Research Article
BibTex RIS Cite

An existence result for integro-differential degenerate sweeping process with convex sets

Year 2025, , 845 - 868, 24.06.2025

Mohamed Kecies

https://doi.org/10.15672/hujms.1449554

Abstract

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a single-valued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy's criterion of the approximate solutions is obtained without any new Gronwall's like inequality.

Keywords

degenerate sweeping process perturbation differential inclusion set-valued map normal cone maximal monotone operator Volterra integro-differential equation

References

  • [1] V. Acary, O. Bonnefon, and B. Brogliato, Nonsmooth modeling and simulation for switched circuits, Lect. Notes Electr. Eng. 69, Springer, 2011.
  • [2] S. Adly, A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics, Springer Briefs in Mathematics, 2018.
  • [3] S. Adly and T. Haddad, Well-posedness of nonconvex degenerate sweeping process via unconstrained evolution problems, Nonlinear Anal. Hybrid Syst. 36, 100832, 2020.
  • [4] H.H. Bauschke and P.L. Combettes,Convex analysis and monotone operator theory in Hilbert spaces, Springer, New York, 2011.
  • [5] A. Bouach, T. Haddad and L. Thibault, On the Discretization of Truncated Integro- Differential Sweeping Process and Optimal Control, J. Optim. Theory Appl. 193, 785-830, 2022.
  • [6] A. Bouach, T. Haddad and L. Thibault, Nonconvex integro-differential sweeping process with applications, SIAM J. Control Optim. 393, 2971-2995, 2022.
  • [7] M. Bounkhel and R. Al-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal. 18, 151-182, 2010.
  • [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
  • [9] B. Brogliato, Nonsmooth mechanics. Models, dynamics and control, Communications and Control Engineering Series. Springer, third edition, 2016.
  • [10] B. Brogliato, A.A. Ten Dam, L. Paoli, F. Gnot, and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, Appl. Mech. Rev. 55 (2), 107-150, 2002.
  • [11] C. Castaing, Version aléatoire du problème de rafle par un convexe variable, C.R. Acad. Sci. Paris, Sér, 277, 1057-1059, 1973.
  • [12] C. Castaing, Equation différentielle multivoque avec contrainte sur l’état dans les espaces de Banach, Sém. Anal. Conv. Montp. Expo. 13, 1978.
  • [13] F.H. Clarke, Optimization and Nonsmooth Analysis, Second edition, Classics in Applied Mathematics 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
  • [14] P. Gidoni, Rate-independent soft crawlers, Quart. J. Mech. Appl. Math, 71, 369-409, 2018.
  • [15] M. Kecies, T. Haddad, and M. Sene, Degenerate sweeping process with a lipschitz perturbation, Appl. Anal. 1-23, 2019.
  • [16] M. Kunze and M. D. P. Monteiro-Marques, Existence of solutions for degenerate sweeping processes, J. Convex Anal. 4 (1), 165-176, 1997.
  • [17] M. Kunze and M. D. P. Monteiro-Marques, On the discretization of degenerate sweeping processes, Portugal. Math. 55 (2), 219-232, 1998.
  • [18] M. Kunze and M. D. P. Monteiro Marques, Degenerate Sweeping Processes, In: Argoul P., Frémond M., Nguyen Q.S. (Eds.) Proc IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics, Paris 1997. Kluwer Acad Press, Dordrecht, 301-307, 1999.
  • [19] B. Maury and J. Venel, Un modèle de mouvements de foule, ESAIM: Proc. 18, 143- 152, 2007.
  • [20] M.D.P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d’évolution dans les espaces de Hilbert, Sém. Anal. Conv. Montp. Expo. 2, 1984.
  • [21] B.S. Mordukhovich, Variational analysis and generalized differentiation I, Grundlehren der Mathematischen Wissenschaften, 330, Berlin: Springer-Verlag, 2006.
  • [22] J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93, 273-299, 1965.
  • [23] J. J. Moreau, Sur l’évolution d’un système élastoplastique, C. R. Acad. Sci. 273, 118-121, 1971.
  • [24] J.J. Moreau, Rafle par un convexe variable I, Sém. Anal. Convexe, Montp, Expo. 15, 1971.
  • [25] J.J. Moreau, Rafle par un convexe variable II, Sém. Anal. Conv. Montp. Expo. 15, 1972.
  • [26] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, JJ. Differ. Equ. 26, 347-374, 1977.
  • [27] J. J. Moreau, Liaisons unilatérales sans frottement et chocs inélastiques, C. R. Acad. Sci. Paris, Sér. II 296, 1473-1476, 1983.
  • [28] J. J. Moreau, Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Eng 177, 329-349, 1999.
  • [29] M. Valadier, Quelques problèmes d’entrainement unilatéral en dimension finie, Sém. Anal. Conv. Montp. Expo. 8, 1988.
  • [30] R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer, Berlin, 1998.
  • [31] R. E. Showalter, Monotone operators in Banach spaces and nonlinear partial differential equations, Providence (RI): American Mathematical Society, 1997.
  • [32] D. E. Stewart, Dynamics with Inequalities: impacts and hard constraints, Society for Industrial and Applied Mathematics, 2011.

