Local exponential H2 stabilization of 2x2 quasilinear hyperbolic systems using backstepping

In this work, we consider the problem of boundary stabilization for a quasilinear $2 imes2$ system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves $H^2$ exponential stability of the closed-loop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type $4 imes4$ system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them