Observer-based boundary control for semi-linear stochastic systems

This paper investigates an observer-based approach to address combined H∞ and passivity-based boundary control for semilinear stochastic partial differential equations (SPDEs) affected by Lèvy-type noise. We present an observer design by employing the Luenberger observer design method to estimate system states. Utilising Lyapunov theory, we establish stability results for the system by deriving a novel set of sufficient conditions that guarantee exponential stability in the absence of external disturbances. Then results are extended to the exponential stability in the presence of external disturbances adhering to combined H∞ and passivity performance. Furthermore, we leverage (Q,S,R)-dissipative theory and sector boundary conditions to linearise and regulate the inherent semilinearity within the system. We propose an observer-based robust boundary controller as a solution to this challenge. To validate our results, we present numerical examples and simulations that conclusively demonstrate the effectiveness and practical applicability of our proposed approach.