Tikhonov theorem for linear hyperbolic systems

A class of linear hyperbolic systems of conservation laws with multiple time scales which are modeled by a perturbation parameter is considered in this paper. By setting the perturbation parameter to zero, two subsystems, the reduced subsystem standing for the slow dynamics and the boundary-layer subsystem 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 counterexample is given to indicate that the stability of the two subsystems does not ensure the full system’s stability. Moreover a new Tikhonov theorem for this class of infinite dimensional systems is stated. The solution of the full system can be approximated by that of the reduced subsystem, and this is proved by Lyapunov techniques. An application to boundary feedback stabilization of gas transport model is used to illustrate the results.