A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics

Given H a real Hilbert space and Phi : H --> R a smooth C-2 function, we study the dynamical inertial system (DIN) x(t) + alphax(t) + betadel(2)Phi(x(t))x(t) + delPhi(x(t)) = 0, where alpha and beta 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 del(2)Phi(x(t))x(t) + delPhi(x(t)) = 0. We show that (DIN) is a well-posed dynamical system. Due to their dissipative aspect, trajectories of (DIN) enjoy remarkable optimization properties. For example, when Phi is convex and argmin Phi not equal theta, then each trajectory of (DIN) weakly converges to a minimizer of Phi. If Phi is real analytic, then each trajectory converges to a critical point of Phi. A remarkable feature of (DIN) is that one can produce an equivalent system which is first-order in time and with no occurrence of the Hessian, namely {x(t) + cdelPhi(x(t) + ax(t) + by(t) = 0, y(t) + ax(t) + by(t) = 0, where a, b, c are parameters which can be explicitly expressed in terms of alpha and beta. This allows to consider (DIN) when Phi is C-1 only, or more generally, nonsmooth or subject to constraints. This is first illustrated by a gradient projection dynamical system exhibiting both viable trajectories, inertial aspects, optimization properties, and secondly by a mechanical system with impact. (C) 2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.