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

Cabot, Alexandre

doi:10.1051/m2an:2002023

Cabot, Alexandre

ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 3, pp. 505-516.

Résumé

Let $Φ : H \to ℝ$ be a $𝒞^{2}$ function on a real Hilbert space and $Σ \subset H \times ℝ$ the manifold defined by $Σ : =$ 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 $t \to + \infty$ . 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.

MR Zbl

DOI : 10.1051/m2an:2002023

Classification : 34A12, 34G20, 37N40, 70Fxx
Mots-clés : mechanics of particles, dissipative dynamical system, optimization, convex minimization, asymptotic behaviour, gradient system, heavy ball with friction

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     author = {Cabot, Alexandre},
     title = {Motion with friction of a heavy particle on a manifold. {Applications} to optimization},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {505--516},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {3},
     year = {2002},
     doi = {10.1051/m2an:2002023},
     mrnumber = {1918942},
     zbl = {1032.34059},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2002023/}
}

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Cabot, Alexandre. Motion with friction of a heavy particle on a manifold. Applications to optimization. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 3, pp. 505-516. doi : 10.1051/m2an:2002023. http://www.numdam.org/articles/10.1051/m2an:2002023/

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