Abstract
In a Hilbert space H, 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 : 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. The dry friction damping function φ : H → R + , which is convex with a sharp minimum at the origin (typically φ(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 ∇ 2 f (x(t))ẋ(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 the use of Leray-Schauder theorem, the other one is based on the Yosida approximation technique and the Mosco convergence. Then, we study the convergence properties of the trajectories as t → +∞, and show their stabilization property in finite time. The convergence results tolerate the presence of perturbations, errors, under the sole assumption of their asymptotic convergence to zero. Then, we extend the study to the case of a nonsmooth convex function f. Mathematics Subject Classifications: 37N40, 34A60, 34G25, 49K24, 70F40.