Abstract

The paper deals with sweeping process measure differential inclusions with data sets subject to the condition of bounded retraction along bounded truncation. Various basic properties are provided, and the existence and uniqueness of solution are established under the convexity of the data sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Lionel Thibault

Université de Montpellier, Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, France
and: Centro de Modelamiento Matematico, Universidad de Chile, Chile

lionel.thibault@univ-montp2.fr

Keywords

  • Sweeping process
  • differential inclusion
  • bounded retraction along truncation
  • mapping of bounded variation
  • Radon measure
  • catching-up algorithm
  • normal cone
  • subdifferential

Mathematics Subject Classification

34A60, 49J52; 49J24, 49J53

References

  1. V. Acary, O. Bonnefon, B. Brogliato: Nonsmooth Modeling and Simulation for Switched Circuits, Lecture Notes in Electrical Engineering 69, Springer, Dordrecht (2011).https://doi.org/10.1007/978-90-481-9681-4Google Scholar
  2. K. Addi, S. Adly, B. Brogliato, D. Goeleven: A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Anal., Hybrid Syst. 1(1) (2007) 30–43.https://doi.org/10.1016/j.nahs.2006.04.001Google Scholar
  3. S. Adly, T. Haddad, L. Thibault: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program. 148 (2014) 5–47.https://doi.org/10.1007/s10107-014-0754-4Google Scholar
  4. S. Adly, F. Nacry and L. Thibault: Discontinuous sweeping process with prox-regular sets, ESAIM Control. Optim. Calc. Var., to appear.Google Scholar
  5. S. Adly, F. Nacry, L. Thibault: Preservation of prox-regularity of sets and constrained optimization, SIAM J. Optim., to appear.Google Scholar
  6. H. Attouch, R. J.-B. Wets: Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991) 695–729.https://doi.org/10.1090/S0002-9947-1991-1018570-0Google Scholar
  7. J.-P. Aubin, I. Ekeland: Applied Nonlinear Analysis, Wiley, New York (1984).Google Scholar
  8. H. Benabdellah: Existence of solutions to the nonconvex sweeping process, J. Differ. Equations 164 (2000) 286–295.https://doi.org/10.1006/jdeq.1999.3756Google Scholar
  9. F. Bernicot, J. Venel: Differential inclusions with proximal normal cones in Banach spaces, J. Convex Analysis. 17 (2010) 451–484.https://doi.org/10.68381/jca17032Google Scholar
  10. F. Bernicot, J. Venel: Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme, J. Differ. Equations 251 (2011) 1195–1224.https://doi.org/10.1016/j.jde.2011.03.010Google Scholar
  11. M. Bounkhel, L. Thibault: On various notions of regularity of sets in nonsmooth analysis, Nonlinear Anal. 48 (2002) 223–246.https://doi.org/10.1016/S0362-546X(00)00183-8Google Scholar
  12. M. Bounkhel, L. Thibault: Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. 6 (2005) 359–374.Google Scholar
  13. H. Brezis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North Holland, Amsterdam (1973).Google Scholar
  14. B. Brogliato, L. Thibault: Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems, J. Convex Analysis 17 (2010) 961–990.https://doi.org/10.68381/jca17060Google Scholar
  15. C. Castaing: Version aléatoire de problème de rafle par un convexe variable, C. R. Acad. Sci., Paris, Sér. A 277 (1973) 1057–1059.Google Scholar
  16. C. Castaing: Equations différentielles. Rafle par un convexe aléatoire à variation continue à droite, C. R. Acad. Sci., Paris, Sér. A 282 (1976) 515–518.Google Scholar
  17. C. Castaing: Equation différentielle multivoque avec contrainte sur l’état dans les espaces de Banach, Sém. Anal. Convexe Montpellier (1978), Exposé 13.Google Scholar
  18. C. Castaing, T. X. Duc Ha, M. Valadier: Evolution equations governed by the sweeping process, Set-Valued Anal. 1 (1993) 109–139.https://doi.org/10.1007/BF01027688Google Scholar
  19. C. Castaing, M. D. P. Monteiro Marques: BV periodic solutions of an evolution problem associated with continuous moving convex sets, Set-Valued Anal. 3 (1995) 381–399.https://doi.org/10.1007/BF01026248Google Scholar
  20. C. Castaing, M. D. P. Monteiro Marques: Evolution problems associated with non-convex closed moving sets with bounded variation, Port. Math. 53 (1996) 73–87.Google Scholar
  21. C. Castaing, M. D. P. Monteiro Marques, P. Raynaud de Fitte: A Skorokhod problem governed by a closed convex moving set, J. Convex Analysis 23 (2016) 387–423.https://doi.org/10.68381/jca23015Google Scholar
  22. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  23. F. Clarke: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics 264, Springer, Berlin (2013).https://doi.org/10.1007/978-1-4471-4820-3Google Scholar
