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

• Volume: 36, Issue: 3, page 505-516
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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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 -

## References

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1. [1] F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space. SIAM J. Control Optim. 38 (2000) 1102–1119. Zbl0954.34053
2. [2] H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method. I The continuous dynamical system. Commun. Contemp. Math. 2 (2000) 1–34. Zbl0983.37016
3. [3] J. Bolte, Exponential decay of the energy for a second-order in time dynamical system. Working paper, Département de Mathématiques, Université Montpellier II.
4. [4] R.E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18 (1975) 15–26. Zbl0319.47041
5. [5] J.K. Hale, Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI (1988). Zbl0642.58013MR941371
6. [6] A. Haraux, Systèmes dynamiques dissipatifs et applications. RMA 17, Masson, Paris (1991). Zbl0726.58001MR1084372
7. [7] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 591–597. Zbl0179.19902

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