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

Alexandre Cabot

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

top
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 ( γ > 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.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.