Constructive finite-dimensional observer-based boundary control of stochastic parabolic PDEs

Recently, a constructive method for the finite-dimensional observer-based control of deterministic parabolic PDEs has been suggested by employing a modal decomposition approach. In the present paper, we aim to extend this method to the stochastic parabolic PDEs with nonlinear multiplicative noise. We consider the Neumann actuation and boundary measurement via dynamic extension. The controller dimension is defined by N
0
unstable modes, whereas the observer may have a larger dimension N. We provide mean-square L
²
stability analysis of the full-order closed-loop system leading to linear matrix inequality (LMI) conditions for finding N. We prove that the LMIs are always feasible for small enough noise intensity and large enough N. A numerical example demonstrates the efficiency of our method.