Abstract
This paper studies, for a differential variational inequality involving a locally prox-regular set, a regularization process with a family of classical differential equations whose solutions converge to the solution of the differential variational inequality. The concept of local prox-regularity will be termed in a quantified way, as \((r,\alpha )\)-prox-regularity.
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Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Differe. Equ. 164, 286–295 (2000)
Borwein, J.M., Giles, J.R.: The proximal normal formula in Banach space. Trans. Am. Math. Soc. 302, 371–381 (1987)
Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox regularity in Hilbert space. J. Nonlinear Convex Anal. 6, 359–374 (2005)
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, no. 5, North-Holland mathematics studies, Elsevier, (1973)
Castaing, C.: Equation differentielle multivoque avec contrainte sur l’état, Sem. Anal. Convexe Montpellier, Exposé no. 13, (1978)
Castaing, C., Duc Ha, T.X., Valadier, M.: Evolution equations governed by the sweeping process. Set Valued Anal. 1, 109–139 (1993)
Castaing, C., Montero Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets. Portugal Math. 53(1), 73–87 (1996)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer, New York (1998)
Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower-\(C^{2}\) property. J. Convex Anal. 2, 117–144 (1995)
Colombo, G., Goncharov, V.V.: The sweeping process without convexity. Set-Valued Anal. 7, 357–374 (1999)
Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis, International Press (2010)
Cornet, B.: Existence of slow solutions for a class of differential inclusions. J. Math Anal. Appl. 96, 130–147 (1983)
Correa, R., Jofré, A., Thibault, L.: Characterization of lower semicontinuous convex functions. Proc. Am. Math. Soc. 116, 67–72 (1992)
Edelstein, M.: On nearest points of sets in uniformly convex Banach spaces. J. Lond. Math. Soc. 43, 375–377 (1968)
Edmond, J.F., Thibault, L.: Relaxation of an optimal control problem involving a perturbed sweeping process. Math Program Ser. B 104, 347–373 (2005)
Edmond, J.F., Thibault, L.: BV solutions of nonconvex sweeping process differential inclusion with perturbation. J. Differ. Equ. 226, 135–179 (2006)
Haddad, T., Jourani, A., Thibault, L.: Reduction of sweeping process to unconstrained differential inclusion. Pac. J. Optim. 4(3), 493–512 (2008)
Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41, 179–186 (1973)
Lau, K.S.: Almost Chebychev subsets in reflexive Banach spaces. Indiana Univ. Math. J 27, 791–795 (1978)
Marcellin, S., Thibault, L.: Evolution problems associated with primal lower nice functions. J. Convex Anal. 2, 385–421 (2006)
Maury, B., Venel, J.: Un modèle de mouvement de foule. ESAIM Proc. 18, 143–152 (2007)
Maury, B., Venel, J.: Handling of Contacts in Crowd Motion Simulations, Traffic and Granular Flow ’07, 171–180, Springer, (2009)
Maury, B., Venel, J.: A mathematical framework for a crowd motion model. C. R. Acad. Sci. Paris Ser. I 346, 1245–1250 (2008)
Mazade, M., Thibault, L.: Differential variational inequalities with locally prox regular sets. J. Convex Anal. 19, 1109–1139 (2012)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I and II, Grundlehren der Mathematischen Wissenschaften [A series of comprehensive studies in mathematics] 330. Springer, Berlin (2005)
Moreau, J.J.: Rafle par un convexe variable I, Sém. Anal. Convexe Montpellier, Exposé 15, (1971)
Moreau, J.J.: Rafle par un convexe variable II, Sém. Anal. Convexe Montpellier, Exposé 3, (1972)
Moreau, J.J.: Multi-applications à rétraction finie. Ann. Scuola Norm. Pisa 1, 169–203 (1974)
Moreau, J.J.: Evolution problems associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26, 347–374 (1977)
Poliquin, R.A.: Integration of subdifferentials of nonconvex functions. Nonlinear Anal. 17(4), 385–398 (1991)
Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231–5249 (2000)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317. Springer, Berlin (1998)
Serea, O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)
Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193, 1–23 (2003)
Thibault, L.: Regularization of nonconvex sweeping process in Hilbert space. Set-valued Anal. 16, 319–333 (2008)
Valadier, M.: Quelques problèmes d’entraînement unilatéral en dimension finie, Sem. Anal. Convexe 18 Montpellier, Exposé no. 8, (1988)
Venel, J.: Integrating strategies in numerical modelling of crowd motion. In: Pedestrian and Evacuation Dynamics, Springer, pp. 641–646 (2010)
Yoshida, K.: Functional Analysis, Classics in Mathematics. Springer, Berlin (1980)
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We thank both referees for their careful reading which allows us to improve the presentation of the paper.
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Mazade, M., Thibault, L. Regularization of differential variational inequalities with locally prox-regular sets. Math. Program. 139, 243–269 (2013). https://doi.org/10.1007/s10107-013-0671-y
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DOI: https://doi.org/10.1007/s10107-013-0671-y
Keywords
- Differential inclusions
- Differential variational inequalities
- Regularization
- Proximal normal cone
- Locally prox-regular set
- Sweeping process
- Perturbation