This study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on \({\mathbb {R}}^d\) representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space.

本研究涉及状态约束与多智能体控制系统的兼容性问题。这样的系统处理的代理数量如此之大，以至于只有统计描述可用。为此，状态变量由\({\mathbb {R}}^d\)上的概率测度描述表示代理的密度并根据所谓的连续性方程进化，该方程是在 Wasserstein 概率测度空间中陈述的方程。本文的目的是为给定的约束（Wasserstein 空间的封闭子集）提供一个与受控连续性方程兼容的充分必要条件。这种新条件在粘度意义上具有如下特征：约束集的距离函数是在 Wasserstein 空间上表述的合适的 Hamilton-Jacobi-Bellman 方程的粘度超解。作为我们方法的副产品和关键成分，我们获得了 Wasserstein 空间中进化 Hamilton-Jacobi 方程的新比较定理。