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

Cabot, Alexandre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 36 (2002) no. 3, pp. 505-516

Résumé

Let $\Phi :H\to ℝ$ be a ${𝒞}^2$ function on a real Hilbert space and $\Sigma \subset H\times ℝ$ the manifold defined by $\Sigma :=$ Graph $\left(\Phi \right)$. We study the motion of a material point with unit mass, subjected to stay on $\Sigma$ 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 $t\to +\infty$. More precisely, we prove the weak convergence of the trajectories when $\Phi$ is convex. When $\Phi$ admits a strong minimum, we show moreover that the mechanical energy exponentially decreases to its minimum.

Détail

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