  We present designs for exponential stabilization of an ODE-heat PDE-ODE
coupled system where the control actuation only acts in one ODE. The
combination of PDE backstepping and ODE backstepping is employed in a
state-feedback control law and in an observer that estimates PDE and two ODE
states only using one PDE boundary measurement. Based on the state-feedback
control law and the observer, the output-feedback control law is then proposed.
The exponential stability of the closed-loop system and the boundedness and
exponential convergence of the control law are proved via Lyapunov analysis.
Finally, numerical simulations validate the effectiveness of this method for
the `sandwiched' system.
