Asymptotic for the perturbed heavy ball system with vanishing damping term

TL;DR

This paper studies the long-term behavior of solutions to a perturbed heavy ball differential equation with vanishing damping, establishing conditions under which solutions converge to a minimizer of a convex function.

Contribution

It provides new asymptotic analysis for the heavy ball system with time-dependent damping and external perturbations, extending understanding of convergence in convex optimization.

Findings

Conditions for weak convergence of trajectories

Conditions for strong convergence of trajectories

Impact of source term g(t) on convergence behavior

Abstract

We investigate the long time behavior of solutions to the differential equation $\ddot{x}(t)+\frac{c}{\left( t+1\right) ^{\alpha}}\dot{x}(t)+\nabla \Phi\left( x(t)\right) =g(t),~t\geq0, $ where $c$ is nonnegative constant, $\alpha\in\lbrack0,1[,$ $\Phi$ is a $C^{1}$ convex function on a Hilbert space $\mathcal{H}$ and $g\in L^{1} (0,+\infty;\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\rightarrow+\infty$ to a minimizer of the function $\Phi$ if one exists.