Singular perturbation approach for linear coupled ODE-PDE 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

This paper focuses on a class of linear coupled ODE-PDE systems whose dynamics evolve in two time scales. The fast time scale modeled by a small positive perturbation parameter is introduced to the dynamics either of the ODE or of the PDE. By setting the perturbation parameter to zero, two subsystems, namely the reduced and the boundary-layer subsystems, are formally computed. Firstly, we propose a sufficient stability condition for the full coupled system. This stability condition implies the stability of both subsystems. Then, we state an approximation of the full coupled ODE-PDE systems by the subsystems based on the singular perturbation method. The error between the solution of the full system and that of the subsystems is the order of the perturbation parameter. Finally, numerical simulations on academic examples illustrate the theoretical results.

Domaines: Optimisation et contrôle [math.OC]; Systèmes et contrôle [cs.SY]; Automatique / Robotique

Ying Tang, Christophe Prieur, Antoine Girard. Singular perturbation approach for linear coupled ODE-PDE systems. Giorgio Valmorbida and Alexandre Seuret and Islam Boussaada and Rifat Sipahi. Delays and Interconnections: Methodology, Algorithms and Applications, 10, Springer, pp.3-17, 2019, Advances in Delays and Dynamics book series, ⟨10.1007/978-3-030-11554-8_1⟩. ⟨hal-02372335⟩