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 -> +infinity of the trajectories of the second-order differential equation (x) over dot + gamma(t)(x) over dot(t) + del Phi(x(t)) = 0, (IGS)(gamma) where del Phi is the gradient of a convex continuously differentiable function Phi : H -> 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(H) Phi and the velocities under general conditions involving only gamma(.) and its derivatives. In particular, in the case gamma(t) = alpha/t, which is directly connected to the Nesterov accelerated gradient method, our approach allows us to cover all the positive values of a, including the subcritical case alpha < 3. Our approach is based on the introduction of a new class of Lyapunov functions.