Agreed, the example presented in the article isn’t even an (actually useful) application of the original problem. SoS is useful for generating certificates of the stability of controls, rather than generating actual trajectories. That latter part is relatively easy via some simple heuristics.

Point being: it is possible to generate trajectories using SoS, but we have much better algorithms to do that. Rather, it’s hard to certify that controls which follow those trajectories are stable, esp. as they become more complicated than simple ones whose stability is easily proven (like PID, for example).