History-dependent operators and prox-regular sweeping processes

Florent Nacry1, Mircea Sofonea1
1Laboratoire de Modélisation Pluridisciplinaire et Simulations, Université Perpignan, 52 Avenue Paul Alduy, Perpignan, (66860), France

Tóm tắt

AbstractWe consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion under appropriate assumptions on the data. The proof is based on the arguments of monotonicity, fixed point, and prox-regularity. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated with prox-regular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.

Từ khóa


Tài liệu tham khảo

Adly, S., Haddad, T.: An implicit sweeping process approach to quasistatic evolution variational inequalities. SIAM J. Math. Anal. 50, 761–778 (2018)

Adly, S., Nacry, F., Thibault, L.: Preservation of prox-regularity of sets with applications to constrained optimization. SIAM J. Optim. 26, 448–473 (2016)

Adly, S., Nacry, F., Thibault, L.: Discontinuous sweeping process with prox-regular sets. ESAIM Control Optim. Calc. Var. 23, 1293–1329 (2017)

Adly, S., Nacry, F., Thibault, L.: Prox-regularity approach to generalized equations and image projection. ESAIM Control Optim. Calc. Var. 24, 677–708 (2018)

Adly, S., Sofonea, M.: Time-dependent inclusions and sweeping processes in contact mechanics. Z. Angew. Math. Phys. 70(2), Paper No. 39, 19 pp. (2019)

Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed sweeping process. J. Differ. Equ. 266, 1003–1050 (2019)

Castaing, C., Ibrahim, A.G., Yarou, M.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10, 1–20 (2009)

Chemetov, N., Monteiro Marques, M.D.P.: Non-convex quasi-variational differential inclusions. Set-Valued Anal. 15, 209–221 (2007)

Clarke, F.H.: Optimization and Nonsmooth Analysis, 2nd edn. Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990)

Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set-Valued Anal. 7, 357–374 (1999)

Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Handbook of Nonconvex Analysis and Applications, pp. 99–182. Int. Press, Somerville (2010)

Edmond, J.F., Thibault, L.: BV solutions of nonconvex sweeping process differential inclusions with perturbation. J. Differ. Equ. 226, 135–179 (2006)

Haddad, T.: Nonconvex differential variational inequality and state-dependent sweeping process. J. Optim. Theory Appl. 159, 386–398 (2013)

Jourani, A., Vilches, E.: Positively α-far sets and existence results for generalized perturbed sweeping processes. J. Convex Anal. 23, 775–821 (2016)

Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser, Basel (1993)

Mordukhovich, B.S.: Variational Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham (2018)

Moreau, J.J.: Rafle par un convexe variable I. In: Travaux Sém. Anal. Convexe Montpellier (1971), Exposé 15

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

Nacry, F., Sofonea, M.: A class of nonlinear inclusions and sweeping processes in solid mechanics. Acta Appl. Math. 171, 16 (2021)

Nacry, F., Sofonea, M.: A history-dependent sweeping processes in contact mechanics. J. Convex Anal. 29 (2022, to appear)

Nacry, F., Thibault, L.: Regularization of sweeping process: old and new. Pure Appl. Funct. Anal. 4, 59–117 (2019)

Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)

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

Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1997)

Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)

Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Pure and Applied Mathematics. Chapman & Hall/CRC Press, Boca Raton–London (2018)

Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193, 1–26 (2003)

Thibault, L.: Unilateral variational analysis in Banach spaces (to appear)

Venel, J.: A numerical scheme for a class of sweeping processes. Numer. Math. 118, 367–400 (2011)