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

Abstract

Given H a real Hilbert space and Φ:H→R a smooth C2 function, we study the dynamical inertial system (DIN)ẍ(t)+αẋ(t)+β∇2Φx(t)ẋ(t)+∇Φx(t)=0, where α and β are positive parameters. The inertial term ẍ(t) acts as a singular perturbation and, in fact, regularization of the possibly degenerate classical Newton continuous dynamical system ∇2Φ(x(t))ẋ(t)+∇Φ(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 Φ is convex and argminΦ≠∅, then each trajectory of (DIN) weakly converges to a minimizer of Φ. If Φ is real analytic, then each trajectory converges to a critical point of Φ.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 ẋ(t)+c∇Φx(t)+ax(t)+by(t)=0,ẏ(t)+ax(t)+by(t)=0, where a, b, c are parameters which can be explicitly expressed in terms of α and β. This allows to consider (DIN) when Φ is C1 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.