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 nonincreasing 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 f ₀^∞; a(t) dt = ∞ guarantees that the energy function converges toward its minimum. The more stringent condition f ₀^∞; e-f₀^t a(s)d^sdt < ∞ 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 nonconvex 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.