Abstract

The heavy ball with friction dynamical system (u) double over dot + gamma (u) over dot + del Phi (u) = 0 is a non-linear oscillator with damping (gamma > 0). In [2], Alvarez proved that when H is a real Hilbert space and Phi : H --> R is a smooth convex function whose minimal value is achieved, then each trajectory t --> u(t) of this system weakly converges towards a minimizer of Phi. We prove a similar result in the convex constrained case by considering the corresponding gradient-projection dynamical system (u) double over dot + gamma (u) over dot + u - proj(C)(u - mu del Phi (u)) = 0, where C is a closed convex subset of H. This result holds when H is a possibly infinite dimensional space, and extends, by using different technics, previous results by Antipin [1].