A coupled system consisting of an evolution inclusion with maximal monotone operators and a prox-regular sweeping process

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Abstract

A coupled system of an evolution inclusion and a sweeping process is considered in a separable Hilbert space. The evolution inclusion is described in terms of maximal monotone operators depending on time and the state variables of both the inclusion and the sweeping process. It involves a multivalued perturbation with nonconvex closed values. In the sweeping process the moving sets are prox-regular, and the perturbation is single-valued. The perturbations in the evolution inclusion and sweeping process are related. An existence theorem is proved for absolutely continuous solutions. As a corollary, an existence theorem if proved for an evolution inclusion with maximal monotone operators. The compactness of the set of solution for convex-valued perturbations is established for the first time. The proof is based ono the author's comparison theorem for evolution inclusions with maximal monotone operators and Fans's fixed point theorem as applied to a direct product of multivalued maps. This approach made it possible to obtain new results.

About the authors

Alexander Alexandrovich Tolstonogov

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia

Email: alexander.tolstonogov@gmail.com
Doctor of physico-mathematical sciences, Professor

References

  1. J. J. Moreau, “On unilateral constraints, friction and plasticity”, New variational techniques in mathematical physics (C.I.M.E., II Ciclo, Bressanone, 1973), C.I.M.E. Summer Sch., 63, Reprint of the 1974 original, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2011, 171–322
  2. J. J. Moreau, “An introduction to unilateral dynamics”, Novel approaches in civil engineering, Lect. Notes Appl. Comput. Mech., 14, Springer-Verlag, Berlin, 2004, 1–46
  3. J. J. Moreau, “Evolution problem associated with a moving convex set in a Hilbert space”, J. Differential Equations, 26:3 (1977), 347–374
  4. M. D. P. Monteiro Marques, Differential inclusions in nonsmooth mechanical problems. Shocks and dry friction, Progr. Nonlinear Differential Equations Appl., 9, Burkhäuser Verlag, Basel, 1993, x+179 pp.
  5. S. Adly, R. Cibulka, H. Massias, “Variational analysis and generalized equations in electronics. Stability and simulation issues”, Set-Valued Var. Anal., 21:2 (2013), 333–358
  6. B. Brogliato, L. Thibault, “Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems”, J. Convex Anal., 17:3-4 (2010), 961–990
  7. D. Goeleven, Complementarity and variational inequalities in electronics, Math. Anal. Appl., Academic Press, London, 2017, xv+192 pp.
  8. S. Adly, M. Sofonea, “Time-dependent inclusions and sweeping processes in contact mechanics”, Z. Angew. Math. Phys., 70:2 (2019), 39, 19 pp.
  9. F. Nacry, M. Sofonea, “A class of nonlinear inclusions and sweeping processes in solid mechanics”, Acta Appl. Math., 171 (2021), 16, 26 pp.
  10. Ba Khiet Le, “On a class of Lur'e dynamical systems with state-dependent set-valued feedback”, Set-Valued Var. Anal., 28:3 (2020), 537–557
  11. B. Brogliato, A. Tanwani, “Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability”, SIAM Rev., 62:1 (2020), 3–129
  12. J. Venel, “A numerical scheme for a class of sweeping processes”, Numer. Math., 118:2 (2011), 367–400
  13. F. Nacry, “Perturbed BV sweeping process involving prox-regular sets”, J. Nonlinear Convex Anal., 18:9 (2017), 1619–1651
  14. S. Adly, F. Nacry, L. Thibault, “Discontinuous sweeping process with prox-regular sets”, ESAIM Control Optim. Calc. Var., 23:4 (2017), 1293–1329
  15. F. Nacry, L. Thibault, “BV prox-regular sweeping process with bounded truncated variation”, Optimization, 69:7-8 (2020), 1391–1437
