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

In this paper, we address the problem of exponential stabilization of $2 \times 2$ hyperbolic PDEs systems with time- and space-varying in-domain coupling coefficients using event-triggered gain scheduling. More precisely, we sample the coupling terms according to a Lyapunov-based event-triggering condition. At each triggering time, we define the control input as the classical static backstepping control law that would stabilize the system, thereby scheduling the gains of the controller according to the triggering mechanism while solely considering the spatial variation of the coefficients. We prove that we avoid the Zeno phenomenon under the even-triggering policy, provided that the coupling coefficients are slowly time-varying. The closed-loop exponential stability is shown using a Lyapunov analysis. Unlike existing results in the literature, the proposed approach does not require solving time-varying backstepping kernel equations in real-time, which implies a smaller computational burden and better applicability.