  In this paper, we present output feedback boundary stabilization for a class
of semilinear parabolic PDEs with a boundary measurement and an actuation
located at the same place. The method uses backstepping transformations, where
the state and error systems are proved to be locally exponentially stable in
the $\mathbb{H}^4$ norm. The stability of the transformed systems are obtained
by constructing a strict Lyapunov function. A numerical example using the
FitzHugh-Nagumo equation shows the proposed control law stabilizes the system
into its equilibrium solution.
