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Existence and Lyapunov Pairs for the Perturbed Sweeping Process Governed by a Fixed Set

  • Emilio Vilches
Article
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Abstract

The aim of this paper is to prove existence results for a class of sweeping processes in Hilbert spaces by using the catching-up algorithm. These processes are governed by ball-compact non autonomous sets. Moreover, a full characterization of nonsmooth Lyapunov pairs is obtained under very general hypotheses. We also provide a criterion for weak invariance. Some applications to hysteresis and crowd motion are given.

Keywords

Sweeping process Lyapunov pair Differential inclusions Invariance Normal cone 

Mathematics Subject Classification (2010)

34A60 49J52 34G25 49J53 93D30 

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Notes

Acknowledgements

The author wishes to express his deep gratitude to Prof. Abderrahim Jourani for his constant encouragement. Moreover, the author wishes to thank the referees for their helpful comments and suggestions which substantially improved the paper.

References

  1. 1.
    Adly, S., Hantoute, A., Théra, M.: Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions. Nonlinear Anal. 75(3), 985–1008 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adly, S., Hantoute, A., Théra, M.: Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain. Math. Program. 157 (2), 349–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aizicovici, S., Staicu, V.: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. NoDEA Nonlinear Diff. Equat. Appl. 14(3), 361–376 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aubin, J.P.: Viability theory. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston (1991)Google Scholar
  5. 5.
    Aubin, J.P., Cellina, A.: Differential Inclusions, Grundlehren Mathematics Wissenschaften, vol. 264. Springer-Verlag, Berlin (1984)Google Scholar
  6. 6.
    Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Diff. Equat. 164(2), 286–295 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernicot, F., Venel, J.: Convergence order of a numerical scheme for sweeping process. SIAM J. Control Optim. 51(4), 3075–3092 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bounkhel, M.: Regularity Concepts in Nonsmooth Analysis. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48(2), 223–246 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6(2), 359–374 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York (1973)Google Scholar
  12. 12.
    Brogliato, B., Daniilidis, A., Lemaréchal, C., Acary, V.: On the equivalence between complementarity systems, projected systems and differential inclusions. Syst. Control Lett. 55(1), 45–51 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cȧrjȧ, O., Monteiro-Marques, M.D.P.: Weak tangency, weak invariance, and Carathéodory mappings. J. Dyn. Control Syst. 8(4), 445–461 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Castaing, C., Duc Ha, T.X., Valadier, M.: Evolution equations governed by sweeping process. Set-Valued Anal. 1, 109–139 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Castaing, C., Monteiro-Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets. Port. Math. 53(1), 73–87 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley Intersciences, New York (1983)zbMATHGoogle Scholar
  17. 17.
    Clarke, F.: Lyapunov functions and feedback in nonlinear control. In: de Queiroz, M., Malisoff, M., Wolenski, P. (eds.) Optimal Control, Stabilization and Nonsmooth Analysis. Springer, Berlin Heidelberg (2004)Google Scholar
  18. 18.
    Clarke, F.: Nonsmooth analysis in systems and control theory. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science. Springer, New York (2009)Google Scholar
  19. 19.
    Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory, Grad Texts in Mathematics, vol. 178. Springer-Verlag, New York (1998)Google Scholar
  20. 20.
    Cojocaru, M.G., Daniele, P., Nagurney, A.: Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications. J. Optim. Theory Appl. 127(3), 549–563 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Colombo, G., Goncharov, V.V.: The sweeping process without convexity. Set-Valued Anal. 7(4), 357–374 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Colombo, G., Palladino, M.: The minimum time function for the controlled moreau’s sweeping process. SIAM J. Control Optim. 54(4), 2036–2062 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cornet, B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96(1), 130–147 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Edmond, J.F., Thibault, L.: BV solutions of nonconvex sweeping process differential inclusion with perturbation. J. Diff. Equat. 226, 135–179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Frankowska, H., Plaskacz, S.: A measurable upper semicontinuous viability theorem for tubes. Nonlinear Anal. Theory Meth. Appl. 26(3), 565–582 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    García-Falset, J., Muñíz Pérez, O.: Projected dynamical systems on Hilbert space. J. Nonlinear Convex Anal. 15(2), 325–344 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control Optim. 49(2), 788–807 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Haddad, T., Jourani, A., Thibault, L.: Reduction of sweeping process to unconstrained differential inclusion. Pac. J. Optim. 4, 493–512 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Hantoute, A., Mazade, M.: Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets. Positivity 21(1), 423–448 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Henry, C.: Differential equations with discontinuous right-hand side for planning procedures. J. Econom. Theory 4(3), 545–551 (1972)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41(1), 179–186 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. Vol. I, Mathematics and Applications, vol. 419. Kluwer Academic Publishers, Dordrecht (1997)CrossRefGoogle Scholar
  33. 33.
    Jourani, A., Vilches, E.: Galerkin-like method for generalized perturbed sweeping process with nonregular sets. SIAM J. Control Optim. 55(4), 2412–2436 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Jourani, A., Vilches, E.: Positively α-far sets and existence results for generalized perturbed sweeping processes. J. Convex Anal. 23(3), 775–821 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Jourani, A., Vilches, E.: Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets. J. Optim. Theory Appl. 173(1), 91–116 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Le, B.K.: On properties of differential inclusions with prox-regular sets. Pac. J. Optim. 13(1), 17–27 (2017)MathSciNetGoogle Scholar
  37. 37.
    Maury, B., Venel, J.: Un modèle de mouvement de foule. ESAIM Proc. 18, 143–152 (2007)CrossRefzbMATHGoogle Scholar
  38. 38.
    Mazade, M., Thibault, L.: Differential variational inequalities with locally prox-regular sets. J. Convex Anal. 19(4), 1109–1139 (2012)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mazade, M., Thibault, L.: Regularization of differential variational inequalities with locally prox-regular sets. Math. Program. 139(1), 243–269 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Moreau, J.J.: Rafle par un convexe variable I, expo. 15. Sém, Anal. Conv. Mont., pp. 1–43 (1971)Google Scholar
  41. 41.
    Moreau, J.J.: Rafle par un convexe variable II, expo. 3. Sém, Anal. Conv. Mont., pp. 1–36 (1972)Google Scholar
  42. 42.
    Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Diff. Equat. 26(3), 347–374 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Noel, J.: Inclusions différentielles d’évolution associées à des ensembles sous lisses. Ph.D. thesis, Université Montpellier II (2013)Google Scholar
  44. 44.
    Noel, J., Thibault, L.: Nonconvex sweeping process with a moving set depending on the state. Vietnam J. Math. 42(4), 595–612 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11), 5231–5249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Recupero, V.: The play operator on the rectifiable curves in a Hilbert space. Math. Methods Appl. Sci. 31(11), 1283–1295 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sene, M., Thibault, L.: Regularization of dynamical systems associated with prox-regular moving sets. J. Nonlinear Convex Anal. 15(4), 647–663 (2014)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Serea, O.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Thibault, L.: Sweeping process with regular and nonregular sets. J. Diff. Equat. 193(1), 1–26 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Thibault, L., Zakaryan, T.: Convergence of subdifferentials and normal cones in locally uniformly convex Banach space. Nonlinear Anal. 98, 110–134 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Venel, J.: A numerical scheme for a class of sweeping processes. Numer. Math. 118(2), 367–400 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de Bourgogne Franche-ComtéDijonFrance
  2. 2.Departamento de Ingeniería MatemáticaUniversidad de ChileSantiagoChile
  3. 3.Instituto de Ciencias de la EducaciónUniversidad de O’HigginsRancaguaChile

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