Q:

On solution of a discrete-time equation

Hello, members. 
I have a problem for the following problem 
when I derive an optimization algorithm for stochastic singular systems
$$S(k+1)=A(k)S(k)A^{T}(k)+R(k)+F(k)S(k+1)F^{T}(k)$$
where $R(k)>=0$
So, how to calculate $S$, is there analytic solution or numerical solution to $S$?
This problem is different from the following one
On solution of a recursion with rectangular matrices
Thanks for your help

A:

you'll want to solve this equation iteratively, considering $S(k)$ as known and $S(k+1)$ as unknown; for $F(k)$ invertible you then have a Sylvester equation, of the form $F^{-1}(k)S(k+1)-S(k+1)F^{T}(k)=C(k)$, which has a unique solution iff $F^{-1}(k)$ and $F(k)$ have no common eigenvalue. The Wikipedia page gives algorithms to solve this equation, implemented in several software packages.

