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

Yıl 2025, Cilt: 54 Sayı: 3, 845 - 868, 24.06.2025

Mohamed Kecies

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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.

Yıl 2025, Cilt: 54 Sayı: 3, 845 - 868, 24.06.2025

Mohamed Kecies

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

Öz

Kaynakça

  • [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.
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalarda Dinamik Sistemler
Bölüm Matematik
Yazarlar

Mohamed Kecies 0000-0001-6928-0918

Erken Görünüm Tarihi 27 Ağustos 2024
Yayımlanma Tarihi 24 Haziran 2025
Gönderilme Tarihi 9 Mart 2024
Kabul Tarihi 6 Temmuz 2024
Yayımlandığı Sayı Yıl 2025 Cilt: 54 Sayı: 3

Kaynak Göster

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. Haziran 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, sy. 3 (Haziran 2025): 845-68. https://doi.org/10.15672/hujms.1449554.
EndNote Kecies M (01 Haziran 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, c. 54, sy. 3, ss. 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 (Haziran 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, c. 54, sy. 3, 2025, ss. 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.

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