Advertisement

Moreau-Yosida Regularization of State-Dependent Sweeping Processes with Nonregular Sets

  • Abderrahim JouraniEmail author
  • Emilio Vilches
Article

Abstract

The existence and the convergence (up to a subsequence) of the Moreau-Yosida regularization for the state-dependent sweeping process with nonregular (subsmooth and positively alpha-far) sets are established. Then, by a reparametrization technique, the existence of solutions for bounded variation continuous state-dependent sweeping processes with nonregular (subsmooth and positively alpha-far) sets is proved. An application to vector hysteresis is discussed, where it is shown that the Play operator with positively alpha-far sets is well defined for bounded variation continuous inputs.

Keywords

Moreau-Yosida regularization Subsmooth sets Sweeping process Positively \(\alpha \)-far sets Measure differential inclusions Play operator 

Mathematics Subject Classification

34A60 49J52 49J53 

Notes

Acknowledgements

The authors wish to thank the referees for providing several helpful suggestions. The research of the second author was supported by CONICYT-PCHA/Doctorado Nacional/2013-21130676.

References

  1. 1.
    Moreau, J.J.: Rafle par un convexe variable I, expo. 15. Sém. Anal. Conv. Mont. 1–43 (1971)Google Scholar
  2. 2.
    Moreau, J.J.: Rafle par un convexe variable II, expo. 3. Sém. Anal. Conv. Mont. 1–36 (1972)Google Scholar
  3. 3.
    Moreau, J.J.: Multiapplications à retraction finie. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1(4), 169–203 (1974)Google Scholar
  4. 4.
    Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26(3), 347–374 (1977)Google Scholar
  5. 5.
    Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3–4), 329–349 (1999)Google Scholar
  6. 6.
    Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth Modeling and Simulation for Switched Circuits. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Maury, B., Venel, J.: Un modéle de mouvement de foule. ESAIM Proc. 18, 143–152 (2007)CrossRefzbMATHGoogle Scholar
  8. 8.
    Krejc̆i, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO Int. Ser. Math. Sci. Appl., vol. 8. Gakkōtosho Co., Ltd, Tokyo (1996)Google Scholar
  9. 9.
    Bounkhel, M.: Regularity Concepts in Nonsmooth Analysis. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Jourani, A., Vilches, E.: Positively \(\alpha \)-far sets and existence results for generalized perturbed sweeping processes. J. Convex Anal. 23(3), 775–821 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kunze, M., Monteiro-Marques, M.: An introduction to Moreau’s sweeping process. In: Brogliato, B. (ed.) Impacts in Mechanical Systems (Grenoble, 1999), Lecture Notes in Phys., vol. 551, pp. 1–60. Springer, Berlin (2000)Google Scholar
  12. 12.
    Chraibi Kaadoud, M.: Étude théorique et numérique de problèmes d’évolution en présence de liaisons unilatérales et de frottement. Ph.D. thesis, USTL, Montpellier (1987)Google Scholar
  13. 13.
    Kunze, M., Monteiro Marques, M.: On parabolic quasi-variational inequalities and state-dependent sweeping processes. Topol. Methods Nonlinear Anal. 12(1), 179–191 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Haddad, T., Haddad, T.: State-dependent sweeping process with perturbation. In: Anastassiou, G.A., Duman, O. (eds.) Advances in Applied Mathematics and Approximation Theory, Springer Proc. Math. Stat., vol. 41, pp. 273–281. Springer, New York (2013)Google Scholar
  15. 15.
    Bounkhel, M., Castaing, C.: State dependent sweeping process in \(p\)-uniformly smooth and \(q\)-uniformly convex Banach spaces. Set-Valued Var. Anal. 20(2), 187–201 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chemetov, N., Monteiro-Marques, M.: Non-convex quasi-variational differential inclusions. Set-Valued Anal. 15(3), 209–221 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chemetov, N., Monteiro-Marques, M., Stefanelli, U.: Ordered non-convex quasi-variational sweeping processes. J. Convex Anal. 15(2), 201–214 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Castaing, C., Ibrahim, A.G., Yarou, M.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10(1), 1–20 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Azzam-Laouir, D., Izza, S., Thibault, L.: Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process. Set-Valued Var. Anal. 22(1), 271–283 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Haddad, T., Kecis, I., Thibault, L.: Reduction of state dependent sweeping process to unconstrained differential inclusion. J. Global Optim. 62(1), 167–182 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Noel, J.: Inclusions différentielles d’évolution associées à des ensembles sous lisses. Ph.D. thesis, Université Montpellier II (2013)Google Scholar
  22. 22.
    Noel, J., Thibault, L.: Nonconvex sweeping process with a moving set depending on the state. Vietnam J. Math. 42(4), 595–612 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Monteiro-Marques, M.: Regularization and graph approximation of a discontinuous evolution problem. J. Differ. Equ. 67(2), 145–164 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Monteiro-Marques, M.: Differential Inclusions in Nonsmooth Mechanical Problems. Prog. Nonlinear Differ. Equ. Appl., vol. 9. Birkhäuser Verlag, Basel (1993)Google Scholar
