Motion with friction of a heavy particle on a manifold. Applications to optimization
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, p. 505-516

Let Φ:H be a 𝒞 2 function on a real Hilbert space and ΣH× 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 + . 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.

DOI : https://doi.org/10.1051/m2an:2002023
Classification:  34A12,  34G20,  37N40,  70Fxx
Keywords: mechanics of particles, dissipative dynamical system, optimization, convex minimization, asymptotic behaviour, gradient system, heavy ball with friction
@article{M2AN_2002__36_3_505_0,
     author = {Cabot, Alexandre},
     title = {Motion with friction of a heavy particle on a manifold. Applications to optimization},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {3},
     year = {2002},
     pages = {505-516},
     doi = {10.1051/m2an:2002023},
     zbl = {1032.34059},
     mrnumber = {1918942},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_3_505_0}
}
Motion with friction of a heavy particle on a manifold. Applications to optimization. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, pp. 505-516. doi : 10.1051/m2an:2002023. http://www.numdam.org/item/M2AN_2002__36_3_505_0/

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