Adaptive boundary control of reaction–diffusion PDEs with unknown distributed delay and unknown parameters

Abstract We study a reaction–diffusion partial differential equation (PDE) system with a distributed input, subject to multiple unknown plant parameters with arbitrarily large uncertainties. Using Lyapunov-based techniques, we design a delay-adaptive predictor feedback controller that ensures local boundedness of system trajectories and asymptotic regulation of the closed-loop system in terms of the plant state. Specifically, we model the input delay as a one-dimensional transport PDE with a spatial variable, effectively transforming the time delay into a spatially distributed shift. For the resulting coupled transport and reaction–advection–diffusion PDE system, we employ a PDE backstepping approach combined with the certainty-equivalence principle to derive an adaptive control law that compensates for both the unknown time delay and the unknown functional parameters. Simulation results are provided to illustrate the feasibility of our control design.