 
Summary: A SECONDORDER GRADIENTLIKE DISSIPATIVE DYNAMICAL
SYSTEM with HESSIAN DRIVEN DAMPING.
Application to Optimization and Mechanics.
F. ALVAREZ
H. ATTOUCH
J. BOLTE
P. REDONT
Abstract
Given H a real Hilbert space and : H R a smooth C2
function, we study the dynamical system
(DIN) ¨x(t) + x(t) + 2
(x(t)) x(t) + (x(t)) = 0
where and are positive parameters. The inertial term ¨x(t) acts as a singular perturbation and, in fact,
regularization of the possibly degenerate classical Newton continuous dynamical system 2
(x(t)) x(t)+ (x(t)) =
0.
We show that (DIN) is a wellposed dynamical system. Due to their dissipative aspect, trajectories of (DIN)
enjoy remarkable optimization properties. For example, when is convex and argmin = , then each trajectory
of (DIN) weakly converges to a minimizer of . If is real analytic, then each trajectory converges to a critical
point of .
