Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity

In a Hilbert space H, we study the asymptotic behavior, as time variable t goes to +infinity, of nonautonomous gradient-like inertial dynamics, with a time-dependent viscosity coefficient. Given Phi : H -> R a convex differentiable function, gamma (.) a time-dependent positive damping term, we consider the second-order differential equation (x) over dot(t) + gamma(t)(x) over dot (t) + del Phi(x(t)) = 0. This system plays a central role in mechanics and physics in the asymptotic stabilization of nonlinear oscillators. Its importance in optimization was recently put to the fore by Su, Boyd, and Candes. They have shown that in the particular case gamma (t) = 3/t, this is a continuous version of the fast gradient method initiated by Nesterov, with Phi(x (t)) - min(H) Phi = O(1/t(2)) as t -> +infinity in the worst case. Recently, in the case gamma (t) = alpha/t with alpha > 3, Attouch and Peypouquet have improved this result by showing the convergence of the trajectories to optimal solutions, and Phi(x (t)) - min(H) Phi = O(1/t(2)) as t -> +infinity. For these questions, and the design of fast optimization methods, the tuning of the damping parameter gamma(t) is a subtle question, which we deal with in this paper in general. We obtain convergence rates for the values, and convergence results of the trajectories under general conditions on gamma(.) which unify, and often improve the results already present in the literature. We complement these results by showing that they are robust with respect to perturbations. (C) 2017 Elsevier Inc. All rights reserved.