Performance-Barrier-Based Event-Triggered Boundary Control of a Class of Reaction-Diffusion PDEs

This paper presents a novel event-triggered boundary control technique named performance-barrier-based event-triggered control for a class of reaction-diffusion PDEs under Neumann actuation of a Robin boundary condition. At its core, rather than insisting on a strictly monotonic decrease of the Lyapunov function of the closed-loop system, we allow it to increase as long as it remains within an established performance barrier. This approach integrates a performance residual-the difference between the performance barrier and the Lyapunov function-into the triggering mechanism. This integration provides the system's Lyapunov function with enhanced flexibility, thereby allowing for longer dwell-times compared to “regular” strategies demanding a monotonic decrease of the Lyapunov function. Notably, while adhering to the performance barrier, the closed-loop system globally exponentially converges to zero in the spatial
$L^{2}$
norm without encountering Zeno phenomenon. We provide numerical simulations to illustrate the proposed technique and to compare it with the regular event-triggered control design, the latter being associated with strictly decreasing Lyapunov functions.