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

In a Hilbert space

, we study the stabilization in finite-time of the trajectories generated by a continuous (in time

t t

) damped inertial dynamic system. The potential function

f\colon {{\mathcal H}}\to \mathbb{R} f  ⁣ : H → 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 f

. The dry friction damping function

\phi\colon {{\mathcal H}}\to \mathbb{R}_+ ϕ  ⁣ : H → R +

, which is convex and continuous with a sharp minimum at the origin (typically

\phi(x) = r \|x\| ϕ ( x ) = r ∥ x ∥

with

r>0 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) ∇ 2 f ( x ( t ) ) 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 f

is Lipschitz continuous on the bounded sets of

{\mathcal H} H

. Then, we study the convergence properties of the trajectories as

t \to +\infty t → + ∞

, 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 f

by using Moreau's envelope.