# 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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topCabot, 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 $g>0$), 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 $\mathbb \{R\}_+$. 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 $g>0$), 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 $\mathbb {R}_+$. 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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- [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] J. Bolte, Exponential decay of the energy for a second-order in time dynamical system. Working paper, Département de Mathématiques, Université Montpellier II.
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- [5] J.K. Hale, Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI (1988). Zbl0642.58013MR941371
- [6] A. Haraux, Systèmes dynamiques dissipatifs et applications. RMA 17, Masson, Paris (1991). Zbl0726.58001MR1084372
- [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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