A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems

• Main Authors: Burzoni, Matteo, Cecchin, Alekos, Cosso, Andrea
• Format: Journal Article
• Language: English
• Published: 23-12-2022
• Subjects: Mathematics - Optimization and Control; Mathematics - Probability
• DOI: 10.48550/arxiv.2212.12293

We provide a new version of the Tikhonov theorem for both two-scale forward
systems and also two-scale forward-backward systems of stochastic differential
equations, which also covers the McKean-Vlasov case. Differently from what is
usually done in the literature, we prove a type of convergence for the ''fast''
variable, which allows the limiting process to be discontinuous. This is
relevant for the second part of the paper, where we present a new application
of this theory to the approximation of the solution of mean field control
problems. Towards this aim, we construct a two-scale system whose ''fast''
component converges to the optimal control process, while the ''slow''
component converges to the optimal state process. The interest in such a
procedure is that it allows to approximate the solution of the control problem
avoiding the usual step of the minimization of the Hamiltonian.