This seminar will be presented in hybrid mode.  The speaker will deliver his talk in person.

Title: Two results on the optimal control for stochastic partial differential equations (SPDE)

Abstract: The talk will consist of two parts: In the first part we present an extension of Peng’s maximum principle for semilinear SPDEs in one space-dimension with non-convex control domains and control-dependent diffusion coefficients to the case of general cost functionals with Nemytskii-type coefficients ([1]). The analysis is based on a new approach to the characterization of the second order adjoint state as the solution of a function-valued backward SPDE. In the second part we discuss a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations ([2]). Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm.

The talk is based on joint work with A. Vogler and L. Wessels.

[1] W. Stannat, L. Wessels: Peng’s Maximum Principle for Stochastic Partial Differential Equations, SIAM J. Control Optim., 59 (2021), 3552-3573

[2] W. Stannat, A. Vogler, L. Wessels: Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations. Preprint, arXiv:2301.11926

The talk will be followed by refreshments in the Huxley Common Room at 4pm.