 
Summary: ON THE MINIMIZING PROPERTY OF A SECOND ORDER
DISSIPATIVE SYSTEM IN HILBERT SPACES
FELIPE ALVAREZ
SIAM J. CONTROL OPTIM. c 2000 Society for Industrial and Applied Mathematics
Vol. 38, No. 4, pp. 11021119
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 if one exists; this convergence
is strong when is even or when the optimal set has a nonempty interior. We introduce a second
order proximallike 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 find conditions on some parameters of the algorithm in order to have a convergence
result similar to the continuous case.
Key words. dissipative system, linear damping, asymptotic behavior, weak convergence, con
vexity, implicit discretization, iterativevariational algorithm
AMS subject classifications. 34G20, 34A12, 34D05, 90C25
PII. S0363012998335802
1. Introduction. Consider the following differential system defined in a real
Hilbert space H:
