Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems
Abstract
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.
Cite
@article{arxiv.2205.03735,
title = {Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems},
author = {Sachin Shivakumar and Amritam Das and Siep Weiland and Matthew Peet},
journal= {arXiv preprint arXiv:2205.03735},
year = {2024}
}