Finite Time Stabilization of Nonlinear Oscillators Subject to dry Friction

Given a smooth function f : ℝn → ℝ and a convex function Φ: ℝn → ℝ, we consider the following differential inclusion: $$ \left( S \right) \ddot x\left( t \right) + \partial \Phi \left( {\dot x\left( t \right)} \right) + \nabla f\left( {x\left( t \right)} \right) \mathrel\backepsilon 0, t \geqslant 0, $$ where ∂Φ denotes the subdifferential of Φ. The term ∂Φ(∂Φ\( \dot x \) ) is strongly related with the notion of friction in unilateral mechanics. The trajectories of (S) are shown to converge toward a stationary solution of (S). Under the additional assumption that 0 ∈ int ∂Φ(0) (case of a dry friction), we prove that the limit is achieved in a finite time. This result may have interesting consequences in optimization.