A SECOND-ORDER GRADIENT-LIKE DISSIPATIVE DYNAMICAL SYSTEM with HESSIAN DRIVEN DAMPING. Application to Optimization and Mechanics

Given H a real Hilbert space and Φ : H → R a smooth C function, we study the dynamical system (DIN) ẍ(t) + αẋ(t) + β∇Φ(x(t))ẋ(t) +∇Φ(x(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 ∇Φ(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 6= ∅, 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∇Φ(x(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 C only, or more generally, non-smooth 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.