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

Abstract

Given γ ⩾ 0 , let us consider the following differential inclusion, (S) x ¨ ( t ) + γ x ˙ ( t ) + ∂ Φ ( x ( t ) ) ∋ 0 , t ∈ R + , where Φ : R d → R ∪ { + ∞ } is a lower semicontinuous convex function such that int ( dom Φ ) ≠ ∅ . The operator ∂ Φ denotes the subdifferential of Φ. When Φ = f + δ K with f : R d → R a smooth convex function and K ⊂ R d 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 x ˙ . 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 | dom Φ 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 + δ K , 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.