Unify and relax compactness hypotheses for Volterra sweeping processes

Determine whether the ball-compactness condition on the moving sets and the measure-of-noncompactness condition on the perturbation can be unified and relaxed through a single dissipative or one-sided Lipschitz structural condition, possibly combined with partial compactness, that encompasses both the upper semicontinuous convex-valued framework and the Lipschitz nonconvex-valued framework for Volterra sweeping processes.

Background

The paper establishes existence for Volterra sweeping processes under either ball-compactness of the moving constraint sets or a quantitative measure-of-noncompactness condition on the upper semicontinuous, convex-valued perturbation. The authors note that these assumptions are complementary rather than nested with the Lipschitz, nonconvex-valued framework developed in related work.

The unresolved problem is to formulate a broader structural hypothesis—possibly involving dissipativity or a one-sided Lipschitz property and partial compactness—that would encompass these distinct analytical settings and clarify the precise balance between regularity and compactness required for well-posedness.

References

A first question is whether the two compactness hypotheses considered here can be unified and relaxed. These alternatives, and the Lipschitz framework of Haddad, Gaouir and Thibault , are complementary rather than nested. It would therefore be desirable to formulate a single structural condition on the perturbation, of dissipative or one-sided Lipschitz type and possibly combined with a partial compactness requirement, that subsumes the upper semicontinuous, convex-valued setting of the present work as well as the Lipschitz, nonconvex-valued setting of , thereby clarifying the exact interplay between regularity and compactness needed for well-posedness.

— Volterra Sweeping Processes with Multivalued Perturbations under Compactness Conditions  (2608.22691 - Jourani et al., 24 Aug 2026) in Section 6, Conclusions and Future Work

We emphasize that the results obtained here concern existence rather than uniqueness: for a merely upper semicontinuous, set-valued perturbation, uniqueness is neither expected nor, in general, true, in contrast with the single-valued case $F\equiv 0$ treated in . Characterizing the perturbations for which the solution set enjoys additional structural properties, such as compactness, connectedness, or continuous dependence with respect to the data, constitutes a further question of independent interest.

— Volterra Sweeping Processes with Multivalued Perturbations under Compactness Conditions  (2608.22691 - Jourani et al., 24 Aug 2026) in Section 6, Conclusions and Future Work