Stochastic maximum principle for optimal control of SPDEs

TL;DR

This paper establishes a Pontryagin maximum principle for controlling stochastic partial differential equations driven by finite-dimensional noise, applicable to a broad class of stochastic parabolic equations with control-dependent diffusion.

Contribution

It introduces a semi-abstract formulation of the maximum principle for SPDEs with control-dependent diffusion, including the development of two adjoint processes, one of which is operator-valued.

Findings

Proves a maximum principle for a wide class of controlled SPDEs.

Handles control-dependent diffusion coefficients.

Introduces a novel second adjoint process in an operator space.

Abstract

We prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a stochastic partial differential equation driven by a finite dimensional Wiener process. The equation is formulated in a semi-abstract form that allows direct applications to a large class of controlled stochastic parabolic equations. We allow for a diffusion coefficient dependent on the control parameter, and the space of control actions is general, so that in particular we need to introduce two adjoint processes. The second adjoint process takes values in a suitable space of operators on $L^4$.