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A TIKHONOV THEOREM FOR MCKEAN--VLASOV TWO-SCALE SYSTEMS AND A NEW APPLICATION TO MEAN FIELD OPTIMAL CONTROL PROBLEMS.

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Title: A TIKHONOV THEOREM FOR MCKEAN--VLASOV TWO-SCALE SYSTEMS AND A NEW APPLICATION TO MEAN FIELD OPTIMAL CONTROL PROBLEMS.
Authors: BURZONI, MATTEO1 matteo.burzoni@unimi.it, CECCHIN, ALEKOS2 alekos.cecchin@unipd.it, COSSO, ANDREA1 andrea.cosso@unimi.it
Source: SIAM Journal on Control & Optimization. 2024, Vol. 62 Issue 5, p2475-2505. 31p.
Subjects: Stochastic control theory, Approximation theory, Stochastic systems
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 one to approximate the solution of the control problem, avoiding the usual step of the minimization of the Hamiltonian. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Control & Optimization is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: 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 one to approximate the solution of the control problem, avoiding the usual step of the minimization of the Hamiltonian. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Control & Optimization is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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        PageCount: 31
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      – SubjectFull: Stochastic control theory
        Type: general
      – SubjectFull: Approximation theory
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      – SubjectFull: Stochastic systems
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      – TitleFull: A TIKHONOV THEOREM FOR MCKEAN--VLASOV TWO-SCALE SYSTEMS AND A NEW APPLICATION TO MEAN FIELD OPTIMAL CONTROL PROBLEMS.
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              M: 09
              Text: 2024
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