Abstract

We study the well-posedness of integro-differential sweeping processes of Volterra type. Using new enhanced versions of Gronwall's inequality, a reparametrization technique, and a fixed point argument for history-dependent operators, we obtain the existence of solutions and provide a fully continuous dependence result for the integro-differential sweeping process. The paper ends with an application to projected dynamical systems.

Author information

Contact details are reproduced from the original publication and may be historical.

Emilio Vilches

(1) Instituto de Ciencias de la Ingeniería, Universidad de O'Higgins, Rancagua, Chile
(2) Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile

emilio.vilches@uoh.cl

Keywords

  • Sweeping process
  • prox-regular sets
  • normal cone
  • history-dependent operator

Mathematics Subject Classification

34A60, 49J52, 49J53

References

  1. V. Acary, O. Bonnefon, B. Brogliato: Nonsmooth Modeling and Simulation for Switched Circuits, Springer, Berlin (2011).https://doi.org/10.1007/978-90-481-9681-4Google Scholar
  2. J.-P. Aubin, A. M. Bayen, P. Saint-Pierre: Viability Theory, 2nd ed., Springer, Berlin (2011).https://doi.org/10.1007/978-3-642-16684-6Google Scholar
  3. J.-P. Aubin, H. Frankowska: Set-valued analysis, viability theory and partial differential inclusions, in: World Congress of Nonlinear Analysts ’92, Vol. I–IV, Tampa 1992, de Gruyter, Berlin (1996) 1039–1058.https://doi.org/10.1515/9783110883237.1039Google Scholar
  4. A. Bouach, T. Haddad, L. Thibault: Nonconvex integro-differential sweeping process with applications, SIAM J. Control Optim. 60/5 (2022) 2971–2995.https://doi.org/10.1137/21M1397635Google Scholar
  5. A. Bouach, T. Haddad, L. Thibault: On the discretization of truncated integro-differential sweeping process and optimal control, J. Optim. Theory Appl. 193/1-3 (2022) 785–830.https://doi.org/10.1007/s10957-021-01991-zGoogle Scholar
  6. Y. Brenier, W. Gangbo, G. Savaré, M. Westdickenberg: Sticky particle dynamics with interactions, J. Math. Pures Appl. (9) 99/5 (2013) 577–617.https://doi.org/10.1016/j.matpur.2012.09.013Google Scholar
  7. H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies No. 5, North-Holland Publishing Co. and American Elsevier Publishing Co., New York (1973).Google Scholar
  8. B. Brogliato: Nonsmooth Mechanics. Models, Dynamics and Control, 3rd ed., Communications and Control Engineering Series, Springer, Cham (2016).https://doi.org/10.1007/978-3-319-28664-8Google Scholar
  9. B. Brogliato, A. Daniilidis, C. Lemaréchal, V. Acary: On the equivalence between complementarity systems, projected systems and differential inclusions, Systems Control Lett. 55/1 (2006) 45–51.https://doi.org/10.1016/j.sysconle.2005.04.015Google Scholar
  10. C. Castaing, L. Thibault: Various perturbations and relaxations of the sweeping process, J. Convex Analysis 30/2 (2023) 659–742.https://doi.org/10.68381/jca30035Google Scholar
  11. F. Clarke, Y. Ledyaev, R. Stern, P. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  12. M. G. Cojocaru, P. Daniele, A. Nagurney: Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, J. Optim. Theory Appl. 127/3 (2005) 549–563.https://doi.org/10.1007/s10957-005-7502-0Google Scholar
  13. G. Colombo, C. Kozaily: Existence and uniqueness of solutions for an integral perturbation of Moreau’s sweeping process, J. Convex Analysis 27/1 (2020) 229–238.https://doi.org/10.68381/jca27013Google Scholar
  14. G. Colombo, L. Thibault: Prox-regular sets and applications, in: Handbook of Nonconvex Analysis and Applications, International Press, Somerville (2010) 99–182.Google Scholar
  15. B. Cornet: Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. 96/1 (1983) 130–147.https://doi.org/10.1016/0022-247X(83)90032-XGoogle Scholar
  16. J. F. Edmond, L. Thibault: Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program. Ser. B 104/2-3 (2005) 347–373.https://doi.org/10.1007/s10107-005-0619-yGoogle Scholar
  17. H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1Google Scholar
  18. A. Jourani, E. Vilches: Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets, J. Optim. Theory Appl. 173/1 (2017) 91–116.https://doi.org/10.1007/s10957-017-1083-6Google Scholar
  19. B. Maury, J. Venel: Un modèle de mouvement de foule, ESAIM Proc. 18 (2007) 143–152.https://doi.org/10.1051/proc:071812Google Scholar
  20. J. J. Moreau: Rafle par un convexe variable I, Exp. 15, Séminaire d’Analyse Convexe, Montpellier (1971) 1–43.Google Scholar
  21. J. J. Moreau: Rafle par un convexe variable II, Exp. 3, Séminaire d’Analyse Convexe, Montpellier (1972) 1–36.Google Scholar
  22. M. Nagumo: Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan (3) 24 (1942) 551–559.Google Scholar
  23. D. Narváez, E. Vilches: Moreau-Yosida regularization of degenerate state-dependent sweeping processes, J. Optim. Theory Appl. 193/1-3 (2022) 910–930.https://doi.org/10.1007/s10957-022-02030-1Google Scholar
  24. J.-P. Penot: Calculus Without Derivatives, Graduate Texts in Mathematics 266, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  25. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352/11 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  26. M. Sofonea, S. Migórski: Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton (2018).https://doi.org/10.1201/9781315153261Google Scholar
  27. L. Thibault: Unilateral Variational Analysis in Banach Spaces. II: Special Classes of Functions and Sets, World Scientific, Singapore (2023).https://doi.org/10.1142/12797Google Scholar

Suggested citation

E. Vilches. “Well-Posedness for Integro-Differential Sweeping Processes of Volterra Type.” Journal of Convex Analysis 31 (2024), No. 4, 1273–1288. https://doi.org/10.68381/jca31063

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press.