There is a nonlinear LQE observer. It's more referred as Extended Kalman Filter(EKF). It's quite easy, just linearize matrix $A$ and $C$ in the estimated vector $\hat{x}$ before you find the kalman gain matrix $K$ from the Riccati equation.
But is ther a nonlinear LQR controller?
Can I just linearize matrix $A$ and $B$ in the estimated vector $\hat{x}$ before I find the control law gain matrix $L$ from the Riccati equation? But still have the same $Q$ and $R$ weighting matrices?