Motion with friction of a heavy particle on a manifold - applications to optimization

ACSIOM, CNRS-FRE 2311, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France. cabot@math.univ-montp2.fr.

Received: 8 October 2001
Revised: 27 February 2002

Abstract

Let Φ : H → R be a C² function on a real Hilbert space and ∑ ⊂ H x R the manifold defined by ∑ := Graph (Φ). We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force (characterized by g>0), the reaction force and the friction force ( is the friction parameter). For any initial conditions at time t=0, we prove the existence of a trajectory x(.) defined on R₊. We are then interested in the asymptotic behaviour of the trajectories when t → +∞. More precisely, we prove the weak convergence of the trajectories when Φ is convex. When Φ admits a strong minimum, we show moreover that the mechanical energy exponentially decreases to its minimum.