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: {equation*}(S) ẍ(t)+a(t)ẋ(t)+ ∇ G(x(t))=0, t≥ 0, {equation*} 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 ∫ ₀^∞ a(t) dt =∞ guarantees that the energy function converges toward its minimum. The more stringent condition ∫ ₀∞ e-∫ ₀ᵗ 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 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.