Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Moreau–Yosida Regularization of Degenerate State-Dependent Sweeping Processes

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we introduce and study degenerate state-dependent sweeping processes with nonregular moving sets (subsmooth and positively \(\alpha \)-far). Based on the Moreau–Yosida regularization, we prove the existence of solutions under the Lipschitzianity of the moving sets with respect to the truncated Hausdorff distance.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth Modeling and Simulation for Switched Circuits. Springer (2011)

  2. Adly, S., Haddad, T.: Well-posedness of nonconvex degenerate sweeping process via unconstrained evolution problems. Nonlinear Anal. Hybrid Syst. 36, 100832, 13 (2020)

  3. Adly, S., Le, B.K.: Unbounded second-order state-dependent Moreau’s sweeping processes in Hilbert spaces. J. Optim. Theory Appl. 169(2), 407–423 (2016)

  4. Aliouane, F., Azzam-Laouir, D., Castaing, C., Monteiro-Marques, M.D.P.: Second-order time and state-dependent sweeping process In Hilbert space. J. Optim. Theory Appl. 182(1), 153–188 (2019)

    Article  MathSciNet  Google Scholar 

  5. Aliprantis, C., Border, K.: Infinite Dimensional Analysis, 3rd edn. Springer (2006)

  6. Aubin, J.P., Cellina, A.: Differential Inclusions, volume 264 of Grundlehren Math. Wiss. Springer-Verlag (1984)

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. Bothe, D.: Multivalued perturbations of \(m\)-accretive differential inclusions. Israel J. Math. 108, 109–138 (1998)

    Article  MathSciNet  Google Scholar 

  9. Brogliato, B.: Nonsmooth Mechanics. Communications and Control Engineering Series, 3rd edn. Springer (2016)

  10. Brogliato, B., Tanwani, A.: Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability. SIAM Rev. 62(1), 3–129 (2020)

    Article  MathSciNet  Google Scholar 

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

  12. Deimling, K.: Multivalued Differential Equations. de Gruyter Ser, vol. 1. Nonlinear Anal. Appl. Walter de Gruyter & Co., Berlin (1992)

  13. Deutsch, F.: Best Approximation in Inner Product Spaces, volume 7 of CMS Books Math./Ouvrages Math. SMC. Springer-Verlag, New York (2001)

  14. Gidoni, P.: Rate-independent soft crawlers. Quart. J. Mech. Appl. Math. 71(4), 369–409 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Haddad, T., Jourani, A., Thibault, L.: Reduction of sweeping process to unconstrained differential inclusion. Pac. J. Optim. 4(3), 493–512 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Haddad, T., Kecis, I., Thibault, L.: Reduction of state dependent sweeping process to unconstrained differential inclusion. J. Global Optim. 62(1), 167–182 (2015)

    Article  MathSciNet  Google Scholar 

  17. Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. vol. I, volume 419 of ath. Appl. Kluwer Academic Publishers, Dordrecht (1997) (Theory)

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

    MathSciNet  MATH  Google Scholar 

  19. Jourani, A., Vilches, E.: Galerkin-like method and generalized perturbed sweeping process with nonregular sets. SIAM J. Control Optim. 55(4), 2412–2436 (2017)

    Article  MathSciNet  Google Scholar 

  20. 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  Google Scholar 

  21. Jourani, A., Vilches, E.: A differential equation approach to implicit sweeping processes. J. Differ. Equ. 266(8), 5168–5184 (2019)

    Article  MathSciNet  Google Scholar 

  22. Kecies, M., Haddad, T., Sene, M.: Degenerate sweeping process with a Lipschitz perturbation. In: Applicable Analysis, pages 1–23 (2019)

  23. Krejc̆i, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations, volume 8 of GAKUTO Internat. Ser. Math. Sci. Appl. Gakkōtosho Co., Ltd., Tokyo (1996)

  24. Kunze, M., Monteiro Marques, M.D.P.: On the discretization of degenerate sweeping processes. Portugal. Math. 55(2), 219–232 (1998)

    MathSciNet  MATH  Google Scholar 

  25. Kunze, M., Monteiro Marques, M.D.P.: Degenerate sweeping processes. In: Argoul, P., Frémond, M., Nguyen, Q.S., (eds) Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), volume 66 of Solid Mech. Appl., pp 301–307. Kluwer Acad. Publ., Dordrecht (1999)

