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

.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 \(L^2\) exponential stability analysis of the full-order closed-loop system. We construct a Lyapunov functional \(V\) 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 \(V\) with Halanay's inequality with respect to the expected value of \(V\) 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).Keywordsstochastic parabolic PDEssampled-data controlobserver-based controlboundary controlLyapunov–Krasovskii methodMSC codes93C5760H1593E15