<3$ . Our approach is based on the introduction of a new class of Lyapunov functions.