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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,...

Let $H$ be a real Hilbert space, ${\Phi }_1:H\to ℝ$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint $S$. A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [5]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to ℝ$ whose critical points coincide with $S$ and a control...

Let be a real Hilbert space, ${\Phi }_1:H\to$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint . A classical approach consists in following the trajectories of the generalized steepest descent system (  Brézis [CITE]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to$ whose critical points coincide with  and a control...

Let Φ : be a function on a real Hilbert space and ∑ ⊂ the manifold defined by ∑ := Graph (Φ). We study the motion of a material point with unit mass, subjected to stay on and which moves under the action of the gravity force (characterized by ), the reaction force and the friction force ($\gamma >0$ is the friction parameter). For any initial conditions at time , we prove the existence of a trajectory defined on . We are then interested in the asymptotic behaviour...