We investigate when a mean field-type control system can fulfill a given constraint. Namely, given a closed set of probability measures on the torus, starting from any initial probability measure belonging to this set, does there exist a solution to the mean field control system remaining in it for any time? This property—the so-called viability property—is equivalently characterized through a property involving normals to the given set of probability measures. We prove that, if the Hamiltonian is nonpositive at any normal distribution to the given set, then the feedback strategy realizing the extremal shift rule provides the approximate viability. This implies the usual viability property. Conversely, the Hamiltonian is nonpositive at any normal distribution if the given set is viable. Our approach enables us to derive generalized feedback laws which ensure the trajectory to fulfill the constraint. This generalized feedback called here extremely shift rule is inspired by constructive motions developed by Krasovskii and Subbotin for differential games.

我们调查平均字段类型的控制系统何时可以满足给定的约束。也就是说，给定圆环上的一组封闭的概率测度，从属于该集合的任何初始概率测度开始，是否存在对平均场控制系统随时存在的解决方案？该属性（所谓的生存力属性）通过涉及给定概率度量集的法线的属性来等效地表征。我们证明，如果哈密顿量在给定集合的任何正态分布上都不为正，则实现极值移位规则的反馈策略将提供近似的生存能力。这意味着通常的生存能力。相反，如果给定集合可行，则哈密顿量在任何正态分布下都是非正值的。我们的方法使我们能够导出广义反馈定律，以确保轨迹满足约束条件。这种称为“极度偏移规则”的广义反馈是受Krasovskii和Subbotin为差分游戏开发的构造运动启发的。