  24. G. Colombo, V. V. Goncharov: The sweeping processes without convexity, Set-Valued Anal. 7 (1999) 357–374.https://doi.org/10.1023/A:1008774529556Google Scholar
  25. G. Colombo, R. Henrion, N. D. Hoang, B. S. Mordukhovich: Discrete approximations of a controlled sweeping process, Set-Valued Var. Anal. 23 (2015) 69–86.https://doi.org/10.1007/s11228-014-0299-yGoogle Scholar
  26. G. Colombo, M. D. P. Monteiro Marques: Sweeping by a continuous prox-regular set, J. Differ. Equations 187 (2003) 46–62.https://doi.org/10.1016/S0022-0396(02)00021-9Google Scholar
  27. G. Colombo, L. Thibault: Prox-regular sets and applications, in: Handbook of Nonconvex Analysis and Applications, D. Y. Gao, D. Motreanu (eds.), International Press, Somerville (2010) 99–182.Google Scholar
  28. B. Cornet: Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. 96 (1983) 130–147.https://doi.org/10.1016/0022-247X(83)90032-XGoogle Scholar
  29. E. Delassus: Mémoire sur la théorie des liaisons finies unilatérales, Ann. Sci. Éc. Norm. Supér. 34 (1917) 95–179.https://doi.org/10.24033/asens.701Google Scholar
  30. N. Dinculeanu: Vector Measures, Pergamon, Oxford (1967).https://doi.org/10.1016/B978-1-4831-9762-3.50004-4Google Scholar
  31. J. F. Edmond, L. Thibault: Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program. 104 (2005) 347–373.https://doi.org/10.1007/s10107-005-0619-yGoogle Scholar
  32. J. F. Edmond, L. Thibault: BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differ. Equations 226 (2006) 135–179.https://doi.org/10.1016/j.jde.2005.12.005Google Scholar
  33. S. Flåm, J.-B. Hiriart-Urruty, A. Jourani: Feasibility in finite time, J. Dyn. Control Syst. 15 (2009) 537–555.https://doi.org/10.1007/s10883-009-9074-zGoogle Scholar
  34. H. Frankowska: A viability approach to the Skorohod problem, Stochastics 14 (1985) 227–244.https://doi.org/10.1080/17442508508833340Google Scholar
  35. T. Haddad, A. Jourani, L. Thibault: Reduction of sweeping process to unconstrained differential inclusion, Pac. J. Optim. 4 (2008) 493–512.Google Scholar
  36. T. Haddad, L. Thibault: Mixed semicontinuous perturbations of nonconvex sweeping processes, Math. Program. 123 (2010) 225–240.https://doi.org/10.1007/s10107-009-0315-4Google Scholar
  37. C. Henry: An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl. 41 (1973) 179–186.https://doi.org/10.1016/0022-247X(73)90192-3Google Scholar
  38. P. Krejčı: Vector hysteresis models, Eur. J. Appl. Math. 2 (1991) 281–292.https://doi.org/10.1017/S0956792500000541Google Scholar
  39. P. Krejčı: Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto International Series, Mathematical Sciences and Applications 8, Gakkōtosho, Tokyo (1997).Google Scholar
  40. P. Krejčı, V. Recupero: Comparing BV solutions of rate independent processes, J. Convex Analysis 21 (2014) 121–146.https://doi.org/10.68381/jca21006Google Scholar
  41. M. Kunze, M. D. P. Monteiro Marques: An introduction to Moreau’s sweeping process, in: Impacts in Mechanical Systems. Analysis and Modelling (Grenoble, 1999), B. Brogliato (ed.), Springer, Berlin (2000) 1–60.https://doi.org/10.1007/3-540-45501-9_1Google Scholar
  42. P. Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, Cambridge (1995).https://doi.org/10.1017/CBO9780511623813Google Scholar
  43. B. Maury, J. Venel: A mathematical framework for a crowd motion model, C. R. Acad. Sci., Paris 346 (2008) 1245–1250.https://doi.org/10.1016/j.crma.2008.10.014Google Scholar
  44. B. Maury, J. Venel: A discrete contact model for crowd motion, ESAIM, Math. Model. Numer. Anal. 45 (2011) 145–168.https://doi.org/10.1051/m2an/2010035Google Scholar
  45. M. Mazade, L. Thibault: Regularization of differential variational inequalities with locally prox-regular sets, Math. Program. 139 (2013) 243–269.https://doi.org/10.1007/s10107-013-0671-yGoogle Scholar
  46. M. D. P. Monteiro Marques: Perturbations convexes semi-continues supérieurement dans les espaces de Hilbert, Sém. Anal. Convexe Montpellier (1984), Exposé 2.Google Scholar
  47. M. D. P. Monteiro Marques: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and dry Friction, Birkhäuser, Basel (1993).Google Scholar
  48. B. S. Mordukohovich: Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften 331, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  49. J. J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. Fr. 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  50. J. J. Moreau: Sur l’évolution d’un système élastoplastique, C. R. Acad. Sci., Paris, Sér. A 273 (1971) 118–121.Google Scholar
