Speaker
Description
Sweeping processes are a class of evolution problems with unilateral constraints which were originally introduced by J.J. Moreau, motivated by problems in elastoplasticity and nonsmooth mechanics. Later they have then found applications in several diverse disciplines: economic theory, electrical circuits, crowd motion modeling, biology.
In his work, Moreau considered moving convex constraints having suitable continuity properties with respect to the asymmetric Hausdorff distance, also called \emph{excess}, which is very natural for evolution problems. His results were later generalized by several authors to non-convex prox-regular constraints, but in the simpler framework of the symmetric Hausdorff distance.
In this talk I will present some recent results where for the first time sweeping processes are driven by non-convex prox-regular constraints having suitable continuity properties with respect to the asymmetric Hausdorff distance. Some of these results were obtained in collaboration with Federico Stra (Politecnico di Torino, Italy).
Bibliography
@article{rucepero_prox-regular_2025,
title = {Prox-regular sweeping processes with bounded retraction},
volume = {32},
issn = {0944-6532, 2363-6394},
url = {https://www.heldermann.de/JCA/JCA32/JCA323/jca32036.htm},
abstract = {Abstract
The aim of this paper is twofold. On one hand we prove that the Moreau's sweeping processes driven by a uniformly prox-regular moving set with local bounded retraction have a unique solution provided that the coefficient of prox-regularity is larger than the size of any jump of the driving set. On the other hand we show how the case of local bounded retraction can be easily reduced to the 1-Lipschitz continuous case: indeed we first solve the Lipschitz continuous case by means of the so called "catching-up algorithm", and we reduce the local bounded retraction case to the Lipschitz one by using a reparametrization technique for functions with values in the family of prox-regular sets.},
language = {en},
number = {3},
urldate = {2026-03-23},
journal = {Journal of Convex Analysis},
author = {Recupero, Vincenzo},
year = {2025},
pages = {731-756},
}
@misc{recupero_excess-continuous_2025,
title = {Excess-continuous prox-regular sweeping processes},
copyright = {arXiv.org perpetual, non-exclusive license},
url = {https://arxiv.org/abs/2507.21646},
doi = {10.48550/ARXIV.2507.21646},
abstract = {In this paper we consider the Moreau's sweeping processes driven by a time dependent prox-regular set \$C(t)\$ which is continuous in time with respect to the asymmetric distance \$e\$ called the excess, defined by \$e(A,B) := {\textbackslash}sup\_\{x {\textbackslash}in A\} d(x,B)\$ for every pair of sets \$A\$, \$B\$ in a Hilbert space. As observed by J.J. Moreau in his pioneering works, the excess provides the natural topological framework for sweeping process. Assuming a uniform interior cone condition for \$C(t)\$, we prove that the associated sweeping process has a unique solution, thereby improving the existing result on continuous prox-regular sweeping processes in two directions: indeed, in the previous literature \$C(t)\$ was supposed to be continuous in time with respect to the symmetric Hausdorff distance instead of the excess and also its boundary \${\textbackslash}partial C(t)\$ was required to be continuous in time, an assumption which we completely drop. Therefore our result allows to consider a much wider class of continuously moving constraints.},
urldate = {2026-03-23},
publisher = {arXiv},
author = {Recupero, Vincenzo and Stra, Federico},
year = {2025},
keywords = {Classical Analysis and ODEs (math.CA), Analysis of PDEs (math.AP), Dynamical Systems (math.DS), FOS: Mathematics, FOS: Mathematics, 34G25, 34A60, 47J20, 74C05},
}
@misc{recupero_prox-regular_2024,
title = {Prox-regular sweeping processes with bounded retraction},
copyright = {arXiv.org perpetual, non-exclusive license},
url = {https://arxiv.org/abs/2407.09354},
doi = {10.48550/ARXIV.2407.09354},
abstract = {The aim of this paper is twofold. On one hand we prove that the Moreau's sweeping process driven by a uniformly prox-regular moving set with local bounded retraction has a unique solution provided that the coefficient of prox-regularity is larger than the size of any jump of the driving set. On the other hand we show how the case of local bounded retraction can be easily reduced to the \$1\$-Lipschitz continuous case: indeed we first solve the Lipschitz continuous case by means of the so called ``catching-up algorithm", and we reduce the local bounded retraction case to the Lipschitz one by using a reparametrization technique for functions with values in the family of prox-regular sets.},
urldate = {2026-03-23},
publisher = {arXiv},
author = {Recupero, Vincenzo},
year = {2024},
keywords = {Dynamical Systems (math.DS), FOS: Mathematics, FOS: Mathematics, 34G25, 34A60, 47J20, 74C05},
}
