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 su cient 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.
International audienceA class of linear systems of conservation laws with a small perturbation parameter is introduced. By setting the perturbation parameter to zero, two subsystems, the reduced system standing for the slow dynamics and the boundary-layer system representing the fast dynamics, are computed. It is first proved that the exponential stability of the full system implies the stability of both subsystems. Secondly, a counter example is given to indicate that the converse is not true. Moreover a new Tikhonov theorem for this class of the infinite dimensional systems is stated. The solution of the full system can be approximated by that of the reduced system, and this is proved by Lyapunov techniques. An application to boundary feedback stabilization of gas transport model is used to illustrate the results
International audienceThis paper is concerned with a coupled ODE-PDE system with two time scales modeled by a perturbation parameter. Firstly, the perturbation parameter is introduced into the PDE system. We show that the stability of the full system is guaranteed by the stability of the reduced and the boundary-layer subsystems. A numerical simulation on a gas flow transport model is used to illustrate the first result. Secondly, an example is used to show that the full system can be unstable even though both subsystems are stable when the perturbation parameter is introduced into the ODE system
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