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Nonconvex Sweeping Process with a Moving Set Depending on the State

Vietnam Journal of Mathematics volume 42, pages 595–612 (2014)Cite this article

Abstract

Recently, a great advance has been made in the study of sweeping process variational inequalities with the papers (Chemetov, Monteiro Marques Set-Valued Anal. 15, 209–221, 2007 Castaing et al. J. Nonlinear Convex Anal. 10, 1–20, 2009 Haddad J. Optim. Theory Appl. 159, 386–398, 2013 Azzam-Laouir et al. Set-Valued Var. Anal. 22, 271–283, 2014 where, for a prox-regular moving set depending on both the time and the state, several existence results are provided. Those authors also studied the case where such a differential inclusion is perturbed by multimapping. The present paper establishes the existence of solutions for such perturbed differential inclusions in some context not considered in the previous papers.

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References

  1. Azzam-Laouir, D., Izza, S., Thibault, L.: Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process. Set-Valued Var. Anal. 22, 271–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balashov, M.V., Ivanov, G.E.: Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images. Math. Notes 80, 461–467 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Differ. Equ. 164, 286–295 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6, 359–374 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Castaing, C., Duc Ha, T.X., Valadier, M.: Evolution equations governed by the sweeping process. Set-Valued Anal. 1, 109–139 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castaing, C., Ibrahim, A.G., Yarou, M.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10, 1–20 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Castaing, C., Monteiro Marques, M.D.P.: Perturbations convexes semi-continues supérieurement de problèmes d’évolution dans les espaces de Hilbert. Sém. Anal. Convexe Montp. 14, Exp. 2 (1984)

  9. Castaing, C., Monteiro Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets with bounded variation. Port. Math. 53, 73–87 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Chemetov, N., Monteiro Marques, M.D.P.: Non-convex quasi-variational differential inclusions. Set-Valued Anal. 15, 209–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  12. Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set-Valued Anal. 7, 357–374 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cornet, B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96, 130–147 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Edmond, J.F., Thibault, L.: Relaxation of an optimal control problem involving a perturbed sweeping process. Math. Program. Ser. B 104, 347–373 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Edmond, J.F., Thibault, L.: BV solutions of nonconvex sweeping process differential inclusion with perturbation. J. Differ. Equ. 226, 135–179 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gamal, A.: Perturbation semi-continues supérieurement de certaines équations d’évolution. Sém. Anal. Convex Montp., Exp. 14 (1981)

  17. Haddad, T.: Nonconvex Differential variational inequality and state dependent sweeping process. J. Optim. Theory Appl. 159, 386–398 (2013)

  18. Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41, 179–186 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Idzik, A.: Almost fixed points theorems. Proc. Am. Math. Soc. 104, 779–784 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. Lond. Math. Soc. 41, 175–192 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kunze, M., Monteiro Marques, M.D.P.: On parabolic quasi-variational inequalities and state-dependent sweeping processes. Topol. Methods Nonlinear Anal. 12, 179–191 (1998)

    MathSciNet  MATH  Google Scholar 

  22. Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems Shocks and Dry Friction. Birkhäuser, Basel (1993)

    Book  MATH  Google Scholar 

  23. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988)

    MATH  Google Scholar 

  24. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory (2006)

  25. Moreau, J.J.: Rafle par un convexe variable I. Sém. Anal. Convexe Montp., Exp. 15 (1971)

  26. Moreau, J.J.: Rafle par un convexe variable II. Sém. Anal. Convexe Montp., Exp. 3 (1972)

  27. Moreau, J.J.: Multi-applications à rétraction finie. Ann. Scuola Norm. Sup. Pisa 1, 169–203 (1974)

  28. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26, 347–374 (1977)

  29. Park, S.: Fixed points of a approximable or Kakutani maps in generalized convex spaces. J. Nonlinear Convex Anal. 7, 1–17 (2006)

  30. Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352, 5231–5249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  32. Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193, 1–26 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Valadier, M.: Quelques problèmes d’entrainement unilatéral en dimension finie. Sém. Anal. Convexe Montp., Exp. 8 (1988)

  34. Valadier, M.: Entrainement unilatéral, lignes de descente, fonction lipschitziennes non pathologiques. C.R. Acad. Sci. Paris Sér. I Math. 308, 241–244 (1989)

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Authors and Affiliations

  1. Département de Mathématiques CC 051, Université Montpellier 2, Place Eugène Bataillon, 34095, Montpellier, France

    Jimmy Noel & Lionel Thibault

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  1. Jimmy Noel
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  2. Lionel Thibault
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Correspondence to Lionel Thibault.

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Noel, J., Thibault, L. Nonconvex Sweeping Process with a Moving Set Depending on the State. Vietnam J. Math. 42, 595–612 (2014). https://doi.org/10.1007/s10013-014-0109-8

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  • DOI: https://doi.org/10.1007/s10013-014-0109-8

Keywords

  • Differential inclusion
  • Sweeping process
  • Normal cone
  • Prox-regular set
  • Subdifferential

Mathematics Subject Classifications (2010)

  • 34A60
  • 49J52
  • 49J24
  • 49J53