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Journal of Convex Analysis 28 (2021), No. 2, 281--310
Copyright Heldermann Verlag 2021

Finite-Time Stabilization of Continuous Inertial Dynamics Combining Dry Friction with Hessian-Driven Damping

Samir Adly
Laboratoire XLIM, Université de Limoges, 87060 Limoges, France

Hedy Attouch
IMAG -- CNRS, Université de Montpellier, 34095 Montpellier, France


\newcommand{\cH}{{\mathcal H}} In a Hilbert space $\cH$, we study the stabilization in finite-time 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 finite-time. 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 second-order differential inclusion. One is based on a fixed point argument and Leray-Schauder 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 finite-time. 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, Hessian-driven damping, finite-time stabilization.

MSC: 37N40, 34A60, 34G25, 49K24, 70F40.

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