Noncoercive convex sweeping processes with velocity constraints

Abstract

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.