A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems
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...
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23-12-2022
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Cosso, Andrea Burzoni, Matteo Cecchin, Alekos |
| Author_xml | – sequence: 1 givenname: Matteo surname: Burzoni fullname: Burzoni, Matteo – sequence: 2 givenname: Alekos surname: Cecchin fullname: Cecchin, Alekos – sequence: 3 givenname: Andrea surname: Cosso fullname: Cosso, Andrea |
| BackLink | https://doi.org/10.48550/arXiv.2212.12293$$DView paper in arXiv |
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| Snippet | 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... |
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| SubjectTerms | Mathematics - Optimization and Control Mathematics - Probability |
| Title | A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems |
| URI | https://arxiv.org/abs/2212.12293 |
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