Viability theorem for deterministic mean field type control systems

TL;DR

This paper establishes a viability theorem for deterministic mean field control systems, characterizing conditions under which a set of probability distributions remains invariant under the system's dynamics in Wasserstein space.

Contribution

It introduces a set of tangent elements to probability sets and formulates a viability theorem specific to mean field type control systems.

Findings

The viability of probability sets is characterized by intersection with feasible tangent distributions.

A new framework for tangent elements in Wasserstein space is developed.

The theorem provides necessary and sufficient conditions for set invariance in mean field control systems.

Abstract

A mean field type control system is a dynamical system in the Wasserstein space describing an evolution of a large population of agents with mean-field interaction under a control of a unique decision maker. We develop the viability theorem for the mean field type control system. To this end we introduce a set of tangent elements to the given set of probabilities. Each tangent element is a distribution on the tangent bundle of the phase space. The viability theorem for mean field type control systems is formulated in the classical way: the given set of probabilities on phase space is viable if and only if the set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.