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 article, 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.