Representation of linear PDEs with spatial integral terms as Partial Integral Equations

TL;DR

This paper introduces a novel PIE representation for linear 1D PDEs with spatial integral terms, enabling stability analysis through convex optimization, and provides explicit transformation maps from PDEs to PIEs.

Contribution

The paper develops a systematic method to convert linear PDEs with integral terms into PIEs, facilitating analysis and control design.

Findings

PIE representation accurately models PDEs with integral terms.

Conversion from PDE to PIE is explicit and systematic.

Stability analysis via convex optimization is demonstrated.

Abstract

In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.