the linear-quadratic-Gaussian regulator
The linear-quadratic-Gaussian (LQG) regulator is the one control problem that solves completely and beautifully in closed form — and for that reason it is the workhorse of engineering control and the template against which everything else is measured. The ingredients are: linear dynamics, dX_t = (A X_t + B u_t) dt + sigma dB_t (state X, control u, Gaussian noise B); a quadratic cost, J = E[ integral_0^T ( X_t^T Q X_t + u_t^T R u_t ) dt + X_T^T M X_T ] with Q, M nonnegative and R positive definite; and Gaussian noise. The aim is to keep the state near zero (the Q term penalises deviation) without spending too much control effort (the R term penalises action) — a regulator.
The solution is the celebrated linear feedback plus Riccati equation. The value function is exactly quadratic, V(t,x) = x^T P(t) x + c(t), where the symmetric matrix P(t) solves the matrix Riccati differential equation -P' = A^T P + P A - P B R^{-1} B^T P + Q, with terminal condition P(T) = M; and the offset c(t) absorbs the noise via c'(t) = -trace( sigma sigma^T P(t) ). The optimal control is the LINEAR state feedback u*_t = -R^{-1} B^T P(t) X_t — a gain matrix times the current state. A striking and genuinely deep fact, the certainty-equivalence (separation) principle, governs the partially-observed case: if you can only see noisy linear observations of X, the optimal controller is to run a Kalman-Bucy filter to produce the best estimate X-hat_t and then apply the SAME feedback gain to the estimate, u*_t = -R^{-1} B^T P(t) X-hat_t. Estimation and control separate cleanly — the optimal control law does not change just because you are uncertain about the state.
LQG matters because it is exactly solvable, scalable to high dimensions (you solve a matrix ODE/algebraic equation, not a PDE), and it is the local approximation to nonlinear control near an operating point — which is why real autopilots, robot controllers, and process regulators are built on it. Honest caveats: the closed form depends critically on the structure — costs must be quadratic and dynamics linear; the moment you have control-dependent noise, state constraints, or non-quadratic costs, the magic ends and you are back to a full HJB equation. The infinite-horizon version requires stabilizability of (A,B) and detectability of (Q^{1/2}, A) for the algebraic Riccati equation to have a stabilizing solution. And certainty equivalence is special to the linear-quadratic-Gaussian structure — it is NOT a general principle; for non-LQG problems estimation and control do not separate, and the controller must account for how its actions affect future information (the dual-control / exploration-exploitation problem).
Steering a satellite to point at a target: the attitude error and angular velocity form the state X, the thruster torques are the control u, you penalise pointing error (Q) and fuel (R). LQG gives a fixed gain K = R^{-1} B^T P from the Riccati equation, and the controller fires thrusters proportional to -K X-hat, where X-hat comes from a Kalman filter blending gyroscope and star-tracker readings. One Riccati solve, one Kalman gain, done.
LQG in practice: Kalman filter estimates the state, fixed Riccati gain turns the estimate into action — separation in action.
Certainty equivalence (estimate then apply the same gain) is SPECIAL to linear-quadratic-Gaussian structure; it is not a general principle. Outside LQG, the controller must account for how its actions shape future information (dual control).