Approximation of deterministic mean field type control systems

TL;DR

This paper presents a method to approximate deterministic mean field control systems using mean field Markov chains, providing explicit constructions and error estimates based on ordinary differential equations.

Contribution

It introduces a novel approximation approach for mean field control systems via Markov chains with explicit construction and Hausdorff distance estimates.

Findings

The dynamics of the approximating system are described by ODEs.

Explicit control construction from Markov chain strategies.

Error bounds for the approximation in Hausdorff distance.

Abstract

The paper is concerned with the approximation of the deterministic the mean field type control system by a mean field Markov chain. It turns out that the dynamics of the distribution in the approximating system is described by a system of ordinary differential equations. Given a strategy for the Markov chain, we explicitly construct a control in the deterministic mean field type control system. Our method is a realization of the model predictive approach. The converse construction is also presented. These results lead to an estimate of the Hausdorff distance between the bundles of motions in the deterministic mean field type control system and the mean field Markov chain. Especially, we pay the attention to the case when one can approximate the bundle of motions in the mean field type system by solutions of a finite systems of ODEs.