General Doubly Stochastic Maximum Principle and Its Applications to Optimal Control of SPDEs

TL;DR

This paper establishes a comprehensive maximum principle for optimal control of systems governed by quasilinear stochastic heat equations, extending to fully coupled forward-backward doubly stochastic systems and their applications.

Contribution

It introduces necessary and sufficient maximum principles for control of SPDEs with control-dependent coefficients, covering fully coupled forward-backward systems.

Findings

Derived NSMPs for quasilinear stochastic heat equations

Applied NSMPs to linear quadratic control problems

Provided an example of optimal control of SPDEs

Abstract

In this paper, we prove the necessary and sufficient maximum principles (NSMPs in short) for the optimal control of systems described by a quasilinear stochastic heat equation within convex control domains, which all the coefficients contain control variables. For that, the optimal control problem of fully coupled forward-backward doubly stochastic system is studied. We apply our NSMPs to treat a kind of forward-backward doubly stochastic linear quadratic optimal control problems and an example of optimal control of stochastic partial differential equations (SPDEs in short) as well.