Singular perturbation in heavy ball dynamics

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.