A Filippov theorem for Volterra sweeping processes with one-sided Lipschitz perturbation
Abstract
We prove a Filippov-type stability theorem for integro-differential sweeping processes of Volterra type with an outer multivalued perturbation, in a separable Hilbert space, under the assumptions that the moving sets are uniformly prox-regular and Lipschitz continuous with respect to the Hausdorff distance and that the perturbation is one-sided Lipschitz. Given an absolutely continuous solution of the system perturbed both in the state argument of the multivalued term and by an outer integrable term, we show the existence of a solution of the original Volterra sweeping process and estimate the distance between the two trajectories explicitly in terms of the data of the problem. The estimate becomes sharper when only the outer perturbation is present, and it recovers the Lipschitz dependence on the initial condition of Filippov's classical theorem. The proof combines a reduction of the constrained dynamics to an unconstrained differential inclusion, measurable selection arguments, and an enhanced version of Grönwall's inequality established in this work. As an application, we obtain the Lipschitz dependence of the attainable set on the initial set with respect to the Hausdorff distance, together with a one-sided estimate quantifying the effect of the perturbations. We also work out a spatially distributed fishery model with ecological memory whose harvesting rule, triggered by the aggregate biomass, is one-sided Lipschitz but is not Lipschitz continuous with respect to the Hausdorff distance, so that the classical framework does not apply. For this model, our estimates produce a stability constant governed only by the biological data.
Related
- Maximal Monotone Differential Inclusions with Volterra and One-Sided Lipschitz Perturbations under Nonlocal Initial Conditions and Applications
- Optimal Control of Prox-Regular Sweeping Processes with Variable Time
- Quasihelix properties of selected Volterra Gaussian processes
- Affine Volterra covariance processes and application to commodity markets
- Conley Index of Limit Cycles in Filippov Vector Fields
BibTeX
@misc{jourani2026filippov,
title = {{A Filippov theorem for Volterra sweeping processes with one-sided Lipschitz perturbation}},
author = {Abderrahim Jourani and Diana Narváez and Emilio Vilches},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14456},
eprint = {2609.14456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14456},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Filippov theorem for Volterra sweeping processes with one-sided Lipschitz perturbation
AU - Abderrahim Jourani
AU - Diana Narváez
AU - Emilio Vilches
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14456
UR - https://arxiv.org/abs/2609.14456
AB - We prove a Filippov-type stability theorem for integro-differential sweeping processes of Volterra type with an outer multivalued perturbation, in a separable Hilbert space, under the assumptions that the moving sets are uniformly prox-regular and Lipschitz continuous with respect to the Hausdorff distance and that the perturbation is one-sided Lipschitz. Given an absolutely continuous solution of the system perturbed both in the state argument of the multivalued term and by an outer integrable term, we show the existence of a solution of the original Volterra sweeping process and estimate the distance between the two trajectories explicitly in terms of the data of the problem. The estimate becomes sharper when only the outer perturbation is present, and it recovers the Lipschitz dependence on the initial condition of Filippov's classical theorem. The proof combines a reduction of the constrained dynamics to an unconstrained differential inclusion, measurable selection arguments, and an enhanced version of Grönwall's inequality established in this work. As an application, we obtain the Lipschitz dependence of the attainable set on the initial set with respect to the Hausdorff distance, together with a one-sided estimate quantifying the effect of the perturbations. We also work out a spatially distributed fishery model with ecological memory whose harvesting rule, triggered by the aggregate biomass, is one-sided Lipschitz but is not Lipschitz continuous with respect to the Hausdorff distance, so that the classical framework does not apply. For this model, our estimates produce a stability constant governed only by the biological data.
ER -