We are interested in the existence results for second-order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal normal cone to a time-dependent set. In order to prove these existence results, we study an extension of the numerical scheme introduced in [10] and prove a convergence result for this scheme.
Frédéric Bernicot 1 ; Aline Lefebvre-Lepot 1
@article{CML_2010__2_4_445_0,
author = {Fr\'ed\'eric Bernicot and Aline Lefebvre-Lepot},
title = {Existence results for nonsmooth second-order differential inclusions, convergence result for a numerical scheme and application to the modeling of inelastic collisions},
journal = {Confluentes Mathematici},
pages = {445--471},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {4},
year = {2010},
doi = {10.1142/S1793744210000247},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744210000247/}
}
TY - JOUR AU - Frédéric Bernicot AU - Aline Lefebvre-Lepot TI - Existence results for nonsmooth second-order differential inclusions, convergence result for a numerical scheme and application to the modeling of inelastic collisions JO - Confluentes Mathematici PY - 2010 SP - 445 EP - 471 VL - 2 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744210000247/ DO - 10.1142/S1793744210000247 LA - en ID - CML_2010__2_4_445_0 ER -
%0 Journal Article %A Frédéric Bernicot %A Aline Lefebvre-Lepot %T Existence results for nonsmooth second-order differential inclusions, convergence result for a numerical scheme and application to the modeling of inelastic collisions %J Confluentes Mathematici %D 2010 %P 445-471 %V 2 %N 4 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744210000247/ %R 10.1142/S1793744210000247 %G en %F CML_2010__2_4_445_0
Frédéric Bernicot; Aline Lefebvre-Lepot. Existence results for nonsmooth second-order differential inclusions, convergence result for a numerical scheme and application to the modeling of inelastic collisions. Confluentes Mathematici, Tome 2 (2010) no. 4, pp. 445-471. doi : 10.1142/S1793744210000247. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000247/
[1] P. Ballard, Arch. Rational Mech. Anal. 154, 199 (2000), DOI: 10.1007/s002050000105.
[2] F. Bernicot and J. Venel, Stochastic perturbations of sweeping process, submitted , arXiv:1001.3128 .
[3] F. Bernicot and J. Venel, Existence of solutions for second-order differential inclusions involving proximal normal cones, submitted , arXiv:1006.2292 .
[4] F. H. Clarke, R. J. Stern and P. R. Wolenski, J. Convex Anal. 2, 117 (1995).
[5] F. H. Clarke et al. , Nonsmooth Analysis and Control Theory ( Springer-Verlag , 1998 ) .
[6] G. Colombo and M. D. P. Monteiro, J. Differential Equation 187, 46 (2003), DOI: 10.1016/S0022-0396(02)00021-9.
[7] R. Dzonou and M. D. P. Monteiro Marques, Eur. J. Mech. A/Solids 26, 474 (2007), DOI: 10.1016/j.euromechsol.2006.07.002.
[8] R. Dzonou, M. D. P. Monteiro Marques and L. Paoli, Nonlinear Dyn. 58, 361 (2009), DOI: 10.1007/s11071-009-9484-1.
[9] A. Lefebvre, Model. Math. Anal. Numer. 43, 53 (2009), DOI: 10.1051/m2an/2008042.
[10] B. Maury, Numer. Math. 102, 649 (2006), DOI: 10.1007/s00211-005-0666-6.
[11] B. Maury and J. Venel , Model. Math. Anal. Numer. , arXiv:0901.0984 .
[12] M. D. P. Monteiro-Marques , PNLDE 9 ( Birkhäuser , 1993 ) .
[13] M. D. P. Monteiro-Marques and L. Paoli, Nonsmooth Mechanics and Analysis, Adv. Mech. Math 12 (Springer, 2006) pp. 279–288.
[14] J. J. Moreau, C. R. Acad. Sci. Ser. I 255, 238 (1962).
[15] J. J. Moreau, C. R. Acad. Sci. Ser. II 296, 1473 (1983).
[16] J. J. Moreau, Standard Inelastic Shocks and the Dynamics of Unilateral Constraints, CISM Courses and Lectures 288 (Springer, 1985) pp. 173–221.
[17] J. J. Moreau, Eur. J. Mech. A/Solids 13, 93 (1994).
[18] L. Paoli and M. Schatzman, Model. Math. Anal. Numer. 6, 673 (1993).
[19] L. Paoli and M. Schatzman, J. Differential Equations 177, 375 (2001), DOI: 10.1006/jdeq.2001.4027.
[20] L. Paoli and M. Schatzman, SIAM J. Numer. Anal. 2, 702 (2002).
[21] L. Paoli and M. Schatzman, Multibody Syst. Dynam. 8, 347 (2002).
[22] L. Paoli, J. Differential Equations 211, 247 (2005), DOI: 10.1016/j.jde.2004.11.008.
[23] L. Paoli , A.R.M.A. .
[24] L. Paoli , A.R.M.A. .
[25] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Trans. Amer. Math. Soc. 352, 5231 (2000), DOI: 10.1090/S0002-9947-00-02550-2.
[26] M. Schatzman, Nonlinear Anal. 2, 355 (1978), DOI: 10.1016/0362-546X(78)90022-6.
[27] M. Schatzman, Math. Comput. Model. 28, 1 (1998), DOI: 10.1016/S0895-7177(98)00104-6.
[28] M. Schatzman, Phil. Trans. Roy. Soc. London A 359, 2429 (2001).
[29] SCoPI Software, presentation available at http://www.projet-plume.org/relier/scopi, and numerical simulations available at http://www.cmap.polytechnique.fr/ lefebvre/SCoPI.htm .
[30] J. Venel, Num. Math. (2011), arXiv:0904.2694.
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