On the minimizing property of a second order dissipative system in Hilbert spaces

Abstract

We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below, then the solution trajectories are minimizing for it and converge weakly towards a minimizer of Phi if one exists; this convergence is strong when Phi is even or when the optimal set has a nonempty interior. We introduce a second order proximal-like iterative algorithm for the minimization of a convex function. It is defined by an implicit discretization of the continuous evolution problem and is valid for any closed proper convex function. We nd conditions on some parameters of the algorithm in order to have a convergence result similar to the continuous case.