Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913262653669376 |
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| author | Shivakumar, Sachin Das, Amritam Weiland, Siep Peet, Matthew |
| author_facet | Shivakumar, Sachin Das, Amritam Weiland, Siep Peet, Matthew |
| contents | It has been shown that the existence of a Partial Integral Equation (PIE) representation of a Partial Differential Equation (PDE) simplifies many numerical aspects of analysis, simulation, and optimal control. However, the PIE representation has not previously been extended to many of the complex, higher-order PDEs such as may be encountered in speculative or data-based models. In this paper, we propose PIE representations for a large class of such PDE models, including higher-order derivatives, boundary-valued inputs, and coupling with Ordinary Differential Equations. The main technical contribution which enables this extension is a generalization of Cauchy's rule for repeated integration. The process of conversion of a complex PDE model to a PIE is simplified through a PDE modeling interface in the open-source software PIETOOLS. Several numerical tests and illustrations are used to demonstrate the results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_03735 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems Shivakumar, Sachin Das, Amritam Weiland, Siep Peet, Matthew Optimization and Control Dynamical Systems It has been shown that the existence of a Partial Integral Equation (PIE) representation of a Partial Differential Equation (PDE) simplifies many numerical aspects of analysis, simulation, and optimal control. However, the PIE representation has not previously been extended to many of the complex, higher-order PDEs such as may be encountered in speculative or data-based models. In this paper, we propose PIE representations for a large class of such PDE models, including higher-order derivatives, boundary-valued inputs, and coupling with Ordinary Differential Equations. The main technical contribution which enables this extension is a generalization of Cauchy's rule for repeated integration. The process of conversion of a complex PDE model to a PIE is simplified through a PDE modeling interface in the open-source software PIETOOLS. Several numerical tests and illustrations are used to demonstrate the results. |
| title | Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems |
| topic | Optimization and Control Dynamical Systems |
| url | https://arxiv.org/abs/2205.03735 |