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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]
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Database: Engineering Source
Description
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]
ISSN:03630129
DOI:10.1137/22M1543070