A Distributed SOS Program For Local Stability Analysis of Polynomial PDEs in the PIE Representation

TL;DR

This paper introduces a distributed sum-of-squares (SOS) framework for analyzing the local stability of polynomial PDEs using the PIE representation, leveraging tensor algebra and SOS parametrization.

Contribution

It develops a novel distributed SOS program for polynomial PDE stability analysis based on the PIE framework and tensor algebra, advancing computational methods in PDE stability.

Findings

Distributed SOS program successfully tests local stability of polynomial PDEs.

Tensor algebra enables compact representation of PDE dynamics in the PIE framework.

The approach facilitates scalable stability analysis for complex polynomial PDEs.

Abstract

It has recently been shown that the evolution of a state, described by a Partial Differential Equation (PDE), can be more conveniently represented as the evolution of the state's highest spatial derivative (the ``fundamental state''), which lies in $L_2$ and has no boundary conditions (BCs) or continuity constraints. For linear PDEs, this yields a Partial Integral Equation (PIE) parametrized by Partial Integral (PI) operators mapping the fundamental state to the PDE state. In this paper, we show that for polynomial PDEs, the dynamics of the fundamental state can instead be compactly expressed as a distributed polynomial in the fundamental state, parametrized by a new tensor algebra of PI operators acting on the tensor product of the fundamental state. We further define a SOS parametrization of the distributed polynomial and use this to construct a distributed SOS program, for testing local stability of polynomial PDEs.