Linear quadratic regulation, or LQR, chooses state feedback by minimizing an explicit trade-off between state deviation and control effort.

For

$$\dot x=Ax+Bu,$$

a common infinite-horizon cost is

$$J=\int_0^\infty \left(x^TQx+u^TRu\right)dt,$$

where $Q$ penalizes undesirable state deviation and $R$ penalizes control effort.

For a controllable system with suitable positive-semidefinite $Q$ and positive-definite $R$, the optimal control law has the form

$$u=-Kx,$$

with

$$K=R^{-1}B^TP,$$

where $P$ is the stabilizing solution of the algebraic Riccati equation

$$A^TP+PA-PBR^{-1}B^TP+Q=0.$$

LQR does not ask the designer to choose every closed-loop pole directly. Instead, the designer specifies relative penalties, and optimization determines a feedback gain consistent with that trade-off.

This makes LQR a bridge between state-space control and quadratic optimization.