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Viscous Approximations of Non-Convex Sweeping Processes in the Space of Regulated Functions

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Abstract

Vanishing viscosity approximations are considered here for discontinuous sweeping processes with non-convex constraints. It is shown that they are well-posed for sufficiently small viscosity parameters, and that their solutions converge pointwise, as the viscosity parameter tends to zero, to the left-continuous solution to the sweeping process in the Kurzweil integral setting. The convergence is uniform if the input is continuous.

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References

  1. Adly, S., Nacry, F., Thibault, L.: Discontinuous sweeping process with prox-regular sets. ESAIM Control Optim. Calc. Var. 23, 1293–1329 (2017)

    Article  MathSciNet  Google Scholar 

  2. Aumann, G.: Reelle Funktionen. Springer, Berlin (1954)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Bernicot, F., Venel, J.: Existence of solutions for second-order differential inclusions involving proximal normal cones. J. Math. Pures Appl. 98, 257–294 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bernicot, F., Venel, J.: Sweeping process by prox-regular sets in Riemannian Hilbert manifolds. J. Differ. Equ. 259, 4086–4121 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bourbaki, N.: Functions of a Real Variable: Elementary Theory. Springer, Berlin (2004)

    Book  Google Scholar 

  7. Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Discrete Contin. Dyn. Syst., Ser. B 18, 331–348 (2013)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower-\(C^{2}\) property. J. Convex Anal. 2, 117–144 (1995)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Colombo, G., Kozaily, C.: Existence and uniqueness of solutions for an integral perturbation of Moreau’s sweeping process. J. Convex Anal. 27(1), 227–236 (2020)

    MathSciNet  Google Scholar 

  12. Colombo, G., Monteiro Marques, M.D.P.: Sweeping by a continuous prox-regular set. J. Differ. Equ. 187, 46–62 (2003)

    Article  MathSciNet  Google Scholar 

  13. de Pinho, M.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal control involving sweeping processes. Set-Valued Var. Anal. 27, 523–548 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Fraňková, D.: Regulated functions with values in Banach space. Math. Bohem. 144, 437–456 (2019)

    Article  MathSciNet  Google Scholar 

  17. Gavioli, C., Krejčí, P.: Control and controllability of PDEs with hysteresis. Appl. Math. Optim. 84(1), 829–847 (2021)

    Article  MathSciNet  Google Scholar 

  18. Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2–15 (1977)

    Article  Google Scholar 

  19. Hartman, P.: Ordinary Differential Equations. Wiley, Baltimore (1973)

    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. Kaltenbacher, B., Krejčí, P.: Analysis of an optimization problem for a piezoelectric energy harvester. Arch. Appl. Mech. 89, 1103–1122 (2019)

    Article  Google Scholar 

  22. Krejčí, P., Liero, M.: Rate independent Kurzweil processes. Appl. Math. 54, 117–145 (2009)

    Article  MathSciNet  Google Scholar 

  23. Krejčí, P., Monteiro, G.A.: What is the best viscous approximation to a rate-independent process? J. Convex Anal. 27(3), 1015–1032 (2020)

    MathSciNet  Google Scholar 

  24. Krejčí, P., Monteiro, G.A., Recupero, V.: Non-convex sweeping processes in the space of regulated functions. Commun. Pure Appl. Anal. 21(9), 2999–3029 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Monteiro, G.A., Slavík, A., Tvrdý, M.: Kurzweil–Stieltjes Integral: Theory and Applications. World Scientific, New Jersey (2019)

    Google Scholar 

  27. Moreau, J.-J.: Rafle par un convexe variable I. Sém. d’Anal. Convexe, Montpellier 1 (1971). Exposé No. 15

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

    Article  MathSciNet  Google Scholar 

  29. Nacry, F., Thibault, L.: BV prox-regular sweeping process with bounded truncated variation. Optimization 69(7–8), 1391–1437 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Schnabel, H.C.: Zur Wohlgestelltheit des Gurson-Modells. PhD Thesis, TU München (2006)

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

    Article  MathSciNet  Google Scholar 

  33. Valadier, M.: Quelques problèmes d’entrainement unilatéral en dimension finie. Sém. Anal. Convexe, Montpellier (1988). Exposé No. 8

  34. Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the referees for their comments that helped to improve the manuscript.

Funding

The research leading to this results have been supported by GAČR Grant No. 20-14736S and by the European Regional Development Fund, Project No. CZ.02.1.01/0.0/0.0/16_019/0000778. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840.

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Authors and Affiliations

  1. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16629, Praha 6, Czech Republic

    Pavel Krejčí

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567, Praha 1, Czech Republic

    Giselle A. Monteiro

  3. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Vincenzo Recupero

Authors
  1. Pavel Krejčí
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  2. Giselle A. Monteiro
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  3. Vincenzo Recupero
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Correspondence to Giselle A. Monteiro.

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Krejčí, P., Monteiro, G.A. & Recupero, V. Viscous Approximations of Non-Convex Sweeping Processes in the Space of Regulated Functions. Set-Valued Var. Anal 31, 34 (2023). https://doi.org/10.1007/s11228-023-00695-y

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  • DOI: https://doi.org/10.1007/s11228-023-00695-y

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