Spectral Boundary Observer for Counter-Flow Heat Exchangers
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909000933572608 |
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| author | Belhadjoudja, Mohamed Camil Maghenem, Mohamed Witrant, Emmanuel |
| author_facet | Belhadjoudja, Mohamed Camil Maghenem, Mohamed Witrant, Emmanuel |
| contents | We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the $L^2$ norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to $L^2$ exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Boundary Observer for Counter-Flow Heat Exchangers Belhadjoudja, Mohamed Camil Maghenem, Mohamed Witrant, Emmanuel Systems and Control Analysis of PDEs We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the $L^2$ norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to $L^2$ exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property. |
| title | Spectral Boundary Observer for Counter-Flow Heat Exchangers |
| topic | Systems and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2604.26863 |