Viability Theory in the $1$-Wasserstein Space

TL;DR

This paper develops viability conditions for solutions of continuity inclusions in the 1-Wasserstein space, based on the regularity of the dynamics and measure constraints, with different criteria for Lipschitz and upper semicontinuous velocities.

Contribution

It introduces necessary and sufficient viability conditions in the 1-Wasserstein space, adapting to different regularity assumptions of the admissible velocities.

Findings

Viability conditions depend on the regularity of the velocity fields.

For Lipschitz velocities, viability requires intersection with the graphical derivative.

For upper semicontinuous velocities, viability involves the infinitesimal behavior of the Aumann integral.

Abstract

In this article, we establish necessary and sufficient viability conditions for continuity inclusions over the 1-Wasserstein space. Depending on the regularity properties of the dynamics, we derive two results which are based on fairly different proof strategies. When the admissible velocities are Lipschitz in the measure variable, we show that it is necessary and sufficient for viable solutions to exist that the latter intersect the graphical derivative of the constraints. On the other hand, when the admissible velocities are merely upper semicontinuous in the measure variable, we provide a sufficient condition for viability involving the infinitesimal behaviour of their Aumann integral over a neighbouring set of measures.