Sampled-data finite-dimensional boundary control of 2D semilinear parabolic stochastic PDEs

This paper addresses the sampled-data boundary stabilization of 2D semilinear parabolic stochastic PDEs with globally Lipschitz nonlinearities. We consider Dirichlet actuation and design a finite-dimensional state-feedback controller with the shape functions in the form of eigenfunctions corresponding to the first N comparatively unstable eigenvalues. We extend the trigonometric change of variables to the 2D case and further improve it that leads to a dynamic extension with the corresponding proportional-integral controller, where sampled-data control is implemented via a generalized hold device. By employing the corresponding Itô formulas for stochastic ODEs and PDEs, respectively, and suggesting a non-trivial stochastic extension of the descriptor method, we derive linear matrix inequalities (LMIs) for finding the controller dimension and gain that guarantees the global mean-square L² exponential stability for the full-order closed-loop system. A numerical example demonstrates the efficiency and advantage of our method.