Extension of the Partial Integral Equation Representation to GPDE Input-Output 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, h...

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Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Shivakumar, Sachin, Das, Amritam, Weiland, Siep, Peet, Matthew
Format: Article
Language:English
Subjects:
Boundary conditions
Formulas (mathematics)
Integral equations
Mathematical models
Operators (mathematics)
Parameterization
Partial differential equations
Representations
Wave equations
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Summary: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.
ISSN:2331-8422