A second-order differential system with hessian-driven damping; application to non-elastic shock laws

Hedy Attouch (1) , Paul-Emile Maingé (2) , Patrick Redont (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 LAMIA - Laboratoire de Mathématiques Informatique et Applications

We consider a second-order differential system with Hessian-driven damping . An interesting property of this system is that, after introduction of an auxiliary variable y , it can be equivalently written as a first-order system in time and space. This allows us to extend the analysis to the case of a convex lower semicontinuous potential and so to introduce constraints in the model. When considering the indicator function of a closed convex set, the subdifferential operator takes account of the contact forces. In this setting, by playing with the geometrical damping parameter, we can describe nonelastic shock laws with restitution coefficient. Taking advantage of the infinite dimensional framework, we introduce a nonlinear hyperbolic PDE describing a damped oscillating system with obstacle. The system is dissipative; in the convex case each trajectory weakly converges to a minimizer of the global potential energy function. Exponential stabilization is obtained under strong convexity assumptions.

Mots clés: impact dynamics; Hessian-driven damping; gradient-like systems; restitution coefficient; nonelastic shocks; nonsmooth potentials; viscoelastic membrane.; asymptotic stabilization; convex variational analysis; dissipative dynamical systems; exponential stabilization; second-order nonlinear differential equations; unilateral mechanics; viscoelastic membrane

Domaines: Optimisation et contrôle [math.OC]

Hedy Attouch, Paul-Emile Maingé, Patrick Redont. A second-order differential system with hessian-driven damping; application to non-elastic shock laws. Differential Equations and Applications, 2012, 4 (1), pp.27-65. ⟨10.7153/dea-04-04⟩. ⟨hal-00803214⟩