Skip to main content
SpringerLink
Log in

Existence and Lyapunov Pairs for the Perturbed Sweeping Process Governed by a Fixed Set

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

Abstract

The aim of this paper is to prove existence results for a class of sweeping processes in Hilbert spaces by using the catching-up algorithm. These processes are governed by ball-compact non autonomous sets. Moreover, a full characterization of nonsmooth Lyapunov pairs is obtained under very general hypotheses. We also provide a criterion for weak invariance. Some applications to hysteresis and crowd motion are given.

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. Adly, S., Hantoute, A., Théra, M.: Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions. Nonlinear Anal. 75(3), 985–1008 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adly, S., Hantoute, A., Théra, M.: Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain. Math. Program. 157 (2), 349–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aizicovici, S., Staicu, V.: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. NoDEA Nonlinear Diff. Equat. Appl. 14(3), 361–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aubin, J.P.: Viability theory. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston (1991)

    Google Scholar 

  5. Aubin, J.P., Cellina, A.: Differential Inclusions, Grundlehren Mathematics Wissenschaften, vol. 264. Springer-Verlag, Berlin (1984)

    Google Scholar 

  6. Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Diff. Equat. 164(2), 286–295 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bernicot, F., Venel, J.: Convergence order of a numerical scheme for sweeping process. SIAM J. Control Optim. 51(4), 3075–3092 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bounkhel, M.: Regularity Concepts in Nonsmooth Analysis. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  9. Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48(2), 223–246 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6(2), 359–374 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York (1973)

  12. Brogliato, B., Daniilidis, A., Lemaréchal, C., Acary, V.: On the equivalence between complementarity systems, projected systems and differential inclusions. Syst. Control Lett. 55(1), 45–51 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cȧrjȧ, O., Monteiro-Marques, M.D.P.: Weak tangency, weak invariance, and Carathéodory mappings. J. Dyn. Control Syst. 8(4), 445–461 (2002)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Castaing, C., Monteiro-Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets. Port. Math. 53(1), 73–87 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Clarke, F.: Optimization and Nonsmooth Analysis. Wiley Intersciences, New York (1983)

    MATH  Google Scholar 

  17. Clarke, F.: Lyapunov functions and feedback in nonlinear control. In: de Queiroz, M., Malisoff, M., Wolenski, P. (eds.) Optimal Control, Stabilization and Nonsmooth Analysis. Springer, Berlin Heidelberg (2004)

  18. Clarke, F.: Nonsmooth analysis in systems and control theory. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science. Springer, New York (2009)

  19. Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory, Grad Texts in Mathematics, vol. 178. Springer-Verlag, New York (1998)

    Google Scholar 

  20. Cojocaru, M.G., Daniele, P., Nagurney, A.: Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications. J. Optim. Theory Appl. 127(3), 549–563 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Colombo, G., Goncharov, V.V.: The sweeping process without convexity. Set-Valued Anal. 7(4), 357–374 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Colombo, G., Palladino, M.: The minimum time function for the controlled moreau’s sweeping process. SIAM J. Control Optim. 54(4), 2036–2062 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cornet, B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96(1), 130–147 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Frankowska, H., Plaskacz, S.: A measurable upper semicontinuous viability theorem for tubes. Nonlinear Anal. Theory Meth. Appl. 26(3), 565–582 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. García-Falset, J., Muñíz Pérez, O.: Projected dynamical systems on Hilbert space. J. Nonlinear Convex Anal. 15(2), 325–344 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control Optim. 49(2), 788–807 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Hantoute, A., Mazade, M.: Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets. Positivity 21(1), 423–448 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Henry, C.: Differential equations with discontinuous right-hand side for planning procedures. J. Econom. Theory 4(3), 545–551 (1972)

    Article  MathSciNet  Google Scholar 

  31. Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41(1), 179–186 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. Vol. I, Mathematics and Applications, vol. 419. Kluwer Academic Publishers, Dordrecht (1997)

    Book  Google Scholar 

  33. Jourani, A., Vilches, E.: Galerkin-like method for generalized perturbed sweeping process with nonregular sets. SIAM J. Control Optim. 55(4), 2412–2436 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Jourani, A., Vilches, E.: Positively α-far sets and existence results for generalized perturbed sweeping processes. J. Convex Anal. 23(3), 775–821 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Jourani, A., Vilches, E.: Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets. J. Optim. Theory Appl. 173(1), 91–116 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Le, B.K.: On properties of differential inclusions with prox-regular sets. Pac. J. Optim. 13(1), 17–27 (2017)

    MathSciNet  Google Scholar 

  37. Maury, B., Venel, J.: Un modèle de mouvement de foule. ESAIM Proc. 18, 143–152 (2007)

    Article  MATH  Google Scholar 

  38. Mazade, M., Thibault, L.: Differential variational inequalities with locally prox-regular sets. J. Convex Anal. 19(4), 1109–1139 (2012)

    MathSciNet  MATH  Google Scholar 

  39. Mazade, M., Thibault, L.: Regularization of differential variational inequalities with locally prox-regular sets. Math. Program. 139(1), 243–269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Moreau, J.J.: Rafle par un convexe variable I, expo. 15. Sém, Anal. Conv. Mont., pp. 1–43 (1971)

  41. Moreau, J.J.: Rafle par un convexe variable II, expo. 3. Sém, Anal. Conv. Mont., pp. 1–36 (1972)

  42. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Diff. Equat. 26(3), 347–374 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  43. Noel, J.: Inclusions différentielles d’évolution associées à des ensembles sous lisses. Ph.D. thesis, Université Montpellier II (2013)

  44. Noel, J., Thibault, L.: Nonconvex sweeping process with a moving set depending on the state. Vietnam J. Math. 42(4), 595–612 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11), 5231–5249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  46. Recupero, V.: The play operator on the rectifiable curves in a Hilbert space. Math. Methods Appl. Sci. 31(11), 1283–1295 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sene, M., Thibault, L.: Regularization of dynamical systems associated with prox-regular moving sets. J. Nonlinear Convex Anal. 15(4), 647–663 (2014)

    MathSciNet  MATH  Google Scholar 

  48. Serea, O.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. Thibault, L.: Sweeping process with regular and nonregular sets. J. Diff. Equat. 193(1), 1–26 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  50. Thibault, L., Zakaryan, T.: Convergence of subdifferentials and normal cones in locally uniformly convex Banach space. Nonlinear Anal. 98, 110–134 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Venel, J.: A numerical scheme for a class of sweeping processes. Numer. Math. 118(2), 367–400 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to express his deep gratitude to Prof. Abderrahim Jourani for his constant encouragement. Moreover, the author wishes to thank the referees for their helpful comments and suggestions which substantially improved the paper.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, Dijon, France

    Emilio Vilches

  2. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    Emilio Vilches

  3. Instituto de Ciencias de la Educación, Universidad de O’Higgins, Rancagua, Chile

    Emilio Vilches

Authors
  1. Emilio Vilches
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Vilches.

Additional information

This research was supported by CONICYT-PCHA/Doctorado Nacional/2013-21130676.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vilches, E. Existence and Lyapunov Pairs for the Perturbed Sweeping Process Governed by a Fixed Set. Set-Valued Var. Anal 27, 569–583 (2019). https://doi.org/10.1007/s11228-018-0480-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-018-0480-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation