Event-triggered gain scheduling of 2 × 2 linear hyperbolic PDEs with time and space varying coupling coefficients

In this paper, we address the problem of exponential stabilization of 2 × 2 hyperbolic PDEs systems with time- and space-varying in-domain coupling coefficients using event-triggered gain scheduling. More precisely, at each triggering time, we apply the control input as the classical static backstepping control law while scheduling the gains of the boundary controller according to the triggering mechanism and by solely considering the spatial variation of the coefficients. We present two different event-triggered gain scheduling strategies. The first strategy relies on a Lyapunov-based event-triggering condition, and the $L^2$ closed-loop exponential stability is shown using a Lyapunov analysis. The second one is a small-gain based event-triggering condition and builds on the $L^{\infty }$-norm. The $L^{\infty }$ closed-loop exponential stability is shown by combining ISS estimates with small-gain arguments. In both cases, we prove that we avoid the Zeno phenomenon, provided that the coupling coefficients are slowly time-varying. Unlike existing results in the literature, the proposed approaches do not require solving time-varying backstepping kernel equations in real-time resulting in reduced computational burden and broader applicability.