Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS

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.