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

Partial Integral Equation (PIE) representation of a Partial Differential Equation (PDE) allows using computationally tractable algorithms for analysis, simulation, and optimal control. However, the PIE representation has not previously been extended to many of the complex, higher-order PDEs that 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 that 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. Numerical tests and illustrations are used to demonstrate the controller synthesis and simulation of PDEs.