Stability analysis of coupled linear ODE-hyperbolic PDE systems with two time scales

This paper is concerned with a class of coupled ODE/PDE systems with two time scales. The fast constant time scale is modeled by a small positive perturbation parameter. First, we state a general sufficient stability condition for such systems. This condition is also sufficient for the stability of the reduced and boundary-layer subsystems. However, counterexamples illustrate that the converse is not true. Next, we study the stability of such systems by taking into account the fact that the perturbation parameter is sufficiently small. For linear ODE coupled with fast hyperbolic PDE systems the stability of both subsystems implies the stability of the full system. On the other hand, a counterexample shows that the full system can be unstable even though the two subsystems are stable for a PDE coupled with fast ODE system. Numerical simulations on academic examples are proposed. Moreover, an application to boundary control of a gas flow transport system is used to illustrate the theoretical result.