  16. A. Tolstonogov, “BV sweeping process involving prox-regular sets and a composed perturbation”, Set-Valued Var. Anal., 32:1 (2024), 2, 27 pp.
  17. H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., 5, Notas Mat., 50, North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1973, vi+183 pp.
  18. R. A. Poliquin, R. T. Rockafellar, L. Thibault, “Local differentiability of distance functions”, Trans. Amer. Math. Soc., 352:11 (2000), 5231–5249
  19. R. T. Rockafellar, R. J.-B. Wets, Variational analysis, Grundlehren Math. Wiss., 317, Springer-Verlag, Berlin, 1998, xiv+733 pp.
  20. R. D. Bourgin, Geometric aspects of convex sets with the Radon–Nikodym property, Lecture Notes in Math., 993, Springer, Berlin, 1983, xii+474 pp.
  21. H. Attouch, “Familles d'operateurs maximaux monotones et mesurabilite”, Ann. Mat. Pura Appl. (4), 120 (1979), 35–111
  22. A. A. Vladimirov, “Nonstationary dissipative evolution equations in a Hilbert space”, Nonlinear Anal., 17:6 (1991), 499–518
  23. M. Benguessoum, D. Azzam-Laouir, Ch. Castaing, “On a time and state dependent maximal monotone operator coupled with a sweeping process with perturbations”, Set-Valued Var. Anal., 29:1 (2021), 191–219
  24. D. Azzam-Laouir, M. Benguessoum, “Multi-valued perturbations to a couple of differential inclusions governed by maximal monotone operators”, Filomat, 35:13 (2021), 4369–4380
  25. K. Dib, D. Azzam-Laouir, “Existence solutions for a couple of differential inclusions involving maximal monotone operators”, Appl. Anal., 102:9 (2023), 2628–2650
  26. S. Saïdi, “Coupled problems driven by time and state-dependent maximal monotone operators”, Numer. Algebra Control Optim., 14:3 (2024), 547–580
  27. F. Nacry, J. Noel, L. Thibault, “On first and second order state-dependent sweeping processes”, Pure Appl. Funct. Anal., 6:6 (2021), 1453–1493
  28. S. Saïdi, “Second-order evolution problems by time and state-dependent maximal monotone operators and set-valued perturbations”, Int. J. Nonlinear Anal. Appl., 14:1 (2023), 699–715
  29. А. А. Толстоногов, “Теоремы сравнения для эволюционных включений с максимально монотонными операторами. $L^2$-теория”, Матем. сб., 214:6 (2023), 110–135
  30. Ky Fan, “Fixed point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad Sci. U.S.A., 38:2 (1952), 121–126
  31. H. Attouch, Variational convergence for functions and operators, Appl. Math. Ser., Pitman (Advanced Publishing Program), Boston, MA, 1984, xiv+423 pp.
  32. Shui-Hung Hou, “On property $(Q)$ and other semicontinuity properties of multifunctions”, Pacific J. Math., 103:1 (1982), 39–56
  33. Z. Denkowski, S. Migorski, N. S. Papageorgiou, An introduction to nonlinear analysis. Theory, Kluwer Acad. Publ., Boston, MA, 2003, xvi+689 pp.
  34. M. Kunze, M. D. P. Monteiro Marques, “BV solutions to evolution problems with time dependent domains”, Set-Valued Anal., 5:1 (1997), 57–72
  35. A. A. Tolstonogov, D. A. Tolstonogov, “$L_p$-continuous extreme selectors of multifunctions with decomposable values: existence theorems”, Set-Valued Anal., 4:2 (1996), 173–203
  36. A. A. Tolstonogov, “Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values”, Topology Appl., 155:8 (2008), 898–905
  37. A. A. Tolstonogov, “Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators”, J. Math. Anal. Appl., 526:1 (2023), 127197, 29 pp.
  38. T. Ważewski, “Systèmes des equations et des inegalites differentielles ordinaires aux deuxièmes membres monotones et leurs applications”, Ann. Soc. Polon. Math., 23 (1950), 112–166
  39. А. А. Толстоногов, Дифференциальные включения в банаховом пространстве, Наука, Новосибирск, 1986, 296 с.

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