
Journal of Convex Analysis 28 (2021), No. 2, [final page numbers not yet available] Copyright Heldermann Verlag 2021 FiniteTime Stabilization of Continuous Inertial Dynamics Combining Dry Friction with HessianDriven Damping Samir Adly Laboratoire XLIM, Université de Limoges, 87060 Limoges, France samir.adly@unilim.fr Hedy Attouch IMAG  CNRS, Université de Montpellier, 34095 Montpellier, France hedy.attouch@umontpellier.fr [Abstractpdf] \newcommand{\cH}{{\mathcal H}} In a Hilbert space $\cH$, we study the stabilization in finitetime of the trajectories generated by a continuous (in time $t$) damped inertial dynamic system. The potential function $f\colon \cH \to \mathbb{R}$ to be minimized is supposed to be differentiable, not necessarily convex. It enters the dynamic via its gradient. The damping results from the joint action of dry friction, viscous friction, and a geometric damping driven by the Hessian of $f$. The dry friction damping function $\phi\colon \cH \to \mathbb{R}_+$, which is convex and continuous with a sharp minimum at the origin (typically $\phi(x) = r \x\$ with $r>0$), enters the dynamic via its subdifferential. It acts as a soft threshold operator on the velocities, and is at the origin of the stabilization property in finitetime. The Hessian driven damping, which enters the dynamics in the form $\nabla^2 f(x(t))\dot{x}(t)$, permits to control and attenuate the oscillations which occur naturally with the inertial effect. We give two different proofs, in a finite dimensional setting, of the existence of strong solutions of this secondorder differential inclusion. One is based on a fixed point argument and LeraySchauder theorem, the other one uses the Yosida approximation technique together with the Mosco convergence. We also give an existence and uniqueness result in a general Hilbert framework by assuming that the Hessian of the function $f$ is Lipschitz continuous on the bounded sets of $\cH$. Then, we study the convergence properties of the trajectories as $t \to +\infty$, and show their stabilization property in finitetime. The convergence results tolerate the presence of perturbations (or errors) under the sole assumption of their asymptotic convergence to zero. The study is extended to the case of a nonsmooth convex function $f$ by using Moreau's envelope. Keywords: Damped inertial dynamics, differential inclusion, dry friction, Hessiandriven damping, finitetime stabilization. MSC: 37N40, 34A60, 34G25, 49K24, 70F40. [ Fulltextpdf (198 KB)] for subscribers only. 