Constructive finite-dimensional boundary control of stochastic 1D parabolic PDEs

Recently, a constructive method for the finite-dimensional observer-based control of deterministic parabolic PDEs was suggested by employing a modal decomposition approach. In this paper, for the first time we extend this method to the stochastic 1D heat equation with nonlinear multiplicative noise. We consider the Neumann actuation and study the observer-based as well as the state-feedback controls via the modal decomposition approach. We employ either trigonometric or polynomial dynamic extension. For observer-based control we consider a noisy boundary measurement. First, we show the well-posedness of strong solutions to the closed-loop systems. Then 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, leading to linear matrix inequality (LMI) conditions for finding the observer dimension and as large as possible noise intensity bound for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity and large enough observer dimension (for observer-based control). We further show that in the case of state-feedback and linear noise, the system is always stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system with deterministic measurement. Numerical simulations are carried out to illustrate the efficiency of our method. For both state-feedback and observer-based controls, the trigonometric extension always allows for a larger noise than the polynomial one in the example.