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Motion with friction of a heavy particle on a manifold -applications to optimization
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- Journal:
- ESAIM: Mathematical Modelling and Numerical Analysis / Volume 36 / Issue 3 / May 2002
- Published online by Cambridge University Press:
- 15 August 2002, pp. 505-516
- Print publication:
- May 2002
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- Article
- Export citation
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Let Φ : H → R be a C 2 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 ( $\gamma>0$
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.
