Sample-and-hold control of the semilinear heat equation via the L2 residue separation

The L2 residue separation method has recently emerged as an effective approach for designing finitedimensional controllers for PDEs while avoiding spillover caused by neglected modes. This paper extends the method to sampled-data control and shows that it enables implementation via a standard zero-order hold in the cases where existing approaches require a less practical generalized hold mechanism. Specifically, we consider a semilinear reaction–diffusion equation with unknown nonlinearity, boundary heat-flux actuation, and temperature measurements taken at the opposite boundary. For this system, we design a finite-dimensional output-feedback controller whose parameters are chosen to eliminate potential spillover. Stability under sample-and-hold actuation is established via a Lyapunov–Krasovskii functional. The framework naturally extends to additional delay effects, including fastvarying input and network-induced delays, through appropriate augmentations of the Lyapunov–Krasovskii functional.