Singular perturbation in heavy ball dynamics

Abstract

Given a $C_{\mathrm{loc}}^{1,1}$ lower bounded function $f:{\mathbb{R}}^n\to \mathbb{R}$ definable in an o-minimal structure on the real field, we show that the singular perturbation $\hat{I}\mu \searrow 0$ in the heavy ball system \begin{equation} \label{eq:P_eps} \tag{$P_{\hat{I}}\mu$} ε\ddot{x}_ε(t) + γ\dot{x}_ε(t) + \nabla f(x_ε(t)) = 0, ~~~ \forall t \geqslant 0, ~~~ x_ε(0) = x_0, ~~~ \dot{x}_ε(0) = \dot{x}_0, \end{equation} preserves boundedness of solutions, where ${\hat{I}}^3>0$ 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.