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. firstname.lastname@example.org.
Revised: 27 February 2002
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.
Mathematics Subject Classification: 34A12 / 34G20 / 37N40 / 70Fxx
Key words: Mechanics of particles / dissipative dynamical system / optimization / convex minimization / asymptotic behaviour / gradient system / heavy ball with friction.
© EDP Sciences, SMAI, 2002