Stabilization of oscillators subject to dry friction: Finite time convergence versus exponential decay results

We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion:

$(\textit{S})\qquad\qquad \ddot{x}(t) + \,\partial \Phi(\dot{x}(t)) + \, \nabla f(x(t)) \ni 0, \qquad t\geq 0,$

where $f:\mathbb{R}^n \to \mathbb{R}$ is a smooth potential and $\Phi:\mathbb{R}^n\to \mathbb{R}$ is a convex function. The friction is modelized by the subdifferential term $-\partial \Phi(\dot{x})$. When $0\in \operatorname{int}(\partial \Phi(0))$ (dry friction condition), it was shown by Adly, Attouch, and Cabot (2006) that the unique solution to $(S)$ converges in a finite time toward an equilibrium state $x_{\infty}$ provided that $-\nabla f(x_{\infty})\in \operatorname{int}(\partial \Phi(0))$. In this paper, we study the delicate case where the vector $-\nabla f(x_{\infty})$ belongs to the boundary of the set $\partial \Phi(0)$. We prove that either the solution converges in a finite time or the speed of convergence is exponential. When $\Phi=a\,\vert\,.\,\vert+\, b\, \vert\,.\,\vert^2/2$, $a>0$, $b\geq 0$, we obtain the existence of a critical coefficient $b_c>0$ below which every solution stabilizes in a finite time. It is also shown that the geometry of the set $\partial \Phi(0)$ plays a central role in the analysis.