Prox-regular sweeping processes continuous w.r.t. the excess semi-distance, or with bounded retraction and hypo-monotone perturbation

A sweeping process is a differential inclusion in a Hilbert space introduced by J.J. Moreau in 1971 which describes the trajectory $x(t)$ of a "lazy" point constrained to remain inside a moving set $C(t)$, i.e. a solution of $-x'(t) \in N_{C(t)}\bigl(x(t)\bigr)$ and $x(t)\in C(t)$, where $N_C$ denotes the exterior normal cone of the set $C$.
In the original formulation the set was requested to be convex and the movement Lipschitz with respect to the Hausdorff distance, but during the years several people contributed various advances to the theory, relaxing the regularity assumptions (in both space and time) of the family of sets under which one can prove well posedness of the Cauchy problem. The problem $-x'(t) \in N_{C(t)}\bigl(x(t)\bigr) + f\bigl(t,x(t)\bigr)$ with a Lipschitz or hypo-monotone perturbation $f$ has also been studied.
In this seminar I present two recent results obtained in collaboration with V. Recupero and exposed in [1] and [2]. In both cases the sets $C(t)$ are assumed to be prox-regular (a relaxation of convexity). In the former we request that their movement is continuous with respect to the excess semi-distance (a relaxation of the Hausdorff continuity) and that they satisfy an inner cone property. In the latter we request that their movement has bounded retraction (a relaxation of bounded variation, related to the excess) and that the perturbation is hypo-monotone (a relaxation of Lipschitzianity).
The main innovation consists in the introduction of new geometric inequalities and other techniques that allow to extend previous approaches to prox-regular sets whose movement is estimated only through the excess.
[1] V. Recupero, F. Stra Excess-continuous prox-regular sweeping processes Submitted, https://arxiv.org/abs/2507.21646
[2] V. Recupero, F. Stra Perturbed prox-regular sweeping processes with bounded retraction Preprint available soon