A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912285215162368 |
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| author | Balogoun, Ismaïla Auriol, Jean Boussaada, Islam Mazanti, Guilherme |
| author_facet | Balogoun, Ismaïla Auriol, Jean Boussaada, Islam Mazanti, Guilherme |
| contents | In this paper, we establish a necessary and sufficient stability condition for a class of two coupled first-order linear hyperbolic partial differential equations. Through a backstepping transform, the problem is reformulated as a stability problem for an integral difference equation, that is, a difference equation with distributed delay. Building upon a Stépán--Hassard argument variation theorem originally designed for time-delay systems of retarded type, we then introduce a theorem that counts the number of unstable roots of our integral difference equation. This leads to the expected necessary and sufficient stability criterion for the system of first-order linear hyperbolic partial differential equations. Finally, we validate our theoretical findings through simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems Balogoun, Ismaïla Auriol, Jean Boussaada, Islam Mazanti, Guilherme Optimization and Control Analysis of PDEs In this paper, we establish a necessary and sufficient stability condition for a class of two coupled first-order linear hyperbolic partial differential equations. Through a backstepping transform, the problem is reformulated as a stability problem for an integral difference equation, that is, a difference equation with distributed delay. Building upon a Stépán--Hassard argument variation theorem originally designed for time-delay systems of retarded type, we then introduce a theorem that counts the number of unstable roots of our integral difference equation. This leads to the expected necessary and sufficient stability criterion for the system of first-order linear hyperbolic partial differential equations. Finally, we validate our theoretical findings through simulations. |
| title | A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2412.13929 |