Singular Perturbation in Heavy Ball Dynamics

Given a \(C^{1,1}_\textrm{loc}\) $C_{\text{loc}}^{1,1}$ lower bounded function \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) $f:{\mathbb{R}}^n\to \mathbb{R}$ definable in an o-minimal structure on the real field, we show that the singular perturbation \(\epsilon \searrow 0\) $ϵ\searrow 0$ in the heavy ball system preserves boundedness of solutions, where \(\gamma >0\) $\gamma >0$ is the friction and \((x_0,{\dot{x}}_0) \in \mathbb {R}^n \times \mathbb {R}^n\) $(x_0,{\dot{x}}_0)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ 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.