Singular perturbation in heavy ball dynamics

TL;DR

This paper investigates the asymptotic behavior of heavy ball dynamics with a small perturbation parameter, showing that solutions remain bounded over time without requiring convexity or coercivity of the potential function.

Contribution

It extends previous results by analyzing the infinite horizon case for heavy ball systems with singular perturbations, under definability in an o-minimal structure.

Findings

Solutions remain bounded as perturbation parameter tends to zero

The analysis applies to non-convex, non-coercive potential functions

Complements finite horizon results by Attouch, Goudou, and Redont

Abstract

Given a $C^{1,1}_\mathrm{loc}$ lower bounded function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ definable in an o-minimal structure on the real field, we show that the singular perturbation $\epsilon \searrow 0$ in the heavy ball system \begin{equation} \label{eq:P_eps} \tag{$P_\epsilon$} \epsilon\ddot{x}_\epsilon(t) + \gamma\dot{x}_\epsilon(t) + \nabla f(x_\epsilon(t)) = 0, ~~~ \forall t \geqslant 0, ~~~ x_\epsilon(0) = x_0, ~~~ \dot{x}_\epsilon(0) = \dot{x}_0, \end{equation} preserves boundedness of solutions, where $\gamma>0$ is the friction and $(x_0,\dot{x}_0) \in \mathbb{R}^n \times \mathbb{R}^n$ is the initial condition. This complements the work of Attouch, Goudou, and Redont which deals with finite time horizons. In other words, this work studies the asymptotic behavior of a ball rolling on a surface subject to gravitation and friction, without assuming convexity nor coercivity.