Motion with friction of a heavy particle on a manifold. Applications to optimization

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 36.3 (2002): 505-516. <http://eudml.org/doc/245547>.

@article{Cabot2002,
abstract = {Let $\Phi : H\rightarrow \mathbb \{R\}$ be a $\{\mathcal \{C\}\}^2$ function on a real Hilbert space and $\Sigma \subset H \times \mathbb \{R\}$ the manifold defined by $\Sigma :=$ Graph $(\Phi )$. 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 $g&gt;0$), the reaction force and the friction force ($\gamma &gt;0$ is the friction parameter). For any initial conditions at time $t=0$, we prove the existence of a trajectory $x(.)$ defined on $\mathbb \{R\}_+$. We are then interested in the asymptotic behaviour of the trajectories when $t\rightarrow +\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.},
author = {Cabot, Alexandre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {mechanics of particles; dissipative dynamical system; optimization; convex minimization; asymptotic behaviour; gradient system; heavy ball with friction},
language = {eng},
number = {3},
pages = {505-516},
publisher = {EDP-Sciences},
title = {Motion with friction of a heavy particle on a manifold. Applications to optimization},
url = {http://eudml.org/doc/245547},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Cabot, Alexandre
TI - Motion with friction of a heavy particle on a manifold. Applications to optimization
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 3
SP - 505
EP - 516
AB - Let $\Phi : H\rightarrow \mathbb {R}$ be a ${\mathcal {C}}^2$ function on a real Hilbert space and $\Sigma \subset H \times \mathbb {R}$ the manifold defined by $\Sigma :=$ Graph $(\Phi )$. 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 $g&gt;0$), the reaction force and the friction force ($\gamma &gt;0$ is the friction parameter). For any initial conditions at time $t=0$, we prove the existence of a trajectory $x(.)$ defined on $\mathbb {R}_+$. We are then interested in the asymptotic behaviour of the trajectories when $t\rightarrow +\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.
LA - eng
KW - mechanics of particles; dissipative dynamical system; optimization; convex minimization; asymptotic behaviour; gradient system; heavy ball with friction
UR - http://eudml.org/doc/245547
ER -