If you don't mind, can you explain why we need this extra (epsilonx'x) term to prove global stability

these conditions means our cost function will not explode. In other words, states will eventually reach optimal solution x*. Am i thinking in the right direction?

Hi, I understand how can we reach upright position by thinking long term instead of brute feedback cancellation. But,i am wondering what if I want to go to position of pi/2 and stays there. Can we still achieve it using optimal control if available max torque is less than mgl ?. Thank you

is there any situation, where a fixed point is attractive but not locally stable i.s.L? I mean why can't attractive fixed point always be a locally stable i.s.L ? thank you

I have an idea how to go about it when the system is control affine. How we deal with it, when dynamic non linear system can't be reduce to control affine configuration?. And also you mention that under actuation can also be caused by state and actuation limits? how do we analyse the system in that case? Thank you

Can we say,the desired response could be reference acceleration of the system. I mean for every desired acceleration there exists a u which produces that.

What is it mean by minimal coordinates? is it same as less state variables in state space?