Computational stability analysis of PDEs with integral terms using the PIE framework

TL;DR

This paper extends the PIE framework to analyze the stability of PDEs with integral terms, enabling conversion to PIEs and stability testing via optimization, demonstrated through numerical examples.

Contribution

It introduces a method to convert PDEs with integral terms into PIEs using boundary conditions and variable changes, expanding the PIE framework's applicability.

Findings

PDEs with integral terms can be converted to PIEs under certain boundary conditions.

Stability analysis reduces to an operator-valued optimization problem.

Numerical examples validate the effectiveness of the proposed method.

Abstract

The Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued optimization problems. Previous works on the PIE framework focused on the analysis of PDEs with spatial derivatives up to $2^{nd}$-order. In this paper, we extend the class of PDEs by including integral terms and performing stability analysis using the PIE framework. More specifically, we show that PDEs with the integral terms where the integration is with respect to the spatial variable and the kernel of the integral operator is matrix-valued polynomials can be converted to PIEs if the boundary conditions satisfy certain criteria. The conversion is performed by using a change of variable where every PDE state is substituted in terms of its highest derivative and boundary values to obtain a new equation (a PIE) in a variable that does not have any continuity requirements. Later, we show that this change of variable can be represented using explicit maps from the parameters of the PDE to the parameters of the PIE and the stability test can be posed as an optimization problem involving these parameters. Lastly, we present numerical examples to demonstrate the simplicity and application of this method.