Transposition Approach to Optimal Control of McKean-Vlasov SPDEs

TL;DR

This paper develops a Pontryagin-type maximum principle for optimal control of McKean-Vlasov SPDEs, extending existing results to infinite-dimensional systems with nonconvex control sets using advanced stochastic analysis techniques.

Contribution

It introduces a novel maximum principle for McKean-Vlasov SPDEs, incorporating Lions derivatives and spike variation methods, extending control theory to infinite-dimensional stochastic systems.

Findings

Established necessary optimality conditions for McKean-Vlasov SPDEs.

Extended stochastic maximum principle to infinite-dimensional setting.

Handled nonconvex control sets in the control problem.

Abstract

In this paper, we investigate an optimal control problem for McKean-Vlasov stochastic partial differential equations, in which the coefficients depend on the law of the state process. For systems with nonconvex control sets, we establish a Pontryagin-type stochastic maximum principle that provides necessary optimality conditions for admissible controls. The analysis is based on the classical spike variation method together with the introduction of an adjoint backward stochastic partial differential equation involving Lions derivatives with respect to probability measures. Our results extend the stochastic maximum principle for McKean-Vlasov controlled stochastic differential equations to the infinite-dimensional SPDE setting.