Motion with friction of a heavy particle on a manifold. Applications to optimization
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, p. 505-516

Let $\Phi :H\to ℝ$ be a ${𝒞}^{2}$ function on a real Hilbert space and $\Sigma \subset H×ℝ$ the manifold defined by $\Sigma :=$ Graph $\left(\Phi \right)$. We study the motion of a material point with unit mass, subjected to stay on $\Sigma$ 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 $x\left(.\right)$ defined on ${ℝ}_{+}$. We are then interested in the asymptotic behaviour of the trajectories when $t\to +\infty$. More precisely, we prove the weak convergence of the trajectories when $\Phi$ is convex. When $\Phi$ admits a strong minimum, we show moreover that the mechanical energy exponentially decreases to its minimum.

DOI : https://doi.org/10.1051/m2an:2002023
Classification:  34A12,  34G20,  37N40,  70Fxx
Keywords: mechanics of particles, dissipative dynamical system, optimization, convex minimization, asymptotic behaviour, gradient system, heavy ball with friction
@article{M2AN_2002__36_3_505_0,
author = {Cabot, Alexandre},
title = {Motion with friction of a heavy particle on a manifold. Applications to optimization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {36},
number = {3},
year = {2002},
pages = {505-516},
doi = {10.1051/m2an:2002023},
zbl = {1032.34059},
mrnumber = {1918942},
language = {en},
url = {http://www.numdam.org/item/M2AN_2002__36_3_505_0}
}

Cabot, Alexandre. Motion with friction of a heavy particle on a manifold. Applications to optimization. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, pp. 505-516. doi : 10.1051/m2an:2002023. http://www.numdam.org/item/M2AN_2002__36_3_505_0/

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