Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift

TL;DR

This paper establishes a stochastic maximum principle for finite horizon optimal control of certain nonlinear SPDEs with dissipative drift, applicable to reaction-diffusion equations and involving general performance functionals.

Contribution

It extends the stochastic maximum principle to a broad class of nonlinear SPDEs with dissipative Nemytskii-type nonlinearities and polynomial growth, including pointwise evaluation in the cost.

Findings

Proved a Pontryagin maximum principle for nonlinear SPDEs with dissipative drift.

Applicable to a wide range of nonlinear parabolic equations like reaction-diffusion.

Allows for general performance functionals depending on point evaluations.

Abstract

We prove a version of the stochastic maximum principle, in the sense of Pontryagin, for the finite horizon optimal control of a stochastic partial differential equation driven by an infinite dimensional additive noise. In particular we treat the case in which the non-linear term is of Nemytskii type, dissipative and with polynomial growth. The performance functional to be optimized is fairly general and may depend on point evaluation of the controlled equation. The results can be applied to a large class of non-linear parabolic equations such as reaction-diffusion equations.