Dry-Friction Inertial Dynamics with Implicit Hessian-Driven Damping: Finite-Time Stabilization, Shadowing, and Proximal Discretization

Abstract

In a real Hilbert space $\mathcal H$, we study the following inertial differential inclusion $$ \ddot x(t)+\gamma\dot x(t)+\partial\phi(\dot x(t)) +\nabla f(x(t)+\beta\dot x(t))\ni0, $$ where $\gamma>0$ is the viscous damping coefficient, and $\phi$ is a convex potential with a sharp minimum at the origin that models the dry friction damping (typically $\phi=r\Vert\cdot\Vert$ where $r>0$ is the dry-friction parameter). The function $f$ represents the smooth potential to be minimized, and the shifted-gradient evaluation $\nabla f(x(t)+\beta\dot x(t))$ is known as the implicit Hessian-driven model. Here $\beta\geq 0$ represents the corresponding Hessian-driven parameter. Both the explicit and the implicit Hessian-driven dynamics are know to attenuate the oscillations that occurs in inertial systems. Our contribution concerns the quantitative analysis of this continuous dynamic and its temporal discretization counter-part under the action of these three combined dampings: viscous damping, dry friction, and implicit Hessian-driven damping. We establish a global well-posedness, an exact Lyapunov analysis adapted to the implicit Hessian-driven damping, finite length of the trajectory and its strong convergence to an approximate critical point $x_\infty$ of $f$ satisfying: $-\nabla f(x_\infty)\in\partial \phi(0)$. We show finite-time stabilization under a strict interior condition on the terminal force. We also compare the explicit and the implicit Hessian-driven dynamics and show that their trajectories differ by $O(\beta^2)$ on finite horizons. A temporal semi-implicit discretization of the dynamic above leads to a proximal implicit Hessian-driven algorithm based on one shifted-gradient evaluation. We derive its discrete Lyapunov analysis, asymptotic convergence and, under a strict terminal margin, finite convergence of the discrete iterates.