On a Time and State Dependent Maximal Monotone Operator Coupled with a Sweeping Process with Perturbations


In this paper, we state, in separable Hilbert spaces, the existence of absolutely continuous solutions for a couple of evolution problems governed by time and state dependent maximal monotone operator and closed convex sweeping process, with perturbations.

This is a preview of subscription content, access via your institution.


  1. 1.

    Adam, L., Outrata, J.: On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete Contn. Dyn. Syst. Ser. B 19(9), 2709–2738 (2014)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Adly, S., Haddad, T., Le, B.K.: State dependent implicit sweeping process in the framework of quasistatic evolution quasi-variational inequalities. J. Optim. Theory Appl. 182, 473–493 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Attouch, H., Cabot, A., Czarnecki, M.O.: Asymptotic behavior of nonautonomous monotone and subgradient evolution equations. Trans. Amer. Math. Soc. 370, 755–790 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Azzam-Laouir, D., Belhoula, W., Castaing, C., Monteiro Marques, M.D.P.: Perturbed evolution problems with absolutely continuous variation in time and applications. J. Fixed Point Theory Appl. 21, 40 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Azzam-Laouir, D., Belhoula, W., Castaing, C., Monteiro Marques, M.D.P.: Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations & Control Theory 9(1), 219–254 (2020)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Azzam-Laouir, D., Boutana, I.: Mixed semicontinuous perturbation to an evolution problem with time-dependent maximal monotone operator. J. Nonlinear and Convex Analysis 20(1), 39–52 (2019)

    MathSciNet  Google Scholar 

  7. 7.

    Azzam-Laouir, D., Castaing, C., Monteiro Marques, M.D.P.: Perturbed evolution proplems with continuous bounded variation in time and applications. Set-valued Var. Anal. 26(3), 693–728 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces Noordhoff Int Publ Leyden (1976)

  9. 9.

    Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces Springer (2010)

  10. 10.

    Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Brezis, H.: Operateurs maximaux monotones et semi-groupes de contraction dans un espace de Hilbert. North Holland (1979)

  12. 12.

    Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Discrete Contn. Dyn. Syst. Ser. B 18(2), 331–348 (2012)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Cao, T.H., Mordukovich, B.S.: Optimal control of nonconvex perturbed sweeping process. J. Diff. Eqs. 266, 1003–1050 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Castaing, C., Monteiro Marques, M.D.P., Raynaud de Fitte, P.: A Skorohod problem governed by a closed convex moving set. Journal of Convex Analysis 23(2), 387–423 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  16. 16.

    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions Lecture Notes on Math, vol. 580. Springer Verlag, Berlin (1977)

    Google Scholar 

  17. 17.

    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. Springer-verlag New York (1998)

  18. 18.

    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Discrete approximations of a controlled sweeping process. Set-valued Var. Anal. 23, 69–86 (2015)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Diff. Eqs. 260, 3397–3447 (2016)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Colombo, G., Palladino, M.: The minimum time function for the controlled Moreau’s sweeping process. Siam. J. Control Optim. 54(4), 2036–2062 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Edmond, J.F., Thibault, L.: Relaxation and optimal control problem involving a perturbed sweeping process. Math. Program, Ser. B 104, 347–373 (2005)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Haddad, T., Noel, J., Thibault, L.: Perturbed sweeping process with a subsmooth set depending on the state. Linear and Nonlinear Analysis 2(1), 155–174 (2013)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Holte, J.M. Discrete Gronwall Lemma and applications. http://homepages.gac.edu/holte/publications/GronwallLemma.pdf

  24. 24.

    Kunze, M., Monteiro Marques, M.D.P.: BV solutions to evolution proplems with time-depentent domains. Set-Valued. Anal. 5, 57–72 (1997)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kunze, M., Monteiro Marques, M.D.P.: On parabolic quasi-variational inequalities and state dependent sweeping process. Topol. Methods Nonlinear Anal. 12(16), 179–191 (1998)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kunze, M., Monteiro Marques, M.D.P.: An Introduction to Moreau’s Sweeping Process. In: Brogliato, B (ed.) Impacts in Mechanical Systems Lecture Notes in Physics, Vol. 551, p 17. Springer-Verlag, Berlin (2000)

  27. 27.

    Le, B.K.: Well-posedeness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions. Optimization. https://doi.org/10.1080/02331934.2019.1686504 (2019)

  28. 28.

    Maticiuc, L., Rascanu, A., Slominski, L., Topolewski, M.: Cadlag Skorohod problem driven by maximal monotone operator. J. Math. Anal. Appl. 429 (2), 1305–1346 (2015)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Monteiro Marques, M.D.P.: Differential inclusions nonsmooth mechanical problems, shocks and dry friction. Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, Vol 9 (1993)

  30. 30.

    Mordukovich, B.S.: Variational analysis and optimization of sweeping processes with controlled moving sets. Revista Investigatió,n Operacional 39(3), 283–302 (2018)

    MathSciNet  Google Scholar 

  31. 31.

    Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Diff. Eqs 26, 347–374 (1977)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Rascanu, A.: Deterministic and stochastic differential equations in Hilbert spaces involving multivalued maximal monotone operators. Panamer. Math. J. 6(3), 83–119 (1996)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Selamnia, F., Azzam-Laouir, D., Monteiro Marques, M.D.P.: Evolution problems involving state-dependent maximal monotone operators. Appl Anal. https://doi.org/10.1080/00036811.2020.1738401 (2020)

  34. 34.

    Tanwani, A., Brogliato, B., Prieur, C.: Well-posedness and output regulation for implicit time-varying evolution variational inequalities, SIAM. J. Control Optim. 56, 751–781 (2018)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Thibault, L.: Submooth functions and sets. Linear and Nonlinear Analysis 4(2), 1–107 (2018)

    MathSciNet  Google Scholar 

  36. 36.

    Tolstonogov, A.A.: Control sweeping process. J. Convex Anal. 23, 1099–1123 (2016)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Tolstonogov, A.A.: BV Continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation Existence theorem. Set-valued Var. Anal, https://doi.org/10.1007/s11228-020-00535-3 (2020)

  38. 38.

    Vladimirov, A.A.: Nonstationnary dissipative evolution equation in Hilbert space. Nonlinear Anal. 17, 499–518 (1991)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Vrabie, I.L.: Compactness methods for nonlinear evolutions equations Pitman Monographs and Surveys in Pure and Applied mathematics, Longman Scientific and Technical, vol. 32. Wiley and Sons Inc., New York (1987)

    Google Scholar 

Download references


The authors thank the referees and the associated editor for the constructive remarks and suggestions that helped to improve the first version of the manuscript.

Author information



Corresponding author

Correspondence to Dalila Azzam-Laouir.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Benguessoum, M., Azzam-Laouir, D. & Castaing, C. On a Time and State Dependent Maximal Monotone Operator Coupled with a Sweeping Process with Perturbations. Set-Valued Var. Anal 29, 191–219 (2021). https://doi.org/10.1007/s11228-020-00544-2

Download citation


  • Absolutely continuous
  • Maximal monotone operator
  • Pseudo-distance
  • Sweeping process

Mathematics Subject Classification 2010

  • 34A60
  • 28C20
  • 28A25