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 (S) ẍ(t) + ∂Φ(ẋ(t)) + ∇f(x(t)) 3 0, t ≥ 0, where f : R → R is a smooth potential and Φ : R → R is a convex function. The friction is modelized by the subdifferential term −∂Φ(ẋ). When 0 ∈ int(∂Φ(0)) (dry friction condition), it is shown in [1] that the unique solution of (S) converges in a finite time toward an equilibrium state x∞ provided that −∇f(x∞) ∈ int (∂Φ(0)). In this paper, we study the delicate case where the vector −∇f(x∞) belongs to the boundary of the set ∂Φ(0). We prove that either the solution converges in a finite time or the speed of convergence is exponential. When Φ = a | . | + b | . |/2, a > 0, b ≥ 0, we obtain the existence of a critical coefficient bc > 0 below which every solution stabilizes in a finite time. It is also shown that the geometry of the set ∂Φ(0) plays a central role in the analysis.