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A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics. (English) Zbl 1036.34072

Summary: Given \(H\) a real Hilbert space and \(\Phi:H\to \mathbb{R}\) a smooth \({\mathcal C}^2\) function, we study the dynamical inertial system \[ \ddot x(t)+\alpha\dot x(t)+ \beta\nabla^2\Phi(x(t))\dot x(t)+ \nabla\Phi(x(t))= 0,\tag{DIN} \] where \(\alpha\) and \(\beta\) are positive parameters. The inertial term \(\ddot x(t)\) acts as a singular perturbation and, in fact, regularization of the possibly degenerate classical Newton continuous dynamical system \(\nabla^2\Phi(x(t))\dot x(t)+ \nabla\Phi(x(t))= 0\).
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 \(\text{argmin\,}\Phi\neq \emptyset\), 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, \[ \dot x(t)+ c\nabla\Phi(x(t))+ ax(t)+ by(t)= 0,\quad \dot y(t)+ ax(t)+ by(t)= 0, \] where \(a\), \(b\), \(c\) are parameters which can be explicitly expressed in terms of \(\alpha\) and \(\beta\). This allows one to consider (DIN) when \(\Phi\) is \({\mathcal C}^1\) 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.

MSC:

34G20 Nonlinear differential equations in abstract spaces
37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems
34H05 Control problems involving ordinary differential equations
70F40 Problems involving a system of particles with friction
90C30 Nonlinear programming
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