Sampled-data finite-dimensional observer-based boundary control of 1D stochastic parabolic PDEs

Recently, finite-dimensional controllers were introduced for 1D stochastic parabolic PDEs via the modal decomposition method. In the present paper, we suggest a sampled-data implementation of a finite-dimensional observer-based boundary controller for 1D stochastic parabolic PDEs under discrete-time non-local measurement, where both the considered system and the measurement are subject to nonlinear multiplicative noise. We provide mean-square L2 exponential stability analysis of the full-order closed-loop system, where we employ Itô's formula. We consider sampled-data control that employs zero-order hold device and use the time-delay approach to the sampled-data system. We construct Lyapunov functional V, and further combine it with Halanay's inequality with respect to expected value of V to compensate sampling in the infinite-dimensional tail. We provide linear matrix inequalities (LMIs) for finding the observer dimension N and upper bounds on sampling intervals and noise intensities that preserve the exponential stability. We prove that the LMIs are always feasible for large enough N and small enough sampling intervals and noise intensities. A numerical example demonstrates the efficiency of our method.