Skip to main content
SpringerLink
Log in

Mixed Semicontinuous Perturbation of Nonconvex State-Dependent Sweeping Process

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

In this paper we prove a result on the existence of solutions of a first-order differential inclusion governed by a class of nonconvex sweeping process. The moving set involved in the process is prox-regular and depends both on the time and on the state.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  4. Castaing, C., Salvadori, A., Thibault, L.: Functional evolution equations governed by nonconvex sweeping process. J. Nonlinear Convex Anal. 2, 217–241 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Lectures Notes in Math., vol. 580. Springer-Verlag, Berlin (1977)

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley and Sons (1983)

  8. Clarke, F.H., Stern, R.L., Wolenski, P.R.: Proximal smoothness and the lower C 2 property. J. Convex Anal. 2, 117–144 (1995)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Colombo, G.,Thibault, L.: Prox-regular sets and applications. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis. International Press (2010)

  11. Diestel, J., Uhl, J.J.: Vector measures. Mathematical Surveys and Monographs, No. 5. AMS (1977)

  12. Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  13. Haddad, T., Jourani, A., Thibault, L.: Reduction of sweeping process to unconstrained differential inclusion. Pac. J. Optim. 4, 493–512 (2008)

    MATH  MathSciNet  Google Scholar 

  14. Haddad, T., Thibault, L.: Mixed semicontinuous perturbation of nonconvex sweeping process. Math. Program. Ser. B 123, 225–240 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Kunze, M., Monteiro Marques, M.D.P.: An introduction to Moreau’s sweeping process. In: Brogliato, B. (ed.) Impacts in Mechanical Systems. Lecture Notes in Physics, vol. 551. Springer-Verlag, Berlin (2000)

  17. Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems- Shocks and Dry Friction. Birkhauser, Basel-Boston-Berlin (1995)

    Google Scholar 

  18. Moreau, J.J.: Rafle par un convexe variable I. Séminaire Analyse Convexe Montpellier, Exposé 15 (1971)

  19. Moreau, J.J.: Rafle par un convexe variable II. Séminaire Analyse Convexe Montpellier, Exposé 3 (1972)

  20. Moreau, J.J.: On unilateral constraints, friction and plasticity. In: New Variational Techniques in Mathematical Physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973), pp. 171–322. Edizioni Cremonese, Rome (1974)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Thibault, L.: Regularisation of nonconvex sweeping process in Hilbert space. Set-Valued Anal. 16, 319–333 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tolstonogov, A.A.: Solutions of a differential inclusion with unbouded right-hand side. (Russian) Sibirsk. Mat. Zh. 29, 212–225, 241 (1998); Transl. Seberian Math. J. 29, 857–868 (1988)

  27. Valadier, M.: Quelques problèmes d’entrainement unilatéral en dimension finie. Séminaire Analyse Convexe Montpellier, Exposé 8 (1988)

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Pures et Appliquées, Université de Jijel, Jijel, Algeria

    Dalila Azzam-Laouir & Sabrina Izza

  2. Département de Mathématiques, Université Montpellier II, 34095, Montpellier, Cedex 5, France

    Lionel Thibault

Authors
  1. Dalila Azzam-Laouir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sabrina Izza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lionel Thibault
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dalila Azzam-Laouir.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Azzam-Laouir, D., Izza, S. & Thibault, L. Mixed Semicontinuous Perturbation of Nonconvex State-Dependent Sweeping Process. Set-Valued Var. Anal 22, 271–283 (2014). https://doi.org/10.1007/s11228-013-0248-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-013-0248-1

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation