Alexandre Cabot

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:

Steepest Descent with Curvature Dynamical System

\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,

Sequential formulae for the normal cone to sublevel sets

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.

On the Asymptotic Behavior of a System of Steepest Descent Equations Coupled by a Vanishing Mutual Repulsion

We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion: (S) x(t) + ∂Φ(x(t)) + ∇f(x(t)) ∋0, t ≥ 0, where f: R n → R is a smooth potential and Φ: R n → R is a convex function. The friction is modelized by the subdifferential term -∂Φ(x). When 0 ∈ int(∂Φ(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.Too 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 /2, a > 0, b > 0, we obtain the existence of a critical coefficient be > 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.

Inclusion of Subdifferentials, Linear Well-Conditioning, and Steepest Descent Equation

AbstractLet H be a real Hilbert space and let <..,.> denote the corresponding scalar product. Given a

Second-order differential equations with asymptotically small dissipation and piecewise flat potentials

\mathcal{C}^2