Singular perturbation approximation by means of a $H^2$ Lyapunov function for linear hyperbolic systems

Ying Tang (1) , Christophe Prieur (2) , Antoine Girard (3)

1 CRAN - Centre de Recherche en Automatique de Nancy
2 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité
3 L2S - Laboratoire des signaux et systèmes

A linear hyperbolic system of two conservation laws with two time scales is considered in this paper. The fast time scale is modeled by a small perturbation parameter. By formally setting the perturbation parameter to zero, the full system is decomposed into two subsystems, the reduced subsystem (representing the slow dynamics) and the boundary-layer subsystem (standing for the fast dynamics). The solution of the full system can be approximated by the solution of the reduced subsystem. This result is obtained by using a H 2 Lyapunov function. The estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to H 2 and satisfying suitable compatibility conditions. Moreover, for a particular subset of initial conditions, more precise estimates are obtained. The main result is illustrated by means of numerical simulations.

Mots clés: Lyapunov methods; distributed parameter systems; singular perturbation

Domaines: Systèmes et contrôle [cs.SY]

Ying Tang, Christophe Prieur, Antoine Girard. Singular perturbation approximation by means of a $H^2$ Lyapunov function for linear hyperbolic systems. Systems and Control Letters, 2016, 88, pp.24-31. ⟨10.1016/j.sysconle.2015.10.015⟩. ⟨hal-01253479⟩