$H_{\mathrm{\infty }}$ boundary control design of spatial 2-D linear stochastic parabolic PDE systems

In practical applications involving spatiotemporal processes, the two-dimensional (2-D) spatial scenario is often more relevant. However, higher spatial dimensionality poses a major challenge for control design, drastically increasing its complexity. This work investigated the $H_{\mathrm{\infty }}$ boundary control problem for a class of 2-D linear stochastic parabolic partial differential equation (PDE) systems subject to multiplicative noise in the state. We used multidimensional Green's formula and alternative inequalities to handle complex terms in 2-D systems. A static output feedback (SOF) control strategy was introduced to achieve stabilization within a stochastic paradigm. By employing Lyapunov-based technique, a constructive method for SOF controller design was established, ensuring that the closed-loop system achieved exponential stability in the mean square sense with $H_{\mathrm{\infty }}$ performance. Numerical simulations of a stochastic thermal process demonstrated the effectiveness and applicability of the proposed control methodology.