Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems

TL;DR

This paper extends the Partial Integral Equation (PIE) representation to complex, higher-order PDEs with boundary inputs and ODE coupling, facilitating analysis and control through a generalized Cauchy's rule and software tools.

Contribution

It introduces a generalized PIE framework for complex PDEs, including higher-order derivatives and boundary inputs, with a new integration rule and software implementation.

Findings

Successfully extended PIE representation to complex PDEs

Demonstrated numerical tests with the extended framework

Simplified PDE to PIE conversion process

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.