Optimal control of SPDEs driven by time-space Brownian motion

In this paper, we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE which models the growth of an ecosystem. Associated to the maximum principle there is an adjoint process represented by a linear backward stochastic partial differential equation (BSPDE) in the plane driven by the Brownian sheet. We give a closed solution formula for general linear BSPDEs in the plane and also for the particular type coming from the adjoint equation. Further, applying time-space white noise calculus we derive sufficient conditions and necessary conditions of optimality of the control. Finally, we illustrate our results by solving a linear quadratic control problem in the plane. We also study possible applications to machine learning.