To improve the success of the SOS optimization, it's important to use stricter enforcement of the constraint s2+c2=1s^2 + c^2 = 1s2+c2=1 as an equality condition within the optimization framework. Additionally, starting with a smaller level set of the LQR value function can help in identifying a valid and more conservative estimate of the region of attraction. It's also recommended to use tools like the Spotless or Drake framework, which support full polynomial parameterization and are well-suited for such SOS programs. Finally, visualizing the sublevel sets where the derivative of the Lyapunov function V˙<0\dot{V} < 0V˙<0 can provide insight into whether the Lyapunov condition holds, and whether the chosen candidate function is appropriate for proving stability.