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XLIM UMR CNRS 6172 Département Mathématiques-Informatique Asymptotics for some vibro-impact problems with a linear dissipation term Alexandre Cabot & Laetitia Paoli Rapport de recherche n° 2006-08 Déposé le 30 juin 2006 Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex Tél. (33) 5 55 45 73 23 - Fax. (33) 5 55 45 73 22
Voir
- qua- dratic term
- trajectories via penalization techniques
- called elastic
- potential function
- friction force
- when ?
- inclusion
- differential inclusion
XLIM
UMR CNRS 6172
Département Mathématiques-Informatique
Asymptotics for some vibro-impact problems
with a linear dissipation term
Alexandre Cabot & Laetitia Paoli
Rapport de recherche n° 2006-08 Déposé le 30 juin 2006
Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex Tél. (33) 5 55 45 73 23 - Fax. (33) 5 55 45 73 22 http://www.xlim.fr http://www.unilim.fr/laco
Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex
Tél. (33) 5 55 45 73 23 - Fax. (33) 5 55 45 73 22
http://www.xlim.fr http://www.unilim.fr/laco
ASYMPTOTICS FOR SOME VIBRO-IMPACT PROBLEMS WITH A LINEAR DISSIPATION TERM
ALEXANDRE CABOT AND LAETITIA PAOLI Abstract.Given
lofeiwoldisnhtreiantnclidingre e,loentsuislcuo0 (S) x(t) +
x˙ (t) +∂(x(t))30, t∈R+, where :Rd→R∪ {+∞}is a lower semicontinuous convex function such that int (dom )6=∅. The operator∂ the subdi eren tial denotes of . When =f+Kwithf:Rd→Ra smooth convex function andKRda closed convex set, inclusion (S) describes the motion of a discrete mechanical system subjected to the perfect unilateral constraintx(t)∈Kand submitted to the conservative forcerf(x) and the viscous friction force
˙x. We the dene notion ofdissipativeand we prove the existence of such solutionssolution to (S) with conservation (resp. loss) of energy at impacts. If
>0 and |dom is locally Lipschitz continuous, any dissipative solution to (S) converges, as t→+∞imaot,cynonolgssrtnei.Whntofmpoinimu,xevsehtdeepfo convergence is exponential. Assuming as above that =f+K, suppose that the boundary ofKis smooth enough and that the normal component of the velocityisreversedandmultipliedbyarestitutioncoecientr∈[0,1] while the tangential component is conserved wheneverx(t)∈bd (K). We prove that any dissipative solution to (S) satisfying the previous impact law withr <1 is contained in the boundary ofK time.after a nite case Ther= 1 is also addressedandleadstoaqualitativelydi erentbehavior.
1.Introduction Throughout the paper, the spaceRdis endowed with the Euclidean inner product (,) and the corresponding norm || . Given
0, let us consider the second-order in time di eren tial inclusion (S) x(t) +
˙x(t) +∂(x(t))30, t∈R+, where :Rd→R∪ {+∞}is a lower semicontinuous convex function such that int(dom)6=∅. The called the potential function and the operator is function ∂ of tial for the subdi eren stands every for the sense of convex analysis: in x∈dom , ∈∂(x)∀⇐⇒y∈Rd,(y)(x) + ( , y x). The nonnegative parameter
is called the friction parameter. the function When is smooth, the subdi eren tial∂ with the gradient coincidesr inclusion and (S) becomes (H BF) x(t) +
x˙ (t) +r(x(t)) = 0, t∈R+, 1991.noamitscuSMtaehssicatibjectCla34A60, 34A12, 70F35, 65L20, 37N05, 37N40. Key words and phrases. inclusion, frictionless vibro-impact problems, dissipativeDi eren tial solution, Newton’s impact law, time-stepping scheme, constrained convex optimization. 1
