Collocated output-feedback stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping

We consider the problem of output feedback stabilization for a quasilinear 2×2 system of first-order hyperbolic PDEs with boundary actuation and measurement. We design an output feedback control law, with actuation and measurement on only one end of the domain, and prove local H² exponential stability of the closed-loop system. The proof of stability is based on the construction of a strict Lyapunov function which includes the observer states. The feedback law and output injection gains are found using the backstepping method for 2 × 2 system of first-order hyperbolic linear PDEs, developed by the authors in a previous work, which is briefly reviewed.