On the Long Time Behavior of Second Order Differential Equations with Asymptotically Small Dissipation

We investigate the asymptotic properties as t → ∞ of the following differential equation in the Hilbert space H (S) ẍ(t) + a(t)ẋ(t) +∇G(x(t)) = 0, t ≥ 0, where the map a : R+ → R+ is non increasing and the potential G : H → R is of class C1. If the coefficient a(t) is constant and positive, we recover the so-called “Heavy Ball with Friction” system. On the other hand, when a(t) = 1/(t + 1) we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function G is convex, the condition ∫ ∞ 0 a(t) dt = ∞ guarantees that the energy function converges toward its minimum. The more stringent condition ∫ ∞ 0 e − ∫ t 0 a(s) dsdt < ∞ is necessary to obtain the convergence of the trajectories of (S) toward some minimum point of G. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general non-convex function G. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.