Motion with friction of a heavy particle on a manifold. Applications to optimization
ESAIM: Modélisation mathématique et analyse numérique, Volume 36 (2002) no. 3, pp. 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: 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
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     title = {Motion with friction of a heavy particle on a manifold. {Applications} to optimization},
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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, Volume 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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