Linear-quadratic regulator

Remarks

The problem is solved. Let $P$ be the solution of the Riccati differential equation $-P' = A^{T} P + P A - P B R^{-1} B^{T} P + Q$ with $P(T) = F$. Then the unique optimal control is given by the linear feedback $u^*(t) = -R^{-1} B^{T} P(t) x^*(t)$, and the optimal cost is $x_0^{T} P(0) x_0$. The proof completes the square: along any trajectory, $\frac{d}{dt} (x^{T} P x) = -x^{T} Q x - u^{T} R u + (u + R^{-1} B^{T} P x)^{T} R (u + R^{-1} B^{T} P x)$. Integrating from $0$ to $T$ gives $J(u) = x_0^{T} P(0) x_0 + \int_0^T \abs{R^{1/2}(u + R^{-1} B^{T} P x)}^2 \, dt$, which is smallest exactly for the feedback above. For the infinite horizon $T = \infty$ and $F = 0$, assume $(A, B)$ stabilizable and $(C, A)$ detectable, where $Q = C^{T} C$. Then the algebraic Riccati equation $A^{T} P + P A - P B R^{-1} B^{T} P + Q = 0$ has a unique symmetric positive semidefinite solution $P$. The feedback $u = -R^{-1} B^{T} P x$ is optimal and makes the closed loop asymptotically stable.