Predictor methods for finite-dimensional observer-based control of stochastic parabolic PDEs

We study output-feedback control of 1D stochastic semilinear heat equations with nonlinear multiplicative noise and uncertain time-varying input/output delays or sawtooth delays (that correspond to network-based control), where the nonlinearities satisfy globally Lipschitz condition. We assume that the input delay has a large constant known part r. We consider Neumann actuation with non-local measurement. To compensate r, we consider a chain of M sub-predictors (conventional sub-predictors as in the deterministic case) and a novel chain of M+1 sub-predictors, respectively, both in the form of ODEs that correspond to the delay fraction r/M. For both cases, we construct Lyapunov functionals that depend on the deterministic and stochastic parts of the finite-dimensional part of the closed-loop systems, and employ the corresponding Itô’s formulas for stochastic ODEs and PDEs, respectively. We provide the mean-square L² exponential stability analysis of the full-order closed-loop system, leading to LMIs that are feasible for any r provided M and the observer dimension are large enough, and Lipschitz constants, as well as the upper bounds of unknown delays, are small enough. For the novel sub-predictors, we add an additional sub-predictor to the chain that leads to the closed-loop system with the stochastic infinite-dimensional tail and the stochastic finite-dimensional part where the delay fraction r/M and the stochastic term appear in separate equations, which essentially simplifies stochastic Lyapunov functional structure and the resulting LMIs. We also consider a classical observer-based predictor for linear heat equations with nonlinear multiplicative noise and show that the corresponding LMI stability conditions are feasible for any r provided the observer dimension is large enough, and the upper bounds of unknown delays and noise intensity are small enough. A numerical example demonstrates that for comparatively large M and upper bound of noise intensity, the introduction of an addition sub-predictor leads to a larger r compared with conventional sub-predictors, whereas for the linear heat equations with nonlinear noise, the classical predictor always allows larger delays.