Here is a description of some of the math of the control

http://www.philipzucker.com/cartpole-maths/

And some more posts during the creation.

http://www.philipzucker.com/cartpole-workin-boyeee/

http://www.philipzucker.com/cart-pole-trajectory-optimization-using-cvxpy/

http://www.philipzucker.com/cart-pole-using-lyapunov-lqr-control-openai-gym/

Here is an attempt at creating an Iterative LQR controller. Quite Slow, questionably working, but interesting.

https://philzook58.github.io/reinforcement/learning/control/2017/12/07/LQR.html

MIT has a great course taught by Russ Tedrake (Lecture videos available online). This is where I learned about the control that we ended up using

http://underactuated.mit.edu/underactuated.html?chapter=acrobot