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We consider the second-order differential system with Hessian-driven damping ¨ u + α ˙ u +β∇ 2 Φ(u)˙ u +∇Φ(u )+ ∇Ψ(u )= 0, where H is a real Hilbert space, Φ,Ψ : H → R are scalar potentials, and α,β are positive parameters. 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 involving only the time derivatives ˙ u, ˙ y and the gradient operators ∇Φ, ∇Ψ. This allows to extend our analysis to the case of a convex lower semicontinuous function Φ : H → R ∪{ +∞}, and so to introduce constraints in our model. When Φ = δK is the indicator function of a closed convex set K ⊆ H , the subdifferential operator ∂Φ takes account of the contact forces, while ∇Ψ takes account of the driving forces. In this setting, by playing with the geometrical damping parameter β , 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 first-order system is dissipative; each trajectory weakly converges to a minimizer of Φ + Ψ, provided that Φ and Φ +Ψ are convex functions. Exponential stabilization is obtained under strong convexity assumptions.

A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws