Journal of Convex Analysis 27 (2020), No. 2, 673--695
Copyright Heldermann Verlag 2020
Compactness of BV Solutions of a Convex Sweeping Process of Measurable Differential Inclusion
Alexander A. Tolstonogov
Matrosov Institute for System Dynamics and Control Theory, Russian Academy of Sciences, Irkutsk 664033, Russia
aatol@icc.ru
A measurable sweeping process with a multivalued perturbation is considered in a separable Hilbert space. The retraction of the sweeping process is a function of bounded variation whose differential measure is absolutely continuous with respect to a positive Radon measure. The values of the multivalued perturbation are convex compact sets. By a solution of the sweeping process we mean a right continuous function of bounded variation whose differential measure is absolutely continuous with respect to some positive Radon measure and the density of the differential measure with respect to the Radon measure satisfies the corresponding differential inclusion. In order to study the existence of solutions and properties of the solution set, we introduce the topology of uniform convergence on the space of right continuous functions of bounded variation and we investigate the compactness of sets in this space. Under the most general assumptions similar to those used to deal with absolutely continuous solutions, we prove the existence of solutions and the compactness of the solution set.
Keywords: Function of bounded variation, Radon measure, differential measure, density of a measure, BV solutions, existence theorem.
MSC: 28B20, 49J53.
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