Discrete-time algebraic Riccati equation

Remarks

For such $X$ the matrix $R + B^T X B$ is positive definite, hence invertible, so the equation makes sense. A solution $X$ is stabilising if every eigenvalue of $A - BK$ has absolute value less than $1$, where $K = (R + B^T X B)^{-1} B^T X A$. If the pair $(A, B)$ is stabilisable and the pair $(Q^{1/2}, A)$ is detectable, the equation has a unique positive semidefinite solution, and it is stabilising. The optimal control of the infinite-horizon linear-quadratic regulator is then $u_k = -K x_k$. Example: for $n = m = 1$ and $A = B = Q = R = 1$, the equation reads $X = X - X^2/(1 + X) + 1$, that is, $X^2 - X - 1 = 0$. Its nonnegative solution is $X = (1 + \sqrt{5})/2$, and then $K = X/(1 + X) = 1/X$ and $A - BK = 1 - 1/X \approx 0.38$, so it is stabilising. Non-example: the other root $X = (1 - \sqrt{5})/2$ satisfies the matrix identity but is negative, so it is not an admissible solution. For $B = 0$ the equation becomes the linear Stein equation $X - A^T X A = Q$.