Global exponential stabilization of 2 × 2 linear hyperbolic PDEs via dynamic event-triggered backstepping control

Abstract

This paper introduces novel dynamic event-triggered control (ETC) mechanisms for 2 × 2 linear hyperbolic PDEs in three configurations: continuous-time event-triggered control (CETC), periodic event-triggered control (PETC), and self-triggered control (STC). These mechanisms ensure global exponential stability (GES) under ETC using PDE backstepping, with stability estimates provided in the spatial $L^2$ norm of the states. The proposed CETC and PETC designs are observer-based and require continuous boundary measurements collocated with the actuation. In contrast, the STC design requires full-state measurements; however, unlike CETC and PETC, it does not require continuous measurements for the triggering mechanism—only measurements taken at event times. In the CETC design, a lower bound on the time between two consecutive events is enforced, and a dynamic variable with appropriately designed switching dynamics is introduced. By employing a novel Lyapunov functional, GES of the closed-loop system is established under zero-order hold implementation of the backstepping control between events. Events are triggered when the dynamic variable crosses zero from the positive side, after which it is immediately reset to an appropriate nonnegative value. Detecting events, therefore, necessitates continuous monitoring of this dynamic variable. To address this limitation, PETC and STC strategies are proposed. The PETC design identifies a suitable triggering condition that requires only periodic checks and derives an upper bound on the allowable sampling period. This PETC approach preserves the GES guaranteed by CETC without requiring continuous monitoring of a triggering condition, although it still relies on continuous measurements. Unlike CETC and PETC, STC requires neither continuous measurements nor monitoring of a triggering condition. Instead, at each event, STC computes the time to the next event — beyond a suitably enforced minimal dwell-time — using only measurements taken at events. Despite relying solely on event-triggered measurements, STC is capable of guaranteeing GES of the closed-loop system. The well-posedness of the closed-loop systems under all three strategies is established. A simulation study is provided to illustrate the theoretical results.