Once we obtain $$P(t) = S^{-1}(t)$$ it seems like we would want to compute the LQR gain matrix $$K(t) = R^{-1} B^\top S(t) = R^{-1} B^\top P^{-1}(t)$$, but we know that $$P^{-1}(t_f)$$ does not exist. Is this OK considering that our control input does not actually matter at time $$t_f$$ (since we have already reached the solution?)

The FiniteHorizonLinearQuadraticRegulatorResult (Drake) shows that the optimal controller is given instead by $$u^* = u_0(t) - K(t) \left(x-x_0(t)\right) - k_0(t).$$

What is $$k_0(t)$$ and is there a way to, for example, ensure $$k_0(t) = 0$$? It seems odd that if $$x-x_0(t)$$ that your controller would not just output $$u^* = u_0(t)$$.

I think equation (47) should be = -1/L (uc + gs)