Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914708413480960 |
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| author | Arias, Gonzalo Marx, Swann Mazanti, Guilherme |
| author_facet | Arias, Gonzalo Marx, Swann Mazanti, Guilherme |
| contents | 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 $\varepsilon$ 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 $\varepsilon = 0$, 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 $\varepsilon$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01977 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE Arias, Gonzalo Marx, Swann Mazanti, Guilherme Analysis of PDEs Optimization and Control 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 $\varepsilon$ 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 $\varepsilon = 0$, 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 $\varepsilon$ 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. |
| title | Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2304.01977 |