  We consider output-feedback stabilization problems for a class of
two-component linear parabolic systems with boundary actuation and measurement.
The state-feedback control laws are obtained using backstepping method and
require measurement of the state at each point in the domain. To this end,
backstepping observers are designed for both anti-collocated and collocated
sensors and actuators. Furthermore, we show the closed-loop systems consisting
of the plant, the backstepping control laws, and the observer is exponentially
stable. The backstepping method is used to obtain both control and observer
kernels. The kernels are the solution of $4\times4$ systems of second-order
hyperbolic linear PDEs whose well-posedness is shown.
