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
https://mathoverflow.net/questions/128204/on-solution-of-a-recursion-with-rectangular-matrices

Thanks for your help