On Tensor-based Polynomial Hamiltonian Systems
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915216357326848 |
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| author | Cui, Shaoxuan Zhang, Guofeng Jardon-Kojakhmetov, Hildeberto Cao, Ming |
| author_facet | Cui, Shaoxuan Zhang, Guofeng Jardon-Kojakhmetov, Hildeberto Cao, Ming |
| contents | It is known that a linear system with a system matrix A constitutes a Hamiltonian system with a quadratic Hamiltonian if and only if A is a Hamiltonian matrix. This provides a straightforward method to verify whether a linear system is Hamiltonian or whether a given Hamiltonian function corresponds to a linear system. These techniques fundamentally rely on the properties of Hamiltonian matrices. Building on recent advances in tensor algebra, this paper generalizes such results to a broad class of polynomial systems. As the systems of interest can be naturally represented in tensor forms, we name them tensor-based polynomial systems. Our main contribution is that we formally define Hamiltonian cubical tensors and characterize their properties. Crucially, we demonstrate that a tensor-based polynomial system is a Hamiltonian system with a polynomial Hamiltonian if and only if all associated system tensors are Hamiltonian cubical tensors-a direct parallel to the linear case. Additionally, we establish a computationally tractable stability criterion for tensor-based polynomial Hamiltonian systems. Finally, we validate all theoretical results through numerical examples and provide a further intuitive discussion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Tensor-based Polynomial Hamiltonian Systems Cui, Shaoxuan Zhang, Guofeng Jardon-Kojakhmetov, Hildeberto Cao, Ming Systems and Control It is known that a linear system with a system matrix A constitutes a Hamiltonian system with a quadratic Hamiltonian if and only if A is a Hamiltonian matrix. This provides a straightforward method to verify whether a linear system is Hamiltonian or whether a given Hamiltonian function corresponds to a linear system. These techniques fundamentally rely on the properties of Hamiltonian matrices. Building on recent advances in tensor algebra, this paper generalizes such results to a broad class of polynomial systems. As the systems of interest can be naturally represented in tensor forms, we name them tensor-based polynomial systems. Our main contribution is that we formally define Hamiltonian cubical tensors and characterize their properties. Crucially, we demonstrate that a tensor-based polynomial system is a Hamiltonian system with a polynomial Hamiltonian if and only if all associated system tensors are Hamiltonian cubical tensors-a direct parallel to the linear case. Additionally, we establish a computationally tractable stability criterion for tensor-based polynomial Hamiltonian systems. Finally, we validate all theoretical results through numerical examples and provide a further intuitive discussion. |
| title | On Tensor-based Polynomial Hamiltonian Systems |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2503.21487 |