# Asymptotic for the perturbed heavy ball system with vanishing damping term

arXiv: https://arxiv.org/abs/1609.00135 · HTML: https://mlsanity.com/paper/1609.00135/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2016.

## Built on by (most cited first, 1 of 1)

- [Diffusion Sampling with Momentum for Mitigating Divergence Artifacts](https://mlsanity.com/paper/2307.11118/) (2023)

## Abstract (arXiv)

We investigate the long time behavior of solutions to the differential equation ddotx(t)+fraccleft( t+1right) αdotx(t)+nabla Phileft( x(t)right) =g(t),~t≥0,  where c is nonnegative constant, α∈lbrack0,1[, Phi is a C1 convex function on a Hilbert space H and g∈ L1 (0,+∞;H). We obtain sufficient conditions on the source term g(t) ensuring the weak or the strong convergence of any trajectory x(t) as trightarrow+∞ to a minimizer of the function Phi if one exists.