Existence results for nonsmooth second-order differential inclusions, convergence result for a numerical scheme and application to the modeling of inelastic collisions
Confluentes Mathematici, Volume 2 (2010) no. 4, pp. 445-471.

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.

Published online:
DOI: 10.1142/S1793744210000247
Frédéric Bernicot 1; Aline Lefebvre-Lepot 1

1
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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, Volume 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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