# Deterministic Control of Stochastic Reaction-Diffusion Equations

## TL;DR

This paper develops a method for controlling stochastic reaction-diffusion equations using deterministic controls, deriving optimality conditions, and applying a conjugate gradient method to approximate solutions, with applications to the Schl"ogl model.

## Contribution

It introduces a novel approach to control stochastic PDEs with deterministic controls, avoiding backward SPDEs, and provides a practical gradient-based optimization method.

## Key findings

- Derived necessary conditions for optimal controls with multiplicative noise.
- Presented a gradient approximation method for stochastic control problems.
- Applied the approach to the stochastic Schl"ogl model.

## Abstract

We consider the control of semilinear stochastic partial differential equations (SPDEs) via deterministic controls. In the case of multiplicative noise, existence of optimal controls and necessary conditions for optimality are derived. In the case of additive noise, we obtain a representation for the gradient of the cost functional via adjoint calculus. The restriction to deterministic controls and additive noise avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schl\"ogl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.