Journal of Convex Analysis 27 (2020), No. 2, 645--672
Copyright Heldermann Verlag 2020
Existence and Relaxation of BV Solutions for a Sweeping Process with a Nonconvex-Valued Perturbation
Sergey A. Timoshin
Fujian Province University, Key Lab. of Computational Science, School of Math. Sciences, Huaqiao University, Quanzhou 362021, P. R. China
and: Matrosov Institute for System Dynamics and Control Theory, Russian Academy of Sciences, 664033 Irkutsk, Russia
Alexander A. Tolstonogov
Matrosov Institute for System Dynamics and Control Theory, Russian Academy of Sciences, Irkutsk 664033, Russia
We study a measurable sweeping process with a multivalued perturbation in a separable Hilbert space. The values of the perturbation are closed, not necessarily convex sets. The retraction of the sweeping process is bounded by a positive Radon measure. A solution of the sweeping process is a pair consisting of a right continuous function of bounded variation whose differential measure is absolutely continuous with respect to some positive Radon measure and an integrable selector of the perturbation considered along this function. The density of the differential measure with respect to the Radon measure above satisfies the corresponding inclusion.
Along with the original perturbation, we consider the perturbation with the convexified values. We prove theorems on the existence and relaxation of solutions. The latter means the density of the solution set of the sweeping process with the original perturbation in the solution set of the sweeping process with the convexified perturbation. These solution sets are considered as subsets of the Cartesian product of the space of right continuous functions of bounded variation and the space of integrable functions. These spaces are endowed with the topology of uniform convergence on an interval and the weak topology, respectively.
Keywords: Function of bounded variation, Radon measure, differential measure, density of a measure, BV solutions, existence theorem, relaxation theorem.
MSC: 28B20, 49J45, 49J53.
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