Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions

TL;DR

This paper investigates non-convex sweeping processes in the space of absolutely continuous functions, establishing strong continuity and local Lipschitz properties of the input-output and solution mappings under certain conditions.

Contribution

It extends the analysis of sweeping processes to non-convex, prox-regular constraints, proving continuity properties of the associated mappings in function spaces.

Findings

Input-output mapping is strongly continuous in the space of absolutely continuous functions.

Under smoothness assumptions, the input-output mapping is locally Lipschitz continuous.

Solution mapping for the state-dependent problem is also locally Lipschitz continuous.

Abstract

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.