  26. Kunze, M., Monteiro Marques, Manuel DP.: Existence of solutions for degenerate sweeping processes. J. Convex Anal. 4(1), 165–176 (1997)

    MathSciNet  MATH  Google Scholar 

  27. Kunze, M., Monteiro-Marques, M.D.P.: Yosida-Moreau regularization of sweeping processes with unbounded variation. J. Differ. Equ. 130(2), 292–306 (1996)

    Article  MathSciNet  Google Scholar 

  28. Kunze, M., Monteiro-Marques, M.D.P.: An introduction to Moreau’s sweeping process. In: Brogliato, B., (ed), Impacts in Mechanical Systems (Grenoble, 1999), volume 551 of Lecture Notes in Phys., pp 1–60. Springer, Berlin (2000)

  29. Maury, B., Venel, J.: Un Modèle de Mouvements de Foule. ESAIM Proc. 18, 143–152 (2007)

    Article  Google Scholar 

  30. Mazade, M., Thibault, L.: Regularization of differential variational inequalities with locally prox-regular sets. Math. Program. 139(1–2, Ser. B), 243–269 (2013)

  31. Monteiro-Marques, M.D.P.: Regularization and graph approximation of a discontinuous evolution problem. J. Differ. Equ. 67(2), 145–164 (1987)

    Article  MathSciNet  Google Scholar 

  32. Monteiro-Marques, M.D.P.: Differential inclusions in nonsmooth mechanical problems. Progr, vol. 9. Nonlinear Differential Equations Appl. Birkhäuser Verlag, Basel (1993)

  33. Moreau, J.J.: Rafle Par Un convexe variable. I. In: Travaux du Séminaire d’Analyse Convexe, vol. I, Exp. No. 15, pp. 1–43. U.É.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier (1971)

  34. Moreau, J.J.: Rafle Par Un convexe variable. II. In: Travaux du Séminaire d’Analyse Convexe, vol. II, Exp. No. 3, pp. 1–36. U.É.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier (1972)

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

    Article  MathSciNet  Google Scholar 

  36. Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Engrg. 177(3–4), 329–349 (1999)

    Article  MathSciNet  Google Scholar 

  37. Nacry, F., Thibault, L.: Regularization of sweeping process: old and new. Pure Appl. Funct. Anal. 4(1), 59–117 (2019)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Recupero, V.: \(BV\) Continuous sweeping processes. J. Differ. Equ. 259(8), 4253–4272 (2015)

    Article  MathSciNet  Google Scholar 

  40. Recupero, V.: Sweeping processes and rate independence. J. Convex Anal. 23(4), 921–946 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Recupero, V.: Convex valued geodesics and applications to sweeping processes with bounded retraction. J. Convex Anal. 27(2), 537–558 (2020)

    MathSciNet  MATH  Google Scholar 

  42. 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 

  43. Thibault, L.: Regularization of nonconvex sweeping process in Hilbert space. Set-Valued Anal. 16(2–3), 319–333 (2008)

    Article  MathSciNet  Google Scholar 

  44. Thibault, L.: Moreau sweeping process with bounded truncated retraction. J. Convex Anal. 23(4), 1051–1098 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Tolstonogov, A.A.: Sweeping process with unbounded nonconvex perturbation. Nonlinear Anal. 108, 291–301 (2014)

    Article  MathSciNet  Google Scholar 

  46. Zeng, S., Vilches, E.: Well-posedness of history/state-dependent implicit sweeping processes. J. Optim. Theory Appl. 186(3), 960–984 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the referees for providing several helpful suggestions. Diana Narváez was supported by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence, Fondecyt Regular N\(^{\circ }\) 1171854, and Fondecyt Regular N\(^{\circ }\) 1190012 from ANID-Chile. Emilio Vilches was supported by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence, Fondecyt de Iniciación N\(^{\circ }\) 11180098 and Fondecyt Regular 1200283 from ANID-Chile.

Author information

Authors and Affiliations

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

    Diana Narváez

  2. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Rancagua, Chile

    Diana Narváez & Emilio Vilches

Authors
  1. Diana Narváez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emilio Vilches
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Vilches.

Additional information

Communicated by Boris S. Mordukhovich.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Narváez, D., Vilches, E. Moreau–Yosida Regularization of Degenerate State-Dependent Sweeping Processes. J Optim Theory Appl 193, 910–930 (2022). https://doi.org/10.1007/s10957-022-02030-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-022-02030-1

Keywords

Mathematics Subject Classification

Search

Navigation