Abstract
Given a smooth function $f : ℝ^n → ℝ$ and a convex function $Φ: ℝ^n → ℝ$, we consider the following differential inclusion:$$(S)\qquad \ddot{x}(t)+∂Φ(\dot{x}(t))+∇f(x(t))∍0,\quad t⩾0,$$where $∂Φ$ denotes the subdifferential of $Φ$. The term $∂Φ(\dot{x})$ 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 ∈ \text{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.