Local existence conditions for sweeping process solutions

A. A. Tolstonogov

Sbornik. Mathematics, Tome 210 (2019) no. 9, pp. 1305-1325

Résumé

A sufficient condition for the existence of an absolutely continuous solution for a sweeping process is given by the absolute continuity, in a definite sense, of the multivalued mapping which generates the process. This property is described in terms of the Hausdorff distance between values of the multivalued mapping. However, there exist multivalued mappings for which the Hausdorff distance between those values is infinite; for instance, mappings which take hyperplanes as values. For such mappings absolute continuity cannot be described in terms of the Hausdorff distance. In this paper we study conditions which provide local absolute continuity of a multivalued mapping. By using these conditions we prove an existence theorem for an absolutely continuous solution of a sweeping process. We apply the results obtained to the study of sweeping processes with nonconvex and with convexified perturbations. For such sweeping processes we prove an existence theorem for solutions and a relaxation theorem. Bibliography: 13 titles.

Détail

Keywords: sweeping process, local Hausdorff distance, nonconvex multivalued perturbation.
Mots-clés : existence of solutions

Bibliographie

[1] J.-P. Aubin, A. Cellina, Differential inclusions. Set-valued maps and viability theory, Grundlehren Math. Wiss., 264, Springer-Verlag, Berlin, 1984, xiii+342 pp. | DOI | MR | Zbl

[2] J. J. Moreau, “Evolution problem associated with a moving convex set in a Hilbert space”, J. Differential Equations, 26:3 (1977), 347–374 | DOI | MR | Zbl

[3] A. A. Tolstonogov, “Sweeping process with unbounded nonconvex perturbation”, Nonlinear Anal., 108 (2014), 291–301 | DOI | MR | Zbl

[4] C. J. Himmelberg, “Measurable relations”, Fund. Math., 87 (1975), 53–72 | DOI | MR | Zbl

[5] H. Attouch, R. J.-B. Wets, “Quantitative stability of variational systems. I. The epigraphical distance”, Trans. Amer. Math. Soc., 328:2 (1991), 695–729 | DOI | MR | Zbl

[6] A. A. Tolstonogov, “Relaxation in nonconvex optimal control problems containing the difference of two subdifferentials”, SIAM J. Control Optim., 54:1 (2016), 175–197 | DOI | MR | Zbl

[7] I. Ekeland, R. Temam, Convex analysis and variational problems, Stud. Math. Appl., 1, North-Holland Publishing Co., Amsterdam–Oxford; American Elsevier Publishing Co., Inc., New York, 1976, ix+402 pp. | MR | MR | Zbl | Zbl

[8] A. A. Tolstonogov, “Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation”, J. Math. Anal. Appl., 447:1 (2017), 269–288 | DOI | MR | Zbl

[9] H. Brézis, Opérateurs 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. | MR | Zbl

[10] 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. | DOI | MR | Zbl

[11] F. Hiai, H. Umegaki, “Integrals, conditional expectations, and martingales of multivalued functions”, J. Multivariate Anal., 7:1 (1977), 149–182 | DOI | MR | Zbl

[12] A. A. Tolstonogov, D. A. Tolstonogov, “$L_p$-continuous extreme selectors of multifunctions with decomposable values: relaxation theorems”, Set-Valued Anal., 4:3 (1996), 237–269 | DOI | MR | Zbl

[13] A. F. Filippov, “Classical solutions of differential equations with multi-valued right-hand side”, SIAM J. Control, 5 (1967), 609–621 | DOI | MR | MR | Zbl