  51. J. J. Moreau: Rafle par un convexe variable I, Sém. Anal. Convexe Montpellier (1971), Exposé 15.Google Scholar
  52. J. J. Moreau: Rafle par un convexe variable II, Sém. Anal. Convexe Montpellier (1972), Exposé 3.Google Scholar
  53. J. J. Moreau: On unilateral constraints, friction and plasticity, in: New Variational Techniques in Mathematical Physics (Bolzano, 1973), G. Capriz, G. Stampacchia (eds.), Edizioni Cremonese, Roma (1974) 173–322.Google Scholar
  54. J. J. Moreau: Multiapplication à rétraction finie, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 1 (1974) 169–203.Google Scholar
  55. J. J. Moreau: Sur les mesures différentielles des fonctions vectorielles à variation bornée, Sém. Anal. Convexe Montpellier (1975), Exposé 17.Google Scholar
  56. J. J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  57. J. J. Moreau: Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Eng. 177 (1999) 329–349.https://doi.org/10.1016/S0045-7825(98)00387-9Google Scholar
  58. J. J. Moreau: Fonctionnelles Convexes, Edizioni del Dipartamento di Ingegneria Civile dell’Università di Roma (2003).Google Scholar
  59. J. J. Moreau, M. Valadier: A chain rule involving vector functions of bounded variation, J. Funct. Anal. 74 (1987) 333–345.https://doi.org/10.1016/0022-1236(87)90029-2Google Scholar
  60. J. S. Pang, D. Stewart: Differential variational inequalities, Math. Program. 113 (2008) 345–424.https://doi.org/10.1007/s10107-006-0052-xGoogle Scholar
  61. L. Paoli: A position-based time-stepping algorithm for vibro-impact problems with a moving set of constraints, Confluentes Math. 3 (2011) 263–290.https://doi.org/10.1142/S179374421100031XGoogle Scholar
  62. L. Paoli: Existence and approximation for vibro-impact problems with a time-dependent set of constraints, Math. Comput. Simul. 118 (2015) 302–309.https://doi.org/10.1016/j.matcom.2014.11.015Google Scholar
  63. J.-P. Penot: Calculus without Derivatives, Graduate Texts in Mathematics 266, Springer, Berlin (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  64. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  65. V. Recupero: BV solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10 (2011) 269–315.https://doi.org/10.2422/2036-2145.2011.2.02Google Scholar
  66. V. Recupero: Extending vector hysteresis operators, J. Phys.: Conf. Ser. 268 (2011) 12–24.https://doi.org/10.1088/1742-6596/268/1/012024Google Scholar
  67. V. Recupero: A continuity method for sweeping processes, J. Differ. Equations 251 (2011) 2125–2142.https://doi.org/10.1016/j.jde.2011.06.018Google Scholar
  68. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  69. R. T. Rockafellar: Conjugate Duality and Optimization, BMS-NSF Regional Conference Series in Applied Mathematics 16, SIAM, Philadelphia (1974).https://doi.org/10.1137/1.9781611970524Google Scholar
  70. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer, New York (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  71. M. Sene, L. Thibault: Regularization of dynamical systems associated with prox-regular moving sets, J. Nonlinear. Convex Anal. 15 (2014) 647–663.Google Scholar
  72. L. Thibault: Sweeping process with regular and nonregular sets, J. Differ. Equations 193 (2003) 1–26.https://doi.org/10.1016/S0022-0396(03)00129-3Google Scholar
  73. L. Thibault: Regularization of nonconvex sweeping process in Hilbert space, Set-Valued Anal. 16 (2008) 319–333.https://doi.org/10.1007/s11228-008-0083-yGoogle Scholar
  74. A. A. Tolstonogov: Convex sweeping process with unbounded nonconvex perturbation, Nonlinear Anal., Theory Methods Appl. 108 (2014) 291–301.https://doi.org/10.1016/j.na.2014.06.002Google Scholar
  75. M. Valadier: Quelques problèmes d’entrainement unilatéral en dimension finie, Sém. Anal. Convexe Montpellier (1988), Exposé 8.Google Scholar
  76. M. Valadier: Lipschitz approximation of the sweeping (or Moreau) process, J. Differ. Equations 88 (1990) 248–264.https://doi.org/10.1016/0022-0396(90)90098-AGoogle Scholar
  77. M. Valadier: Rafle et viabilité, Sém. Anal. Convexe Montpellier (1992), Exposé 17.Google Scholar
  78. J. Venel: A numerical scheme for a class of sweeping processes, Numer. Math. 118 (2011) 367–400.https://doi.org/10.1007/s00211-010-0329-0Google Scholar
  79. A. Visintin: Differential Models of Hysteresis, Applied Mathematical Sciences 111, Springer, Berlin (1994).https://doi.org/10.1007/978-3-662-11557-2Google Scholar

Suggested citation

L. Thibault. “Moreau Sweeping Process with Bounded Truncated Retraction.” Journal of Convex Analysis 23 (2016), No. 4, 1051–1098. https://doi.org/10.68381/jca23039

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press LLC.