Sampled-Data Finite-Dimensional Observer-Based Control of 1D Stochastic Parabolic PDEs

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 297-325, February 2024.
Abstract. Sampled-data control of PDEs has become an active research area; however, existing results are confined to deterministic PDEs. Sampled-data controller design of stochastic PDEs is a challenging open problem. In this paper we suggest a solution to this problem for 1D stochastic diffusion-reaction equations under discrete-time nonlocal measurement via the modal decomposition method, where both the considered system and the measurement are subject to nonlinear multiplicative noise. We present two methods: a direct one with sampled-data controller implemented via zero-order hold device, and a dynamic-extension-based one with sampled-data controller implemented via a generalized hold device. For both methods, we provide mean-square [math] exponential stability analysis of the full-order closed-loop system. We construct a Lyapunov functional [math] that depends on both the deterministic and stochastic parts of the finite-dimensional part of the closed-loop system. We employ corresponding Itô’s formulas for stochastic ODEs and PDEs, respectively, and further combine [math] with Halanay’s inequality with respect to the expected value of [math] to compensate for sampling in the infinite-dimensional tail. We provide linear matrix inequalities (LMIs) for finding the observer dimension and upper bounds on sampling intervals and noise intensities that preserve the mean-square exponential stability. We prove that the LMIs are always feasible for large enough observer dimension and small enough bounds on sampling intervals and noise intensities. A numerical example demonstrates the efficiency of our methods. The example shows that for the same bounds on noise intensities, the dynamic-extension-based controller allows larger sampling intervals, but this is due to its complexity (generalized hold device for sample-data implementation compared to zero-order hold for the direct method).