My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from.

Kalman filter allows us to estimate state at time $t$ as multivariate normal distribution and the formulas for kalman filter (see wikipedia) says what is the mean and the covariance of this distribution.

The covariance matrix of estimated state doesn't depend on values of observations (but depends on their covariance matrix). The situation is even simper when this covariance and any other the parameters of Kalman filter doesn't change over time.

My question is: What is limit covariance of state estimate after infinite time has passed?

This question is equivalent of asking what is the limit of $\lim_{x\to\infty} P_t$ where

$$P'_t = F P_{t-1} F^T + Q$$

$$P_t = (I-P'_t H^T (R+H P'_t H^T)^{-1} H) P'_t$$

$I$ is unit matrix.

Under some conditions the question is equivalent to solving the above equation with $P_t = P_{t-1}$.

I believe that the series converges under reasonable assumtions to the solution of the equation.

Note that all $P_t$, $Q$ and $R$ are symetric possitive definite and all eigenvalues of F are less or equal to one.

How do I solve the equation or find limit of $P_t$?