  This paper deals with the exponential stability of systems made of a
hyperbolic PDE coupled with an ODE with different time scales, the dynamics of
the PDE being much faster than that of the ODE. Such a difference of time
scales is modeled though a small parameter $\varepsilon$ multiplying the time
derivative in the PDE, and our stability analysis relies on the singular
perturbation method. More precisely, we define two subsystems: a reduced order
system, representing the dynamics of the full system in the limit $\varepsilon
= 0$, and a boundary-layer system, which represents the dynamics of the PDE in
the fast time scale. Our main result shows that, if both the reduced order and
the boundary-layer systems are exponentially stable, then the full system is
also exponentially stable for $\varepsilon$ small enough, and our strategy is
based on a spectral analysis of the systems under consideration. Our main
result improves a previous result in the literature, which was proved using a
Lyapunov approach and required a stronger assumption on the boundary-layer
system to obtain the same conclusion.
