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

  • Alexandre Cabot (a1)

Abstract

Let Φ : H → R be a C 2 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.

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[1] Alvarez, F., On the minimizing property of a second order dissipative system in Hilbert space. SIAM J. Control Optim. 38 (2000) 1102-1119.
[2] Attouch, H., Goudou, X. and Redont, P., 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.
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ESAIM: Mathematical Modelling and Numerical Analysis
  • ISSN: 0764-583X
  • EISSN: 1290-3841
  • URL: /core/journals/esaim-mathematical-modelling-and-numerical-analysis
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