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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. 

Abstract This paper presents dynamic design techniques—namely, continuous-time event-triggered control (CETC), periodic event-triggered control (PETC) and self-triggered control (STC)—for a class of unstable one-dimensional reaction-diffusion partial differential equations (PDEs) with boundary control and an anti-collocated sensing mechanism. For the first time, global exponential stability (GES) of the closed-loop system is established using a PDE backstepping control design combined with dynamic event-triggered mechanisms for parabolic PDEs—a result not previously achieved even under full-state measurement. When the emulated continuous-time backstepping controller is implemented on the plant using a zero-order hold, our design guarantees $$ L^{2} $$-GES through the integration of novel switching dynamic event triggers and a newly developed Lyapunov functional. While CETC requires the continuous monitoring of the triggering function to detect events, PETC only requires the periodic evaluation of this function. The STC design assumes full-state measurements and, unlike CETC, does not require continuous monitoring of any triggering function. Instead, it computes the next event time at the current event time using only full-state measurements available at the current event time and the immediate previous event time. Thus, STC operates entirely with event-triggered measurements, in contrast to CETC and PETC, which rely on continuous measurements. The well-posedness of the closed-loop systems under all three strategies is established, and simulation results are provided to illustrate the theoretical results.