Adaptive event-triggered boundary control for a class of parabolic PDE-ODE systems with serious uncertainties

This paper investigates the adaptive event-triggered stabilisation for a class of uncertain parabolic PDE-ODE systems. Remarkably, different from the related literature where uncertainties are severely constrained, more serious uncertainties are involved since both the system matrices in ODE subsystems are linearly parameterised by unknown constants while the diffusion coefficient and the reaction coefficient matrix in PDE subsystems are unknown. Such an ingredient leads to the incapability of the traditional methods. For this, a novel control strategy is proposed by a skilful combination of infinite-dimensional backstepping method, adaptive technique with the constructive method for dynamic event-triggered mechanism. Specifically, a couple of infinite-dimensional backstepping transformations are first introduced to change the original system into a new one. For the new system, an adaptive controller is explicitly designed joint with an event-triggered mechanism in which the updating law of a pivotal dynamic threshold is smartly chosen. Then, much effort has to be taken to show its stability, i.e. all the states of the resulting closed-loop system is bounded while the original system states converge to zero along with the avoidance of the infinitely fast sampling/execution. Finally, an example is given to validate the effectiveness of the proposed results.