Lyapunov Stability of Differential Inclusions with Lipschitz Cusco Perturbations of Maximal Monotone Operators

  • Samir Adly
  • Abderrahim HantouteEmail author
  • Bao Tran Nguyen


We give new criteria for weak and strong invariant closed sets for differential inclusions in \(\mathbb {R}^{n}\), and which are simultaneously governed by Lipschitz Cusco mapping and by maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions and pairs. The resulting conditions only depend on the data of the system, while the invariant sets are assumed to be closed, and the Lyapunov pairs are assumed to be only lower semi-continuous.


Differential inclusions Cusco mappings Maximal monotone operators a-Lyapunov pairs Invariant sets 

Mathematics Subject Classification (2010)

37B25 47J35 93B05 


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The authors would like to thank the referees for providing valuable comments and suggesting new references, which allowed to improve the manuscript and the presentation of the results.


  1. 1.
    Adly, S., Goeleven, D.: A stability theory for second order non-smooth dynamical systems with application to friction problems. J. Math. Pures Appl. 83(1), 17–51 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adly, S., Hantoute, A., Nguyen, B.T.: Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operator. J. Math. Anal. Appl. 457(2), 1017–1037 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adly, S., Hantoute, A., Nguyen, B.T.: Lyapunov stability of differential inclusions involving prox-regular sets via maximal monotone operators. Will appear in J. Optim. Theory Appl.,
  4. 4.
    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
  5. 5.
    Adly, S., Hantoute, A., Théra, M.: Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain. Math. Program., Ser. B 157(2), 349–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aizicovici, S., Staicu, V.: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. Nonlinear Differ. Equ. Appl. NoDEA 14(3-4), 361–376 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Artstein, Z.: Extensions of Lipschitz selections and an application to differential inclusion. Nonlinear Anal. 16(7-8), 701–704 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Aubin, J.P., Frankowska, H.: Set-valued Analysis. Birkhäuser Boston, Inc., Boston (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York (2010)CrossRefGoogle Scholar
  10. 10.
    Barbu, V., Pavel, N.H.: Flow-invariant closed sets with respect to nonlinear semigroup flows. Nonlinear Differ. Equ. Appl. NoDEA 10(1), 57–72 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brézis, H.: Operateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert. Amsterdam, North-Holland (1973)zbMATHGoogle Scholar
  12. 12.
    Cârja, O., Motreanu, D.: Characterization of Lyapunov pairs in the nonlinear case and applications. Nonlinear Anal. 70(1), 352–363 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cârjă, O., Necula, M., Vrabie, I.I.: Necessary and sufficient conditions for viability for nonlinear evolution inclusions. Set-Valued Anal. 16(5-6), 701–731 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer, Berlin (1977)CrossRefzbMATHGoogle Scholar
  15. 15.
    Clarke, F.: Lyapunov functions and feedback in nonlinear control. In: de Queiroz, M.S. et al. (eds.) Optimal Control, Stabilization and Nonsmooth Analysis, Lecture Notes in Control and Information Sciences, vol. 301, pp 267–282. Springer, Berlin (2004)Google Scholar
  16. 16.
    Clarke, F.H., Ledyaev, Y., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)zbMATHGoogle Scholar
  17. 17.
    Clarke, F.H., Ledyaev, Y.S., Radulescu, M.L.: Approximate invariance and differential inclusions in Hilbert spaces. J. Dynam. Control Systems 3(4), 493–518 (1997)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Discrete approximations of a controlled sweeping process. Set-Valued Var. Anal. 23(1), 69–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    Crandall, M.G., Liggett, T.M.: Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93, 265–298 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Donchev, T.: Functional differential inclusions with monotone right-hand side. Nonlinear Anal. 16, 543–552 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Donchev, T.: Properties of the reachable set of control systems. Syst. Control Lett. 46, 379–386 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Donchev, T., Ríos, V., Wolenski, P.: A characterization of strong invariance for perturbed dissipative systems. In: de Queiroz, M.S. et al. (eds.) Optimal Control, Stabilization and Nonsmooth Analysis, Lecture Notes in Control and Information Sciences, vol. 301, pp 343–349. Springer, Berlin (2004)Google Scholar
  24. 24.
    Donchev, T., Ríos, V., Wolenski, P.: Strong invariance and one-sided Lipschitz multifunctions. Nonlinear Anal. 60(5), 849–862 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Donchev, T., Ríos, V., Wolenski, P.: Strong invariance for discontinuous Hilbert space differential inclusions. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 51(2), 265–279 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Filippov, A.F.: Differential equations with multi-valued discontinuous right-hand side. (Russian) Dokl. Akad. Nauk SSSR 151, 65–68 (1963)MathSciNetGoogle Scholar
  27. 27.
    Frankowska, H., Plaskacz, S.: A measurable upper semicontinuous viability theorem for tubes. Nonlinear Anal. 26(3), 565–582 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mazade, M., Thibault, L.: Differential variational inequalities with locally prox-regular sets. J. Convex Anal. 19(4), 1109–1139 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26(3), 347–374 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kocan, M., Soravia, P.: Lyapunov functions for infinite-dimensional systems. J. Funct. Anal. 192(2), 342–363 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pazy, A.: The Lyapunov method for semigroups of nonlinear contractions in Banach space. J. Anal. Math. 40(1), 239–262 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11), 5231–5249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ríos, V., Wolenski, P.: A characterization of strongly invariant systems for a class of non-Lipschitz multifunctions. In: Decision and Control Proceedings. 42nd IEEE Conference on 3, pp. 2593–2594 (2003)Google Scholar
  34. 34.
    Sadi, S., Yarou, M.: Control problems governed by time-dependent maximal monotone operators. ESAIM Control Optim. Calc. Var. 23(2), 455–473 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  • Samir Adly
    • 1
  • Abderrahim Hantoute
    • 2
    Email author
  • Bao Tran Nguyen
    • 3
  1. 1.XLIM UMR-CNRS 7252Université de LimogesLimogesFrance
  2. 2.Center for Mathematical Modeling (CMM)Universidad de ChileSantiagoChile
  3. 3.Universidad de O’HigginsRancaguaChile

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