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

TL;DR

This paper studies the long-term behavior of solutions to second-order differential equations with decreasing damping in Hilbert spaces, revealing conditions for energy convergence and trajectory stability, especially in convex and non-convex potentials.

Contribution

It extends analysis of asymptotic properties of second-order equations with time-dependent damping, including non-convex potentials and general conditions for convergence.

Findings

Energy function converges to the minimum when potential is convex and damping is non-integrable.

Trajectories tend to a minimum point under certain conditions, with convergence described in one-dimensional cases.

The set of initial conditions leading to convergence to a local minimum is open and dense in the one-dimensional setting.

Abstract

We investigate the time-asymptotic properties of solutions of the differential equation x''(t) + a(t)x'(t) + g(x(t)) = 0 in a Hilbert space, where a(.) is non-increasing and g is the gradient of a potential G. If the coefficient a(.) 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 potential G is convex and the coeffient a is non-integrable at infinity, the energy function converges to its minimum. A more stringent condition is required to obtain the convergence of the trajectories of toward some minimum point of the potential. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general coercive non-convex potentials with many local minima and maxima. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.