  25. 25.
    Kunze, M., Monteiro-Marques, M.: Yosida-Moreau regularization of sweeping processes with unbounded variation. J. Differ. Equ. 130(2), 292–306 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Thibault, L.: Regularization of nonconvex sweeping process in Hilbert space. Set-Valued Anal. 16(2–3), 319–333 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mazade, M., Thibault, L.: Regularization of differential variational inequalities with locally prox-regular sets. Math. Program. 139(1–2, Ser. B), 243–269 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sene, M., Thibault, L.: Regularization of dynamical systems associated with prox-regular moving sets. J. Nonlinear Convex Anal. 15(4), 647–663 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Recupero, V.: A continuity method for sweeping processes. J. Differ. Equ. 251(8), 2125–2142 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Recupero, V.: \(BV\) continuous sweeping processes. J. Differ. Equ. 259(8), 4253–4272 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Recupero, V.: Sweeping processes and rate independence. J. Convex Anal 23(4), 921–946 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory. Grad. Texts Math., vol. 178. Springer-Verlag, New York (1998)Google Scholar
  33. 33.
    Haddad, T., Jourani, A., Thibault, L.: Reduction of sweeping process to unconstrained differential inclusion. Pac. J. Optim. 4(3), 493–512 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Poliquin, R., Rockafellar, R., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231–5249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Federer, H.: Geometric Measure Theory. Grundlehren Math. Wiss., vol. 153. Springer, New York (1969)Google Scholar
  36. 36.
    Recupero, V.: \(BV\) solutions of rate independent variational inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10(2), 269–315 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Thibault, L.: Moreau sweeping process with bounded truncated retraction. J. Convex Anal. 23(4), 1051–1098 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Deimling, K.: Multivalued Differential Equations. de Gruyter Ser. Nonlinear Anal. Appl., vol. 1. Walter de Gruyter & Co., Berlin (1992)Google Scholar
  39. 39.
    Bothe, D.: Multivalued perturbations of \(m\)-accretive differential inclusions. Isr. J. Math. 108, 109–138 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gasiński, L., Papageorgiou, N.: Nonlinear Analysis. Ser. Math. Anal. Appl., vol. 9. Chapman & Hall/CRC, Boca Raton (2006)Google Scholar
  41. 41.
    Borwein, J., Fitzpatrick, S., Giles, J.: The differentiability of real functions on normed linear space using generalized subgradients. J. Math. Anal. Appl. 128(2), 512–534 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Hiriart-Urruty, J.B.: Ensembles de Tchebychev vs. ensembles convexes: l’état de la situation vu via l’analyse convexe non lisse. Ann. Sci. Math. Québec 22(1), 47–62 (1998)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Penot, J.P.: Calculus without derivatives, Grad. Texts in Math., vol. 266. Springer, New York (2013)Google Scholar
  44. 44.
    Aliprantis, C., Border, K.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  45. 45.
    Dinculeanu, N.: Vector Measures, Int. Ser. Monogr. Pure Appl. Math., vol. 95. Pergamon Press, Berlin (1967)Google Scholar
  46. 46.
    Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. Vol. I, Math. Appl., vol. 419. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  47. 47.
    Aubin, J., Cellina, A.: Differential Inclusions. Grundlehren Math. Wiss., vol. 264. Springer, Berlin (1984)Google Scholar
  48. 48.
    Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Differ. Equ. 164(2), 286–295 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Colombo, G., Goncharov, V.: The sweeping processes without convexity. Set-Valued Anal. 7(4), 357–374 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193(1), 1–26 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Edmond, J., Thibault, L.: \(BV\) solutions of nonconvex sweeping process differential inclusion with perturbation. J. Differ. Equ. 226(1), 135–179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Recupero, V.: The play operator on the rectifiable curves in a Hilbert space. Math. Methods Appl. Sci. 31(11), 1283–1295 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control Optim. 49(2), 788–807 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Krasnosel’skiĭ, M., Pokrovskiĭ, A.: Systems with Hysteresis. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  55. 55.
    Krejc̆i, P., Recupero, V.: Comparing BV solutions of rate independent processes. J. Convex Anal. 21(1), 121–146 (2014)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Mielke, A., Roubc̆ek, T.: Rate-Independent Systems, Appl. Math. Sci., vol. 193. Springer, New York (2015)Google Scholar
  57. 57.
    Bivas, M., Ribarska, N.: Projection process with definable right-hand side. SIAM J. Control Optim. 53(5), 2819–2834 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de Bourgogne Franche-ComtéDijonFrance
  2. 2.Departamento de Ingeniería MatemáticaUniversidad de ChileSantiagoChile

Personalised recommendations