In
this work, we consider the problem of boundary stabilization for a
quasilinear 2X2 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 closedloop
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 4X4
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.