# Attainability property for a probabilistic target in Wasserstein spaces

## TL;DR

This paper proves an attainability result for the minimum time function in a control problem within Wasserstein spaces, modeling multiagent systems with nonlocal constraints on the dynamics.

## Contribution

It introduces a novel attainability theorem for control problems in Wasserstein spaces with nonlocal nonholonomic constraints on the vector field.

## Key findings

- Establishes attainability of the minimum time function in Wasserstein spaces.
- Models multiagent systems with nonlocal interaction constraints.
- Provides a framework for control in probabilistic measure spaces.

## Abstract

In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called \emph{multiagent system} made of several possible interacting agents.