Module stationary_lq 

Infinite-horizon scalar linear-quadratic regulator.

A validating instance of crate::numeric::finite_difference::elliptic::EllipticControlProblem with a closed-form stationary solution obtained from the algebraic Riccati equation.

Dynamics

dx = (a x + b u) dt + c dW

Objective

Minimize the expected discounted running quadratic cost

J(u) = E[ integral_0^inf e^{-rho t} (q x^2 + r u^2) dt ]

which is maximized in negated form by crate::models::control::ControlProblem. A positive discount rho makes the stationary value finite and the associated elliptic operator well posed.

HJB equation

rho V = sup_u { -(q x^2 + r u^2) + (a x + b u) V_x + 0.5 c^2 V_xx }

Exact solution

The stationary value is V(x) = -P x^2 - d, where

d = c^2 P / rho

and the optimal control is

u*(x) = -(b / r) P x

P is the positive solution of the scalar algebraic Riccati equation

(b^2 / r) P^2 + (rho - 2a) P - q = 0

A stable closed loop selects the positive root. The additive constant d is the noise-induced correction; it vanishes when c = 0.

Structs

StationaryLqRegulator

Infinite-horizon scalar linear-quadratic regulator with constant coefficients.