Abstract: Determining nonlinear stability of steady states for complex dynamical systems are notoriously difficult problems, even for the seemingly simplest cases. For example, it is expected that the standard steady state shear profile, 2D planar Couette flow, is globally stable for all Reynolds numbers, however the state-of-the-art analysis is decades old and only proves the stability for relatively low Reynolds numbers. In recent years, a promising computational approach uses polynomial sum-of-squares optimization to find Lyapunov functions based on low-mode projections onto an orthogonal basis of $L^2 \cap H^1$. Critically, physical symmetries inherent in the system can be exploited to reduce the complexity of the resulting optimization problem. We will present on rigorous and practical extensions of this work.