Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system

We consider the problem of boundary stabilization and state estimation for a 2×2 system of first-order hyperbolic linear PDEs with spatially varying coefficients. First, we design a full-state feedback law with actuation on only one end of the domain and prove exponential stability of the closed-loop system. Then, we construct a collocated boundary observer which only needs measurements on the controlled end and prove convergence of observer estimates. Both results are combined to obtain a collocated output feedback law. The backstepping method is used to obtain both control and observer kernels. The kernels are the solution of a 4 × 4 system of first-order hyperbolic linear PDEs with spatially varying coefficients of Goursat type, whose well-posedness is shown.