Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.05376 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911489280966656 |
|---|---|
| author | Garrido, Juan Guillermo Vilches, Emilio |
| author_facet | Garrido, Juan Guillermo Vilches, Emilio |
| contents | We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones.
Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes.
We further establish a Brézis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05376 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integral Formulation and the Brézis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes Garrido, Juan Guillermo Vilches, Emilio Optimization and Control Dynamical Systems We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones. Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes. We further establish a Brézis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting. |
| title | Integral Formulation and the Brézis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes |
| topic | Optimization and Control Dynamical Systems |
| url | https://arxiv.org/abs/2603.05376 |