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

Abstract:

We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion: \begin{equation*} (\textit {S})\qquad \qquad \ddot {x}(t) + \partial \Phi (\dot {x}(t)) + \nabla f(x(t)) \ni 0, \qquad t\geq 0, \end{equation*} 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 $\Phi =a | . |+ b | . |^2/2$, $a>0$, $b\geq 0$, 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.