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Mat. Sb., 2019, Volume 210, Number 9, Pages 107–128 (Mi msb9122)  

Local existence conditions for sweeping process solutions

A. A. Tolstonogov

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

Abstract: 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.

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

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00026-а
This research was supported by the Russian Foundation for Basic Research (grant no. 18-01-00026-а).


DOI: https://doi.org/10.4213/sm9122

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English version:
Sbornik: Mathematics, 2019, 210:9, 1305–1325

Bibliographic databases:

UDC: 517.911.5
MSC: Primary 34G25; Secondary 49J53
Received: 10.04.2018 and 06.12.2018

Citation: A. A. Tolstonogov, “Local existence conditions for sweeping process solutions”, Mat. Sb., 210:9 (2019), 107–128; Sb. Math., 210:9 (2019), 1305–1325

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