Finite-Dimensional Boundary Control for Stochastic Semilinear 2D Parabolic PDEs

In this paper, we consider state-feedback global stabilization of stochastic semilinear 2D parabolic PDEs with nonlinear multiplicative noise, where the nonlinearities satisfy globally Lipschitz condition. We consider the Dirichlet actuation and design the controller with the shape functions in the form of eigenfunctions corresponding to the first comparatively unstable N eigenvalues. We extend the trigonometric change of variables to the 2D case and further improve it, leading to homogeneous boundary conditions. Employing N-dimensional dynamic extension with the corresponding proportional-integral controller and using modal decomposition, we derive stochastic nonlinear ODEs for the modes of the state with the first N-dimensional part being controllable. By using a direct Lyapunov method and Itô's formula for stochastic ODEs and PDEs, we provide mean-square L² exponential stability analysis of the full-order closed-loop system. We provide linear matrix inequality (LMI) conditions for finding N and the controller gain. We prove that the LMIs are always feasible provided the Lipschitz constants are small enough and N is large enough. Numerical examples demonstrate the efficiency of our method and show that the employment of the suggested dynamic extension allows for larger Lipschitz constants than the previously used dynamic extensions.