# Motion with friction of a heavy particle on a manifold - applications to optimization

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 36.3 (2010): 505-516. <http://eudml.org/doc/194114>.

@article{Cabot2010,

abstract = {
Let Φ : H → R be a C2 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.
},

author = {Cabot, Alexandre},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Mechanics of particles; dissipative
dynamical system; optimization; convex minimization;
asymptotic behaviour; gradient system; heavy ball with friction.; mechanics of particles; dissipative dynamical system; asymptotic behaviour; heavy ball with friction},

language = {eng},

month = {3},

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/194114},

volume = {36},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 3

SP - 505

EP - 516

AB -
Let Φ : H → R be a C2 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.

LA - eng

KW - Mechanics of particles; dissipative
dynamical system; optimization; convex minimization;
asymptotic behaviour; gradient system; heavy ball with friction.; mechanics of particles; dissipative dynamical system; asymptotic behaviour; heavy ball with friction

UR - http://eudml.org/doc/194114

ER -

## References

top- F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space. SIAM J. Control Optim.38 (2000) 1102-1119.
- 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.
- J. Bolte, Exponential decay of the energy for a second-order in time dynamical system. Working paper, Département de Mathématiques, Université Montpellier II.
- R.E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal.18 (1975) 15-26.
- J.K. Hale, Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI (1988).
- A. Haraux, Systèmes dynamiques dissipatifs et applications. RMA 17, Masson, Paris (1991).
- Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc.73 (1967) 591-597.

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