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

International audienceThis 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. Next, we study the stability for ODE/fast PDE and PDE/fast ODE systems based on the singular perturbation method respectively. In the first case, we consider a linear ODE coupled with a fast hyperbolic PDE system. The stability of both reduced and boundary-layer subsystems implies the stability of the full system. On the contrary, a counter-example shows that the full system can be unstable even though the two subsystems are stable for a PDE coupled with a 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