Finite-dimensional boundary control of 2D linear parabolic stochastic PDEs under boundary measurement

This paper addresses finite-dimensional observerbased boundary control of 2D linear stochastic heat equation with multiplicative noise on a bounded connected set. We consider the Neumann actuation and boundary measurement. We design the controller with the shape functions in the form of eigenfunctions corresponding to the unstable N0 eigenvalues. We suggest an appropriate change of variables leading to homogeneous boundary conditions and employ N0-dimensional dynamic extension with the corresponding proportional-integral controller. Both the observer and controller are designed based on the first N(N ≥q N0) modes. By suggesting a direct Lyapunov method and employing Itô 's formula, we provide mean-square L2 exponential stability analysis of the full-order closed-loop system. We derive linear matrix inequality (LMI) conditions for finding observer dimension, the controller and observer gains, and the maximum admissible noise intensity. Numerical example demonstrates the efficiency of our method and shows that controller design from the first N modes allows larger noise intensity than the design from the first N0 modes as studied in previous works on stochastic PDEs.