Local exponential H2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping

We consider the problem of boundary stabilization for a quasilinear 2×2 system of first-order hyperbolic PDEs. We design a full-state feedback control law, with actuation 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. The feedback law is found using the recently developed backstepping method for 2 × 2 system of first-order hyperbolic linear PDEs, developed by the authors in a previous work, which is briefly reviewed.