Asymptotic for a second order evolution equation with convex potential and vanishing damping term

TL;DR

This paper presents a new method to analyze the asymptotic behavior of solutions to a second order convex potential evolution equation with vanishing damping, improving convergence rate results.

Contribution

It introduces an alternative proof for weak convergence and refines the convergence rate of the potential function to its minimum.

Findings

Solutions weakly converge as t approaches infinity.

Improved rate of convergence for the potential function.

Method offers a different perspective from previous work.

Abstract

In this short note, we recover by a different method the new result due to Attouch, Peyrouqet and Redont concerning the weak convergence as $t\rightarrow+\infty$ of solutions $x(t)$ to the second order differential equation \[ x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nabla\Phi(x(t))=0, \] where $K>3$ and $\Phi$ is a smooth convex function defined on an Hilbert Space $\mathcal{H}.$ Moreover, we improve slightly their result on the rate of convergence of $\Phi(x(t))-\min\Phi.$