Boundary exponential stabilisation design for spatial 2-D linear parabolic PDE systems with mobile sensors

When examining real-world physical spatiotemporal processes, a two-dimensional (2-D) spatial model is more relevant to reality. However, the control design methodologies developed for 1-D spatial configurations cannot be straightforwardly adapted to 2-D frameworks. This paper focuses on the observer-based boundary control of linear distributed parameter systems over 2-D spatial domains described by parabolic partial differential equations (PDEs), utilising spatially noncollocated mobile sensors. In practical applications, it is physically infeasible to position sensors and actuators at identical points, and adopting a noncollocated configuration can offer significant advantages in certain operational scenarios. Initially, a Luenberger-type state observer is constructed to estimate the system state, coupled with a sensor guidance scheme that constrains each sensor to move within assigned subregions, thereby effectively avoiding inter-sensor collisions. Within this approach, the 2-D domain is segmented into multiple subdomains based on the number of mobile sensors. Then, by applying Lyapunov stability analysis, an observer-based boundary control law is formulated to ensure the exponential stability of the resulting closed-loop system. In final, the efficacy of the proposed methodology is validated through numerical simulations.