Deterministic Control of Stochastic Reaction-Diffusion Equations

Wilhelm Stannat, Lukas Wessels

Published 2019-05-22Version 1

We consider the control of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise via deterministic controls. Existence of optimal controls and necessary conditions for optimality are derived. Using adjoint calculus, we obtain a representation for the gradient of the cost functional. The restriction to deterministic controls 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.