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

Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE