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Preprints, Working Papers, ... Year : 2023

Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE

Abstract

This paper deals with the exponential stability of systems made of a hyperbolic PDE coupled with an ODE with different time scales, the dynamics of the PDE being much faster than that of the ODE. Such a difference of time scales is modeled though a small parameter ε multiplying the time derivative in the PDE, and our stability analysis relies on the singular perturbation method. More precisely, we define two subsystems: a reduced order system, representing the dynamics of the full system in the limit ε = 0, and a boundary-layer system, which represents the dynamics of the PDE in the fast time scale. Our main result shows that, if both the reduced order and the boundary-layer systems are exponentially stable, then the full system is also exponentially stable for ε small enough, and our strategy is based on a spectral analysis of the systems under consideration. Our main result improves a previous result in the literature, which was proved using a Lyapunov approach and required a stronger assumption on the boundary-layer system to obtain the same conclusion.
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Dates and versions

hal-04059956 , version 1 (05-04-2023)
hal-04059956 , version 2 (06-09-2023)

Identifiers

  • HAL Id : hal-04059956 , version 1

Cite

Gonzalo Arias, Guilherme Mazanti, Swann Marx. Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE. 2023. ⟨hal-04059956v1⟩

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