Journal of Convex Analysis 32 (2025), No. 3, 851--876
Copyright Heldermann Verlag 2025
BV Solutions of a State-Dependent Prox-Regular Sweeping Process with Nonconvex Perturbations
Alexander Tolstonogov
Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia
aatol@icc.ru
In a separable Hilbert space, we consider a sweeping process with perturbation. The values of the moving set are prox-regular sets dependent on time and state. The variation of the moving set is controlled in the state in a Lipschitz way and in time by a positive Radon measure. The perturbation is the sum of two multivalued mappings with different semicontinuity properties in state variable. The first perturbation with closed, possibly, nonconvex values is lower semicontinuous. The second perturbation with closed convex values has weakly sequentially closed graph. We prove the existence of a right continuous solution of bounded variation. Various a priori estimates for the solution are given. The proof is based on a new method that does not use any version of the catching-up algorithm. We use classical approaches based on a priori estimates, the compactness of sets in the space of right continuous regular functions of bounded variation, and a fixed point theorem for multivalued mappings. Our theorem is new and it comprises known results for this class of solutions of sweeping processes.
Keywords: BV-solution, prox-regular sets, bounded variation, Radon measures.
MSC: 34A60, 46B50, 54C65, 49J52, 49J53.
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