Département Mathématiques Informatique

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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. [phone] - Fax. [phone]

• qua- dratic term


• trajectories via penalization techniques


• called elastic


• potential function


• friction force


• when ?


• inclusion


• differential inclusion

Sujets

• Université de Limoges
• Potential function
• Inclusión
• Differential inclusion

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

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. [phone] - Fax. [phone] http://www.xlim.fr http://www.unilim.fr/laco

Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex

Tél. [phone] - Fax. [phone]

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 lofeiwoldisnhtreiantnclidingree,loentsuislcuo0 (S) x(t) + x˙ (t) +∂(x(t))30, t∈R + , where  :R d →R∪ {+∞}is a lower semicontinuous convex function such that int (dom )6=∅. The operator∂ the subdieren tial denotes of . When  =f+ K withf:R d →Ra smooth convex function andKR d a 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 dene 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 addressedandleadstoaqualitativelydierentbehavior.

1.Introduction Throughout the paper, the spaceR d is endowed with the Euclidean inner product (
,
) and the corresponding norm ||
. Given 0, let us consider the second-order in time dieren tial inclusion (S) x(t) + ˙x(t) +∂(x(t))30, t∈R + , where  :R d →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 subdieren stands every for the sense of convex analysis: in x∈dom , ∈∂(x)∀⇐⇒y∈R d ,(y)(x) + ( , y   x). The nonnegative parameter is called the friction parameter. the function When  is smooth, the subdieren tial∂ with the gradient coincidesr inclusion and (S) becomes (H BF) x(t) + x˙ (t) +r(x(t)) = 0, t∈R + , 1991.noamitscuSMtaehssicatibjectCla34A60, 34A12, 70F35, 65L20, 37N05, 37N40. Key words and phrases. inclusion, frictionless vibro-impact problems, dissipativeDieren tial solution, Newton’s impact law, time-stepping scheme, constrained convex optimization. 1