Year 2025, , 845 - 868, 24.06.2025

Mohamed Kecies

https://doi.org/10.15672/hujms.1449554

Abstract

References

  • [1] V. Acary, O. Bonnefon, and B. Brogliato, Nonsmooth modeling and simulation for switched circuits, Lect. Notes Electr. Eng. 69, Springer, 2011.
  • [2] S. Adly, A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics, Springer Briefs in Mathematics, 2018.
  • [3] S. Adly and T. Haddad, Well-posedness of nonconvex degenerate sweeping process via unconstrained evolution problems, Nonlinear Anal. Hybrid Syst. 36, 100832, 2020.
  • [4] H.H. Bauschke and P.L. Combettes,Convex analysis and monotone operator theory in Hilbert spaces, Springer, New York, 2011.
  • [5] A. Bouach, T. Haddad and L. Thibault, On the Discretization of Truncated Integro- Differential Sweeping Process and Optimal Control, J. Optim. Theory Appl. 193, 785-830, 2022.
  • [6] A. Bouach, T. Haddad and L. Thibault, Nonconvex integro-differential sweeping process with applications, SIAM J. Control Optim. 393, 2971-2995, 2022.
  • [7] M. Bounkhel and R. Al-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal. 18, 151-182, 2010.
  • [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
  • [9] B. Brogliato, Nonsmooth mechanics. Models, dynamics and control, Communications and Control Engineering Series. Springer, third edition, 2016.
  • [10] B. Brogliato, A.A. Ten Dam, L. Paoli, F. Gnot, and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, Appl. Mech. Rev. 55 (2), 107-150, 2002.
  • [11] C. Castaing, Version aléatoire du problème de rafle par un convexe variable, C.R. Acad. Sci. Paris, Sér, 277, 1057-1059, 1973.
  • [12] C. Castaing, Equation différentielle multivoque avec contrainte sur l’état dans les espaces de Banach, Sém. Anal. Conv. Montp. Expo. 13, 1978.
  • [13] F.H. Clarke, Optimization and Nonsmooth Analysis, Second edition, Classics in Applied Mathematics 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
  • [14] P. Gidoni, Rate-independent soft crawlers, Quart. J. Mech. Appl. Math, 71, 369-409, 2018.
  • [15] M. Kecies, T. Haddad, and M. Sene, Degenerate sweeping process with a lipschitz perturbation, Appl. Anal. 1-23, 2019.
  • [16] M. Kunze and M. D. P. Monteiro-Marques, Existence of solutions for degenerate sweeping processes, J. Convex Anal. 4 (1), 165-176, 1997.
  • [17] M. Kunze and M. D. P. Monteiro-Marques, On the discretization of degenerate sweeping processes, Portugal. Math. 55 (2), 219-232, 1998.
  • [18] M. Kunze and M. D. P. Monteiro Marques, Degenerate Sweeping Processes, In: Argoul P., Frémond M., Nguyen Q.S. (Eds.) Proc IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics, Paris 1997. Kluwer Acad Press, Dordrecht, 301-307, 1999.
  • [19] B. Maury and J. Venel, Un modèle de mouvements de foule, ESAIM: Proc. 18, 143- 152, 2007.
  • [20] M.D.P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d’évolution dans les espaces de Hilbert, Sém. Anal. Conv. Montp. Expo. 2, 1984.
  • [21] B.S. Mordukhovich, Variational analysis and generalized differentiation I, Grundlehren der Mathematischen Wissenschaften, 330, Berlin: Springer-Verlag, 2006.
  • [22] J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93, 273-299, 1965.
  • [23] J. J. Moreau, Sur l’évolution d’un système élastoplastique, C. R. Acad. Sci. 273, 118-121, 1971.
  • [24] J.J. Moreau, Rafle par un convexe variable I, Sém. Anal. Convexe, Montp, Expo. 15, 1971.
  • [25] J.J. Moreau, Rafle par un convexe variable II, Sém. Anal. Conv. Montp. Expo. 15, 1972.
  • [26] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, JJ. Differ. Equ. 26, 347-374, 1977.
  • [27] J. J. Moreau, Liaisons unilatérales sans frottement et chocs inélastiques, C. R. Acad. Sci. Paris, Sér. II 296, 1473-1476, 1983.
  • [28] J. J. Moreau, Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Eng 177, 329-349, 1999.
  • [29] M. Valadier, Quelques problèmes d’entrainement unilatéral en dimension finie, Sém. Anal. Conv. Montp. Expo. 8, 1988.
  • [30] R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer, Berlin, 1998.
  • [31] R. E. Showalter, Monotone operators in Banach spaces and nonlinear partial differential equations, Providence (RI): American Mathematical Society, 1997.
  • [32] D. E. Stewart, Dynamics with Inequalities: impacts and hard constraints, Society for Industrial and Applied Mathematics, 2011.
There are 32 citations in total.

