Linear-quadratic regulation and Riccati equations

Consider real finite-dimensional ẋ=Ax+Bu over a fixed interval ending at T, with unrestricted square-integrable controls. Use the cost integral of xᵀQx+uᵀRu plus terminal x(T)ᵀFx(T), with Q and F symmetric positive semidefinite and R symmetric positive definite. There is no factor one half in this convention. Seeking V(t,x)=xᵀP(t)x and minimizing the local quadratic in u gives u*=−R^{-1}BᵀP(t)x. The positive input penalty makes this pointwise minimizer unique.

The matrix P satisfies a differential Riccati equation integrated backward from P(T)=F. With continuous coefficients on a finite interval and R uniformly positive definite, the nonnegative-cost finite-horizon problem has the corresponding positive semidefinite Riccati solution; for the constant matrices used here these hypotheses are immediate. Controllability is not required merely to define this finite-horizon problem with a soft terminal cost. A hard specified terminal state is a different boundary constraint and can be infeasible, so it cannot be silently substituted for the terminal penalty.

For constant matrices on the infinite horizon, assume (A,B) stabilizable and (Q^{1/2},A) detectable, with Q≥0 and R>0. Then there is a unique symmetric positive semidefinite stabilizing solution of the algebraic Riccati equation. Its feedback makes A−BR^{-1}BᵀP Hurwitz and gives the finite optimal value x₀ᵀPx₀. Stabilizability permits stable uncontrollable modes; detectability requires every state direction invisible to the running state penalty to be strictly stable. These are weaker than full controllability and a positive-definite Q, respectively.

The word stabilizing is essential because the algebraic equation can have other roots. If detectability is dropped, zero state penalty can make an unstable trajectory cost nothing, so a cost-minimizing law need not stabilize. For example, A=1,B=1,Q=0,R=1 permits u=0 with zero cost while x grows. Under the stated hypotheses, completing the square in the cost supplies the optimality certificate and the Riccati identity links P to a closed-loop Lyapunov equation. The full theorem is being stated here; its detectability step is not replaced by an eigenvalue guess.

Increasing an input penalty generally discourages aggressive commands, but it does not impose a hard amplitude bound. A quadratic regulator can request an arbitrarily large command when the initial state is large. Saturating that command changes the closed-loop system and its exact optimality proof. Units matter as well: weighting a position and a temperature with identical numeric coefficients is not a dimensionless comparison unless the coordinates have been normalized deliberately. The matrices express a chosen tradeoff, not an objective law supplied by the plant itself.

Finite-horizon gains depend on remaining time and terminal penalty. Receding-horizon use of a finite-horizon controller adds repeated optimization and needs a separate stability argument if the terminal construction is changed. Conversely, replacing the finite-horizon Riccati function by an algebraic solution assumes an infinite-horizon stationary problem. Estimation can later be combined with regulation under suitable linear-Gaussian assumptions, but the deterministic full-state calculation here does not by itself establish performance when the state is noisy or hidden. Keep the optimization problem attached to the gain it produced.