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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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., https://doi.org/10.1007/s10957-018-1446-7

  4. Adly, S., Hantoute, A., Théra, M.: Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions. Nonlinear Anal. 75(3), 985–1008 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Artstein, Z.: Extensions of Lipschitz selections and an application to differential inclusion. Nonlinear Anal. 16(7-8), 701–704 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aubin, J.P., Frankowska, H.: Set-valued Analysis. Birkhäuser Boston, Inc., Boston (2009)

    Book  MATH  Google Scholar 

  9. Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York (2010)

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brézis, H.: Operateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert. Amsterdam, North-Holland (1973)

    MATH  Google Scholar 

  12. Cârja, O., Motreanu, D.: Characterization of Lyapunov pairs in the nonlinear case and applications. Nonlinear Anal. 70(1), 352–363 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  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)

  16. Clarke, F.H., Ledyaev, Y., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Donchev, T.: Functional differential inclusions with monotone right-hand side. Nonlinear Anal. 16, 543–552 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Donchev, T.: Properties of the reachable set of control systems. Syst. Control Lett. 46, 379–386 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  24. Donchev, T., Ríos, V., Wolenski, P.: Strong invariance and one-sided Lipschitz multifunctions. Nonlinear Anal. 60(5), 849–862 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  26. Filippov, A.F.: Differential equations with multi-valued discontinuous right-hand side. (Russian) Dokl. Akad. Nauk SSSR 151, 65–68 (1963)

    MathSciNet  Google Scholar 

  27. Frankowska, H., Plaskacz, S.: A measurable upper semicontinuous viability theorem for tubes. Nonlinear Anal. 26(3), 565–582 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mazade, M., Thibault, L.: Differential variational inequalities with locally prox-regular sets. J. Convex Anal. 19(4), 1109–1139 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26(3), 347–374 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kocan, M., Soravia, P.: Lyapunov functions for infinite-dimensional systems. J. Funct. Anal. 192(2), 342–363 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pazy, A.: The Lyapunov method for semigroups of nonlinear contractions in Banach space. J. Anal. Math. 40(1), 239–262 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  32. Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11), 5231–5249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  34. Sadi, S., Yarou, M.: Control problems governed by time-dependent maximal monotone operators. ESAIM Control Optim. Calc. Var. 23(2), 455–473 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. XLIM UMR-CNRS 7252, Université de Limoges, 87060, Limoges, France

    Samir Adly

  2. Center for Mathematical Modeling (CMM), Universidad de Chile, Santiago, Chile

    Abderrahim Hantoute

  3. Universidad de O’Higgins, Rancagua, Chile

    Bao Tran Nguyen

Authors
  1. Samir Adly
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  2. Abderrahim Hantoute
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  3. Bao Tran Nguyen
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Correspondence to Abderrahim Hantoute.

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Research partially supported by Conicyt grants Proyecto/Grant PIA AFB-170001, Fondecyt no. 1151003, 1190012, Conicyt-Redes no. 150040, Mathamsud 17-MATH-06, and Conicyt-Pcha/Doctorado Nacional/2014-63140104.

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Adly, S., Hantoute, A. & Nguyen, B.T. Lyapunov Stability of Differential Inclusions with Lipschitz Cusco Perturbations of Maximal Monotone Operators. Set-Valued Var. Anal 28, 345–368 (2020). https://doi.org/10.1007/s11228-019-00513-4

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