Details

Primary Language English
Subjects Dynamical Systems in Applications
Journal Section Mathematics
Authors

Mohamed Kecies 0000-0001-6928-0918

Early Pub Date August 27, 2024
Publication Date June 24, 2025
Submission Date March 9, 2024
Acceptance Date July 6, 2024
Published in Issue Year 2025

Cite

APA Kecies, M. (2025). An existence result for integro-differential degenerate sweeping process with convex sets. Hacettepe Journal of Mathematics and Statistics, 54(3), 845-868. https://doi.org/10.15672/hujms.1449554
AMA Kecies M. An existence result for integro-differential degenerate sweeping process with convex sets. Hacettepe Journal of Mathematics and Statistics. June 2025;54(3):845-868. doi:10.15672/hujms.1449554
Chicago Kecies, Mohamed. “An Existence Result for Integro-Differential Degenerate Sweeping Process With Convex Sets”. Hacettepe Journal of Mathematics and Statistics 54, no. 3 (June 2025): 845-68. https://doi.org/10.15672/hujms.1449554.
EndNote Kecies M (June 1, 2025) An existence result for integro-differential degenerate sweeping process with convex sets. Hacettepe Journal of Mathematics and Statistics 54 3 845–868.
IEEE M. Kecies, “An existence result for integro-differential degenerate sweeping process with convex sets”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 3, pp. 845–868, 2025, doi: 10.15672/hujms.1449554.
ISNAD Kecies, Mohamed. “An Existence Result for Integro-Differential Degenerate Sweeping Process With Convex Sets”. Hacettepe Journal of Mathematics and Statistics 54/3 (June 2025), 845-868. https://doi.org/10.15672/hujms.1449554.
JAMA Kecies M. An existence result for integro-differential degenerate sweeping process with convex sets. Hacettepe Journal of Mathematics and Statistics. 2025;54:845–868.
MLA Kecies, Mohamed. “An Existence Result for Integro-Differential Degenerate Sweeping Process With Convex Sets”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 3, 2025, pp. 845-68, doi:10.15672/hujms.1449554.
Vancouver Kecies M. An existence result for integro-differential degenerate sweeping process with convex sets. Hacettepe Journal of Mathematics and Statistics. 2025;54(3):845-68.

For more information about the journal, please visit: https://dergipark.org.tr/en/pub/hujms