  This paper presents a backstepping solution for the output feedback control
of general linear heterodirectional hyperbolic PDE-ODE systems with
spatially-varying coefficients. Thereby, the coupling in the PDE is in-domain
and at the uncontrolled boundary, whereby the ODE is coupled with the latter
boundary. For the state feedback design a two-step backstepping approach is
developed, that yields the conventional kernel equations and additional
decoupling equations of simple form. The latter can be traced back to simple
Volterra integral equations of the second kind, which are directly solvable
with a successive approximation. In order to implement the state feedback
controller, the design of observers for the ODE-PDE systems in question is
considered, whereby anticollocated measurements are assumed. Simple conditions
for the existence of the resulting observer-based compensator are formulated,
that can be evaluated in terms of the plant transfer behaviour. The resulting
systematic compensator design is illustrated for a 4x4 heterodirectional
hyperbolic system coupled with a third order ODE modelling a dynamic boundary
condition.
