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Delay perturbed sweeping process. (English) Zbl 1122.34060

A multivalued sweeping process is studied for the class of mappings satisfying Lipschitz conditions. An unique result is proved. Note that the presented proofs are technically complicated but the used methods are rather standard.

MSC:

34K30 Functional-differential equations in abstract spaces
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[1] Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Differential Equations 164, 286–295 (2000) · Zbl 0957.34061
[2] Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. Forum 48, 223–246 (2002) · Zbl 1012.49013
[3] Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6, 359–374 (2005) · Zbl 1086.49016
[4] Bounkhel, M., Yarou, M.: Existence results for first and second order nonconvex sweeping process with delay. Port. Math. 61(2), 2007–2030 (2004) · Zbl 1098.49016
[5] Castaing, C., Duc Ha, T.X., Valadier, M.: Evolution equations governed by the sweeping process. Set-Valued Anal. 1, 109–139 (1993) · Zbl 0813.34018
[6] Castaing, C., Monteiro Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets. Port. Math. 53, 73–87 (1996) · Zbl 0848.35052
[7] Castaing, C., Monteiro Marques, M.D.P.: Topological properties of solution sets for sweeping processes with delay. Port. Math. 54, 485–507 (1997) · Zbl 0895.34053
[8] Castaing, C., Salvadori, A., Thibault, L.: Functional evolution equations governed by nonconvex sweeping Process. J. Nonlinear Convex Anal. 2, 217–241 (2001) · Zbl 0999.34062
[9] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[10] Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, Berlin Heidelberg New York (1998) · Zbl 1047.49500
[11] Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower- \(C^2\) property. J. Convex Anal. 2, 117–144 (1995) · Zbl 0881.49008
[12] Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set-Valued Anal. 7, 357–374 (1999) · Zbl 0957.34060
[13] Edmond, J.F.: Problèmes d’évolution associés à des ensembles prox-réguliers. Inclusions et intégration de sous-différentiels. Thèse de Doctorat, Université Montpellier II, 2004
[14] Edmond, J.F., Thibault, L.: Relaxation of an optimal control problem involving a perturbed sweeping process. Math. Programming 104(2–3), 347–373 (2005) · Zbl 1124.49010
[15] Edmond, J.F., Thibault, L.: BV solution of nonconvex sweeping process differential inclusion with perturbation. J. Differential Equations 226(1), 135–139 (2006) · Zbl 1110.34038
[16] Monteiro Marques, M.D.P.: Differential inclusions in Nonsmooth Mechanical Problems, Shocks and Dry Friction. Birkhäuser, Basel (1993) · Zbl 0802.73003
[17] Moreau, J.J.: Evolution problem associated with a moving convex set in a hilbert space. J. Differential Equations 26, 347–374 (1977) · Zbl 0356.34067
[18] Moreau, J.J.: Sur l’évolution d’un système élasto-viscoplastique. C.R. Acad. Sci. Paris Sér. A–B 273, A118–A121 (1971) · Zbl 0245.73029
[19] Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) New Variational Techniques in Mathematical Physics, pp. 173–322. C.I.M.E. II Ciclo 1973, Edizioni Cremonese, Roma (1974)
[20] Moreau, J.J.: Standard inelastic shocks and the dynamics of unilateral constraints. In: del Piero, G., Maceri, F. (eds.) Unilateral Problems in Structural Analysis, C.I.S.M. Courses and Lectures no. 288, 173–221. Springer, Berlin Heidelberg New York (1985) · Zbl 0619.73115
[21] Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352, 5231–5249 (2000) · Zbl 0960.49018
[22] Thibault, L.: Sweeping process with regular and nonregular sets. J. Differential Equations 193, 1–26 (2003) · Zbl 1037.34007
[23] Valadier, M.: Quelques problèmes d’entrainement unilatéral en dimension finie. Sém. Anal. Convexe Montpellier, Exposé No. 8 (1988)
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