Sampled-data Boundary Control of a Class of Reaction-Diffusion PDEs: A Lyapunov-based Approach

This paper proposes an observer-based sampled-data boundary control approach for a class of reaction-diffusion PDEs with anti-collocated sensing and actuation. The PDE backstepping design is employed as the underlying control method. Using Lyapunov arguments, it is proven that continuous-time feedback boundary control, applied in a sample-and-hold manner, preserves closed-loop global exponential stability, provided that the sampling period is sufficiently small. The introduction of a state-independent dynamic reset variable in the Lyapunov function, to the best of our knowledge not previously employed in PDE control, enables the use of Lyapunov arguments to derive stability properties under sampled-data boundary control with PDE backstepping—previously only derived using small-gain arguments. The careful handling of this state-independent dynamic reset variable within the Lyapunov function absorbs the effect of input sampling, thereby preserving closed-loop global exponential stability. The results obtained provide stability estimates for the spatial L² norms of the plant and observer states. Furthermore, robustness to perturbations in the sampling schedule is guaranteed by establishing a suitable maximum upper diameter for the sampling interval. A numerical simulation is provided to illustrate the theoretical results.