Stability Analysis of Parabolic Linear PDEs with Two Spatial Dimensions Using Lyapunov Method and SOS

TL;DR

This paper develops a numerical method using Lyapunov functionals and SOS optimization to analyze the stability and decay rates of 2D parabolic PDEs with polynomial coefficients, validated on biological models.

Contribution

It introduces a polynomial parametrization approach combined with SOS techniques for stability analysis of 2D parabolic PDEs, including decay rate estimation.

Findings

Successfully applied to biological population growth PDE model

Can estimate decay rates of PDE solutions in L2 norm

Provides a computational framework for stability verification

Abstract

In this paper, we address stability of parabolic linear Partial Differential Equations (PDEs). We consider PDEs with two spatial variables and spatially dependent polynomial coefficients. We parameterize a class of Lyapunov functionals and their time derivatives by polynomials and express stability as optimization over polynomials. We use Sum-of-Squares and Positivstellensatz results to numerically search for a solution to the optimization over polynomials. We also show that our algorithm can be used to estimate the rate of decay of the solution to PDE in the L2 norm. Finally, we validate the technique by applying our conditions to the 2D biological KISS PDE model of population growth and an additional example.