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

Author: Alexandre Cabot
Journal: Trans. Amer. Math. Soc. 360 (2008), 103-121
MSC (2000): Primary 34C15, 34A60; Secondary 70F40, 37N05
Published electronically: July 20, 2007
MathSciNet review: 2341995
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Abstract: We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion:

$\displaystyle (\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.

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Additional Information

Alexandre Cabot
Affiliation: Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex, France

Keywords: Differential inclusion, dry friction, nonlinear oscillator, finite time convergence, exponential decay, convex analysis
Received by editor(s): December 15, 2004
Received by editor(s) in revised form: August 6, 2005
Published electronically: July 20, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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