Asymptotics for some vibro-impact problems with a linear dissipation term

Given γ ≥ 0, let us consider the following differential inclusion (S) ẍ(t) + γ ẋ(t) + ∂Φ(x(t)) 3 0, t ∈ R+, where Φ : Rd → R ∪ {+∞} is a lower semicontinuous convex function such that int(domΦ) 6= ∅. The operator ∂Φ denotes the subdifferential of Φ. When Φ = f + δK with f : Rd → R a smooth convex function and K ⊂ Rd a closed convex set, inclusion (S) describes the motion of a discrete mechanical system subjected to the perfect unilateral constraint x(t) ∈ K and submitted to the conservative force −∇f(x) and the viscous friction force −γ ẋ. We define the notion of dissipative solution to (S) and we prove the existence of such solutions with conservation (resp. loss) of energy at impacts. If γ > 0 and Φ|domΦ is locally Lipschitz continuous, any dissipative solution to (S) converges, as t→ +∞, to a minimum point of Φ. When Φ is strongly convex, the speed of convergence is exponential. Assuming as above that Φ = f+δK , suppose that the boundary of K is smooth enough and that the normal component of the velocity is reversed and multiplied by a restitution coefficient r ∈ [0, 1] while the tangential component is conserved whenever x(t) ∈ bd(K). We prove that any dissipative solution to (S) satisfying the previous impact law with r < 1 is contained in the boundary of K after a finite time. The case r = 1 is also addressed and leads to a qualitatively different behavior.