Output regulation for a reaction-diffusion system with input delay and unknown frequency

This work tackles the output regulation problem for a reaction–diffusion system subject to input delay and unknown multi-channel disturbances with unknown frequencies and amplitudes. A control framework is developed by combining modal decomposition with a dual-observer scheme, which enables real-time estimation of states and disturbances through a state observer and an adaptive disturbance estimator. Unlike existing methods that address delay compensation or disturbance rejection in isolation, the proposed tracking-error-based control law achieves both simultaneously, thereby guaranteeing exponential convergence of the output to the reference signal. Numerical simulations confirm the effectiveness of the output-feedback strategy.